Difference methods for singular perturbation problems

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics 140 DIFFERENCE METHODS FOR SINGULAR PERTURB...

Author: Grigory I. Shishkin | Lidia P. Shishkina

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CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

140

DIFFERENCE METHODS FOR SINGULAR PERTURBATION PROBLEMS

© 2009 by Taylor & Francis Group, LLC

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics Main Editors

H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board

R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgässner, Universität Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide


CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

140

DIFFERENCE METHODS FOR SINGULAR PERTURBATION PROBLEMS

Grigory I. Shishkin Lidia P. Shishkina


Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-459-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Shishkin, G. I. Difference methods for singular perturbation problems / Grigory I. Shishkin, Lidia P. Shishkina. p. cm. -- (Chapman & Hall/CRC monographs and surveys in pure and applied mathematics) Includes bibliographical references and index. ISBN 978-1-58488-459-0 (hardback : alk. paper) 1. Singular perturbations (Mathematics) 2. Difference equations--Numerical solutions. 3. Algebra, Abstract. I. Shishkina, Lidia P. II. Title. III. Series. QC20.7.P47S55 2008 515’.392--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com


2008025636

Dedication Dedicated to the memory of academicians Alexandr Andreevich Samarskii and Nikolai Sergeevich Bakhvalov

v © 2009 by Taylor & Francis Group, LLC

Contents

Preface

I

Grid approximations of singular perturbation partial differential equations

1 Introduction 1.1 The development of numerical methods for singularly perturbed problems . . . . . . . . . . . . . . . . . . . . . . 1.2 Theoretical problems in the construction of difference schemes 1.3 The main principles in the construction of special schemes 1.4 Modern trends in the development of special difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The contents of the present book . . . . . . . . . . . . . . . 1.6 The present book . . . . . . . . . . . . . . . . . . . . . . . 1.7 The audience for this book . . . . . . . . . . . . . . . . . . 2 Boundary value problems for elliptic reaction-diffusion equations in domains with smooth boundaries 2.1 Problem formulation. The aim of the research . . . . . . . 2.2 Estimates of solutions and derivatives . . . . . . . . . . . . 2.3 Conditions ensuring ε-uniform convergence of difference schemes for the problem on a slab . . . . . . . . . . . . . . 2.3.1 Sufficient conditions for ε-uniform convergence of difference schemes . . . . . . . . . . . . . . . . . . . 2.3.2 Sufficient conditions for ε-uniform approximation of the boundary value problem . . . . . . . . . . . . . . 2.3.3 Necessary conditions for distribution of mesh points for ε-uniform convergence of difference schemes. Construction of condensing meshes . . . . . . . . . . 2.4 Monotone finite difference approximations of the boundary value problem on a slab. ε-uniformly convergent difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Problems on uniform meshes . . . . . . . . . . . . . 2.4.2 Problems on piecewise-uniform meshes . . . . . . . . 2.4.3 Consistent grids on subdomains . . . . . . . . . . . . 2.4.4 ε-uniformly convergent difference schemes . . . . . .


xiii

1 3 3 6 8 10 11 12 16 17 17 19 26 26 29

33

38 38 44 51 57

2.5

Boundary value problems in domains with curvilinear boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 A domain-decomposition-based difference scheme for the boundary value problem on a slab . . . . . . . . 2.5.2 A difference scheme for the boundary value problem in a domain with curvilinear boundary . . . . . . . .

3 Boundary value problems for elliptic reaction-diffusion equations in domains with piecewise-smooth boundaries 3.1 Problem formulation. The aim of the research . . . . . . . 3.2 Estimates of solutions and derivatives . . . . . . . . . . . . 3.3 Sufficient conditions for ε-uniform convergence of a difference scheme for the problem on a parallelepiped . . . . . . . . . 3.4 A difference scheme for the boundary value problem on a parallelepiped . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Consistent grids on subdomains . . . . . . . . . . . . . . . 3.6 A difference scheme for the boundary value problem in a domain with piecewise-uniform boundary . . . . . . . . .

58 58 67 75 75 76 85 89 97 102

4 Generalizations for elliptic reaction-diffusion equations 109 4.1 Monotonicity of continual and discrete Schwartz methods . 109 4.2 Approximation of the solution in a bounded subdomain for the problem on a strip . . . . . . . . . . . . . . . . . . . . . 112 4.3 Difference schemes of improved accuracy for the problem on a slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.4 Domain-decomposition method for improved iterative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5 Parabolic reaction-diffusion equations 5.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . 5.2 Estimates of solutions and derivatives . . . . . . . . . . . . 5.3 ε-uniformly convergent difference schemes . . . . . . . . . 5.3.1 Grid approximations of the boundary value problem 5.3.2 Consistent grids on a slab . . . . . . . . . . . . . . . 5.3.3 Consistent grids on a parallelepiped . . . . . . . . . 5.4 Consistent grids on subdomains . . . . . . . . . . . . . . . 5.4.1 The problem on a slab . . . . . . . . . . . . . . . . . 5.4.2 The problem on a parallelepiped . . . . . . . . . . .

133 133 134 145 146 147 154 158 158 161

6 Elliptic convection-diffusion equations 6.1 Problem formulation . . . . . . . . . . . 6.2 Estimates of solutions and derivatives . 6.2.1 The problem solution on a slab . 6.2.2 The problem on a parallelepiped

165 165 166 166 169


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6.3

6.4

On construction of ε-uniformly convergent difference schemes under their monotonicity condition . . . . . . . . . . . . . . 6.3.1 Analysis of necessary conditions for ε-uniform convergence of difference schemes . . . . . . . . . . . 6.3.2 The problem on a slab . . . . . . . . . . . . . . . . . 6.3.3 The problem on a parallelepiped . . . . . . . . . . . Monotone ε-uniformly convergent difference schemes . . . .

7 Parabolic convection-diffusion equations 7.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . 7.2 Estimates of the problem solution on a slab . . . . . . . . . 7.3 Estimates of the problem solution on a parallelepiped . . . 7.4 Necessary conditions for ε-uniform convergence of difference schemes . . . . . . . . . . . . . . . . . . . . . . . 7.5 Sufficient conditions for ε-uniform convergence of monotone difference schemes . . . . . . . . . . . . . . . . . . . . . . . 7.6 Monotone ε-uniformly convergent difference schemes . . . .

II

Advanced trends in difference methods

ε-uniformly convergent


177 180 183 185 191 191 192 199 206 210 213

219

8 Grid approximations of parabolic reaction-diffusion equations with three perturbation parameters 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem formulation. The aim of the research . . . . . . . 8.3 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . 8.4 Grid approximations of the initial-boundary value problem 9 Application of widths for construction of difference schemes for problems with moving boundary layers 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 9.2 A boundary value problem for a singularly perturbed parabolic reaction-diffusion equation . . . . . . . . . 9.2.1 Problem (9.2), (9.1) . . . . . . . . . . . . . . 9.2.2 Some definitions . . . . . . . . . . . . . . . . 9.2.3 The aim of the research . . . . . . . . . . . . 9.3 A priori estimates . . . . . . . . . . . . . . . . . . . 9.4 Classical finite difference schemes . . . . . . . . . . 9.5 Construction of ε-uniform and almost ε-uniform approximations to solutions of problem (9.2), (9.1) . 9.6 Difference scheme on a grid adapted in the moving boundary layer . . . . . . . . . . . . . . . . . . . . . 9.7 Remarks and generalizations . . . . . . . . . . . . .

176

221 221 222 224 230

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235 235

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237 237 238 240 241 243

. . . .

246

. . . . . . . .

251 254

. . . . . .

. . . . . .

10 High-order accurate numerical methods for singularly perturbed problems 259 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 259 10.2 Boundary value problems for singularly perturbed parabolic convection-diffusion equations with sufficiently smooth data 261 10.2.1 Problem with sufficiently smooth data . . . . . . . . 261 10.2.2 A finite difference scheme on an arbitrary grid . . . . 262 10.2.3 Estimates of solutions on uniform grids . . . . . . . 263 10.2.4 Special ε-uniform convergent finite difference scheme 263 10.2.5 The aim of the research . . . . . . . . . . . . . . . . 264 10.3 A priori estimates for problem with sufficiently smooth data 265 10.4 The defect correction method . . . . . . . . . . . . . . . . . 266 10.5 The Richardson extrapolation scheme . . . . . . . . . . . . 270 10.6 Asymptotic constructs . . . . . . . . . . . . . . . . . . . . . 273 10.7 A scheme with improved convergence for finite values of ε . 275 10.8 Schemes based on asymptotic constructs . . . . . . . . . . 277 10.9 Boundary value problem for singularly perturbed parabolic convection-diffusion equation with piecewise-smooth initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 10.9.1 Problem (10.56) with piecewise-smooth initial data . 280 10.9.2 The aim of the research . . . . . . . . . . . . . . . . 281 10.10 A priori estimates for the boundary value problem (10.56) with piecewise-smooth initial data . . . . . . . . . . . . . . 282 10.11 Classical finite difference approximations . . . . . . . . . . 285 10.12 Improved finite difference scheme . . . . . . . . . . . . . . . 287 11 A finite difference scheme on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem formulation. The aim of the research . . . . . . . 11.3 Grid approximations on locally refined grids that are uniform in subdomains . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Difference scheme on a priori adapted grid . . . . . . . . . 11.5 Convergence of the difference scheme on a priori adapted grid 11.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 290 293 297 303 307

12 On conditioning of difference schemes and their matrices for singularly perturbed problems 309 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 309 12.2 Conditioning of matrices to difference schemes on piecewiseuniform and uniform meshes. Model problem for ODE . . . 311 12.3 Conditioning of difference schemes on uniform and piecewiseuniform grids for the model problem . . . . . . . . . . . . . 316


12.4

On conditioning of difference schemes and their matrices for a parabolic problem . . . . . . . . . . . . . . . . . . . . . .

323

13 Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters 327 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 327 13.2 Problem formulation. The aim of the research . . . . . . . 328 13.3 Compatibility conditions. Some a priori estimates . . . . . 330 13.4 Derivation of a priori estimates for the problem (13.2) under the condition (13.5) . . . . . . . . . . . . . . . . . . . . . . 333 13.5 A priori estimates for the problem (13.2) under the conditions (13.4), (13.6) . . . . . . . . . . . . . . . . . . . . . . . . . . 341 13.6 The classical finite difference scheme . . . . . . . . . . . . . 343 13.7 The special finite difference scheme . . . . . . . . . . . . . 345 13.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 348 14 Survey 349 14.1 Application of special numerical methods to mathematical modeling problems . . . . . . . . . . . . . . . . . . . . . . . 349 14.2 Numerical methods for problems with piecewise-smooth and nonsmooth boundary functions . . . . . . . . . . . . . . . . 351 14.3 On the approximation of solutions and derivatives . . . . . 352 14.4 On difference schemes on adaptive meshes . . . . . . . . . . 354 14.5 On the design of constructive difference schemes for an elliptic convection-diffusion equation in an unbounded domain . . 357 14.5.1 Problem formulation in an unbounded domain. The task of computing the solution in a bounded domain 357 14.5.2 Domain of essential dependence for solutions of the boundary value problem . . . . . . . . . . . . . . . . 359 14.5.3 Generalizations . . . . . . . . . . . . . . . . . . . . . 363 14.6 Compatibility conditions for a boundary value problem on a rectangle for an elliptic convection-diffusion equation with a perturbation vector parameter . . . . . . . . . . . . . . . . 364 14.6.1 Problem formulation . . . . . . . . . . . . . . . . . . 365 14.6.2 Compatibility conditions . . . . . . . . . . . . . . . . 366 References


371

Preface

The present book is devoted to the development of difference schemes that converge ε-uniformly in the maximum norm for a representative class of singularly perturbed problems. It also deals with the justification of their convergence, and surveys new directions and approaches developed recently, which are of importance for further progress in numerical methods. The¡ book was intended to be an English translation of the Russian book [138] Shishkin G.I. (1992). Discrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Section, ¢ Ekaterinburg (in Russian) that was initiated by John J.H. Miller. The translation was made by Zora Uzelac, but we decided not to publish this version of the book. The very dense nature of this book, that allowed us to cover a large class of singularly perturbed boundary value problems in little space, was too difficult for most readers and also created problems in the implementation of the results. Since the appearance of the book [138], new results and ideas have appeared that are dealt with in the present book. First, I would like to thank my teachers. My scientific interests in computational mathematics were formed and matured under the influence of the scientific schools of the Academicians of the Russian Academy of Science. A.M. Il’in, A.A. Samarskii, N.S. Bakhvalov, G.I. Marchuk and their influence led to the appearance of my second doctoral thesis. This thesis became the basis of [138], and is a continuing influence on my work. It is with pleasure that I note the long-term and fruitful collaboration with the Irish and Dutch mathematicians in the groups of J. Miller and P. Hemker. This collaboration began in 1990, and yielded progress in the development of numerical methods for problems with boundary layers, and led to new results that were published in numerous joint papers and in two books [87] and [33]. The Russian scientists K.V. Emelianov, V.D. Liseikin, P.N. Vabishchevich, V.B. Andreev, V.F. Butuzov, A.V. Gulin, I.G. Belukhina, N.V. Kopteva, V.V. Shaidurov, B.M. Bagaev, E.D. Karepova, M.M. Lavrentiev, Jr, Yu.M. Laevsky, A.I. Zadorin, A.D. Ljashko and I.B. Badriev also influenced much of the detail of the approaches initiated in [138]. The idea to translate into English the book [138] began during my collaboration over the last dozen years with mathematicians and their students, namely, J.J.H. Miller, E. O’Riordan, A.F. Hegarty, M. Stynes, A. Ansari (Ireland), P.W. Hemker, J. Maubach, P. Wesseling (the Netherlands), P.A. Farrell (USA), F. Lisbona, C. Clavero, J.L. Gracia, J.C. Jorge (Spain), D. Creamer, Lin Pin (Singapore), and through discussions of papers (based on ideas from

xiii © 2009 by Taylor & Francis Group, LLC

xiv [138]) on international conferences with, among others, I.P. Boglaev (New Zealand), R.E. O’Malley, R.B. Kellogg (USA), L. Tobiska, H.-G. Roos, G. Lube, T. Linß (Germany), Wang Song (Australia), L.G. Vulkov, I.A. Bra˘ ianov (Bulgaria), R. Ciegis (Lithuania), and P.P. Matus (Belarus). Numerous ideas from [138] were extended and published in many papers. My thanks especially to L.P. Shishkina, my better half, and main assistantcolleague and mathematician for participation as co-author in writing this book, for enormous scientific and technical support. She has prepared the present book including all stages: the clarification of results by numerous discussions, preparation in LaTeX, the translation, compiling the Index, and reviewing the page-proofs. Significant assistance in the preparation of the English version of the present book, in the translation from Russian-English to idiomatic English, was made by M. Stynes (Part I, and fragments of Part II) and M. Mortell (the Preface and the Introduction) to whom I would like to express my deepest thanks. My thanks to our assistant-colleague I.V. Tselishcheva for support in the process of preparing the book, participation in the translation of some chapters from Part II, of the Introduction, of the Survey, and many other tasks. I am grateful for financial and material support (scientific books, computational technique) to the Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia; Institute for Numerical Computation and Analysis, Dublin (INCA), Ireland; Department of Mathematics at Trinity College Dublin, Ireland; CWI (Research Institute of the Stichting Mathematisch Centrum), Amsterdam, the Netherlands; School of Mathematical Sciences, Dublin City University, Ireland; Department of Mathematics and Statistics at the University of Limerick, Ireland; National Research Institute for Mathematics and Computer Science (NUS), Singapore; School of Mathematical Sciences at the National University of Ireland in Cork (UCC), Ireland; Boole Centre for Research in Informatics at the UCC, Ireland; Mathematics Applications Consortium for Science and Industry in Ireland (MACSI) under the Science Foundation Ireland (SFI) Mathematics Initiative; and to Chapman & Hall for friendly cooperation.


xv In particular, the research work of G.I. Shishkin and L.P. Shishkina was supported by the Russian Foundation for Basic Research under grants No. 01-01-01022, 04-01-00578, 07-01-00729; grant RFBR-NWO (RFFI-NWO) 04-01-89007-NWO a); Dutch Research Organisation NWO under grants No. 047.008.007, 047.016.008. The research work of G.I. Shishkin was also supported by the International Collaboration Programme of Forbairt, Dublin, Ireland No. IC/97/057; Enterprise Ireland Basic Research grants SC–98–612, SC–2000–070.

G.I. Shishkin


Part I

Grid approximations of singular perturbation partial differential equations

1 © 2009 by Taylor & Francis Group, LLC

Chapter 1 Introduction

1.1

The development of numerical methods for singularly perturbed problems

The wide use of computing techniques, combined with the demands of scientific and technical practices, has stimulated the development of numerical methods to a great extent, and in particular, methods for solving differential equations. The efficiency of such methods is governed by their accuracy, simplicity in computing the discrete solution and also their relative insensitivity to parameters in the problem. At present, numerical methods for solving partial differential equations, in particular, finite difference schemes, are well developed for wide classes of boundary value problems (see, for example, [79, 108, 100, 214, 91, 216]). Among boundary value problems, a considerable class includes problems for singularly perturbed equations, i.e., differential equations whose highest-order derivatives are multiplied by a (perturbation) parameter ε. The perturbation parameter ε may take arbitrary values in the open-closed interval (0, 1] (see, e.g., [211, 210, 57, 94, 62]). Solutions of singularly perturbed problems, unlike regular problems, have boundary and/or interior layers, that is, narrow subdomains specified by the parameter ε on which the solutions vary by a finite value. The derivatives of the solution in these subdomains grow without bound as ε tends to zero. In the case of singularly perturbed problems, the use of numerical methods developed for solving regular problems leads to errors in the solution that depend on the value of the parameter ε. Errors of the numerical solution depend on the distribution of mesh points and become small only when the effective mesh-size in the layer is much less than the value of the parameter ε (see, e.g., [138, 87, 106, 33]). Such numerical methods turn out to be inapplicable for singularly perturbed problems. Due to this, there is an interest in the development of special numerical methods where solution errors are independent of the parameter ε and defined only by the number of nodes in the meshes used, i.e., numerical methods (in particular, finite difference schemes) that converge ε-uniformly. When the solutions by such methods are ε-uniformly convergent, we will call these methods and solutions robust (as in [33]). At present, only several books are devoted to the development of numerical methods for solving singularly


4

Introduction

perturbed problems. Grid methods for boundary value problems for partial differential equations are considered in the books [138, 87, 33, 75]; see also [26, 13, 14, 76] for ordinary differential equations. In the book [106], the authors give a number of results and also a comprehensive bibliography on numerical methods for solving singularly perturbed problems for partial differential equations and for ordinary differential equations. The present book was intended to be an English translation of the book [138]. A variety of ideas and approaches from [138] have since been further developed. New approaches and trends appear, which require further investigation. In the present book, we elaborate on approaches to the development of ε-uniformly convergent numerical methods for several boundary value problems from [138] and discuss some new trends in the development of other methods, which have appeared recently. Quite often solutions of boundary value problems, their grid solutions, and also their convergence are considered using maximum norms. The use of either the energy norm or L1 , L2 -norms is inadequate to describe the solutions of singularly perturbed problems and their approximations. For example, in the case of problems with a parabolic boundary layer, the boundary-layer function (that is finite in the maximum norm) tends to zero in the norms mentioned above as ε → 0 [87, 33]. In this book, maximum norms are consistently used. As a rule, we avoid reference to works where problems for singularly perturbed ordinary differential equations are considered since such results and techniques cannot, in general, be carried over to problems for partial differential equations. The first ε-uniformly convergent difference schemes constructed for singularly perturbed problems used two main approaches: fitted operator method and condensing mesh (grid) method/fitted mesh (grid) method. Schemes based on the fitted operator method were constructed in [2] and, independently, constructed and justified in [56] (for ordinary differential equations); in [15], a scheme for the condensing mesh method was constructed and justified (for an elliptic equation). For schemes using condensing meshes, ε-uniform convergence of the solution of a difference scheme to the solution of the boundary value problem is guaranteed by a special choice of the distribution of mesh points (for the given number of nodes). Restrictions on the choice of difference equations approximating singularly perturbed problems (for ensuring the εuniform convergence of the scheme) are, in general, not imposed. In fitted operator methods, ε-uniform convergence of the solution of the difference scheme is achieved by a special choice of coefficients of the difference equations approximating the differential problem. Restrictions on the distribution of mesh points for ensuring the ε-uniform convergence of the scheme are not imposed. We mention also an approach related to additive splitting of a singularity suggested for singularly perturbed problems in [12] (see also [11]). In this method, basic functions include special functions approximating the singular component of the solution of the problem. In the case of singularly perturbed


Numerical methods for singularly perturbed problems

5

problems for partial differential equations, this approach was not widely used because the singular components of solutions of boundary value problems have the form too complicated for the effective construction of a system of basis functions. The difference schemes based on the method of additive splitting of a singularity and constructed in [12, 11] converge ε-uniformly in the energy norm. After the publications [15, 56], there was a large effort to develop fitted operator methods. The first book [26] is completely devoted to the development of such methods for ordinary differential equations. Later, fitted operator methods continued to be intensively developed (see, for example, a series of variants of the fitted operator schemes for elliptic equations in [103]). A comprehensive bibliography on numerical methods for singularly perturbed problems is given in [106]. After [15, 56], in the case of partial differential equations, the first finite difference schemes that converge ε-uniformly in the maximum norm are constructed in [29] (see also [30, 1] for the fitted operator scheme, and [74, 118] for the schemes on condensing meshes). Note that fitted operator methods (see their description, e.g., in [26, 87, 33, 106]) have an advantage in simplicity because meshes used are uniform, and this contributed to their more rapid progress compared with condensing mesh methods. However, fitted operator methods have a restricted domain of applicability for constructing ε-uniformly convergent numerical methods. It was first established in [124] that there are no ε-uniformly convergent schemes based on the fitted operator method in the case for singularly perturbed elliptic convection-diffusion equations in domains where parts of the boundary are characteristics of a reduced equation and parabolic boundary layers appear. In the same paper, a scheme was constructed that converges ε-uniformly, using both the fitted operator method for the approximation of derivatives along characteristics of the reduced equation and the condensing mesh method for the approximation of derivatives in the direction orthogonal to the characteristics. The resulting discrete solutions also make it possible to approximate the normalized derivatives ε-uniformly. For parabolic equations with parabolic layers it is proved that there exist no schemes using the fitted operator method that converge ε-uniformly in any of the papers: [130] in the case of a parabolic boundary layer and [148] in the case of a parabolic initial layer. When constructing schemes for nonlinear problems, the situation is much more complicated. In [137] (for reaction-diffusion equations) and [139] (for convection-diffusion equations), it was established that even for semilinear ordinary differential equations there exist no schemes based on the fitted operator method that converge ε-uniformly. Similar difficulties related to the use of fitted operator methods in numerical methods are discussed in later publications (see, e.g., [138, 87, 86] for partial differential equations and [84, 32] for semilinear ordinary differential equations). Numerical experiments that demonstrate the inconsistency of the fitted operator method for semilinear


6

Introduction

ordinary differential equation are considered in [33]. Thus, for a wide class of boundary value problems there are no schemes using fitted operator methods that converge ε-uniformly in the maximum norm; independently of these, schemes may be constructed using classical finite difference approximations or finite element or finite volume methods. Having summarizing the approaches to the construction of ε-uniformly convergent numerical methods, we make one more remark. In works [134, 140] (see also discussions in [87, 83]), a class of problems is distinguished for elliptic and parabolic equations that degenerate on the boundary of the domain whose solutions contain initial and parabolic boundary layers. It is shown that for such problems there are no schemes of the condensing mesh method (for rectangular meshes) that converge ε-uniformly. But the application of both approaches—fitted mesh and fitted operator methods— makes it possible to construct schemes that converge ε-uniformly. Up to now, practically all singularly perturbed partial differential equations, for which difference schemes that converge ε-uniformly in the maximum norm have been constructed, do not contain mixed derivatives. Boundary value problems have been considered only for the simplest subdomains of dimension not higher than two, and the elliptic operator in the differential equations is the Laplace operator. In the case of an elliptic equation with mixed derivatives considered only on a rectangle, the ε-uniformly convergent scheme obtained appears to be too complex and can not be extended to other dimensions in geometry [28]. Problems in domains with curvilinear boundaries are considered only in few publications. It is one of the goals of the present book to overcome such an existing unsatisfactory state in the area of development of ε-uniformly convergent difference schemes. Another goal of the book is, on model problems, to consider some modern trends in the development of numerical methods for singularly perturbed problems that require further investigation.

1.2 Theoretical problems in the construction of difference schemes We discuss some basic principles related to the foundations of the theory of finite difference schemes, which arise in the development of ε-uniformly convergent numerical methods. The behaviour of derivatives of a solution to singularly perturbed boundary value problems motivates the type of grid and the difference schemes on that. The derivatives of the solution of the boundary value problem in a domain with a sufficiently smooth boundary are ε-uniformly bounded in most of the domain and grow without bound only in a narrow subdomain (the boundary layer) whose width tends to zero with ε. The essential anisotropy of the directional


Theoretical problems in the construction of difference schemes

7

derivatives is exhibited. The maximal values of derivatives of the solution in a neighborhood of the boundary layer are found only for the derivative along the normal to the boundary of the domain; the ¡ k-th ¢ normal derivative in the neighborhood of the boundary is of order O ε−k , and moreover, the derivative decreases exponentially to finite values as the value ε−k ρ grows, where ρ is the distance to the boundary. The behaviour of the solution of the boundary value problem is particularly complicated in the case of domains with piecewise-smooth boundaries. The observed anisotropy in the behaviour of the derivatives of the solution and the unbounded growth of the derivatives for ε → 0 motivates the use of essentially anisotropic meshes condensing in a neighborhood of the boundary layer. Note that the terms of a differential convection-diffusion equation, involving spatial derivatives across the boundary layer, become unbounded as ε → 0. So, very close to the boundary, in order to approximate the solution of the boundary value problem by a solution of a finite difference scheme, it is necessary to use meshes whose step-size along the normal to the boundary is of order o(ε) [138]. Under this requirement on the mesh-size, errors resulting in the approximation of the terms of the differential equation, when the derivatives in the equation are replaced by difference derivatives, also grow without bound [138]. The difference scheme no longer approximates the boundary value problem ε-uniformly. This unbounded behaviour of terms in the differential equation and their difference approximations gives rise to difficulties in the proof of ε-uniform convergence of the relevant schemes. The violation of the monotonicity property (see [108]) of a boundary value problem when grid approximations are constructed even for the simplest problems can lead to both large errors and nonphysical results. For example, in the case of an ordinary differential convection-diffusion equation, when centered difference approximations on uniform meshes are used, the error in the discrete solution oscillates and grows without bound if the parameter ε is much less than the mesh-size [33]. In the presence of mixed derivatives in differential equations and also in the case of problems in domains with curvilinear boundaries, significant difficulties arise in the construction of monotone discrete approximations when meshes that condense gradually in the boundary layer are used (for example, Bakhvalov’s meshes [15]). Also, for difference schemes that do not approximate the boundary value problems ε-uniformly, additional difficulties appear when justifying the ε-uniform convergence of solutions in the maximum norm. Conventional approaches to overcoming the specific problems mentioned above that arise in the construction of ε-uniformly convergent numerical methods turn out to be ineffective. So, in convection-diffusion problems the use of an artificial viscosity in a numerical method in order to suppress oscillations in the grid solution leads to essential errors in a neighborhood of the boundary layer [214, 55]. In the case of singularly perturbed problems the methods for the derivation


8

Introduction

of a priori estimates developed for regular problems (see, e.g., [67, 68, 69, 37]) result in estimates for the derivatives that do not distinguish the different behaviours of derivatives in different directions. The requirements imposed on the data of a regular problem for obtaining a priori estimates by this approach are close to being necessary. But estimates obtained in this way for singularly perturbed problems turn out to be unable to justify ε-uniform convergence. Methods based on asymptotic expansions in powers of the parameter ε allow one to construct approximations to the solutions with an accuracy up to a sufficiently high order of the parameter ε (see, e.g., [211, 210, 208, 59, 62, 209, 94, 57] and references therein). But the detailed behaviour of the derivatives of the solution in various directions is, in general, not considered. To construct asymptotic expansions it is required that the data of the boundary value problem possess sufficient smoothness. This then restricts the class of boundary value problems for which ε-uniformly convergent numerical methods can be developed and justified. Thus, the above approaches turn out to be of little use for the construction of a priori estimates required for developing ε-uniformly convergent numerical methods.

1.3

The main principles in the construction of special schemes

We now outline the main principles underlying the development of the εuniformly convergent difference schemes which are used in this book. Even when numerical methods are constructed for regular problems that have a complicated solution, it is necessary to use grid approximations that inherit the monotonicity property of the boundary value problem [108] in order to prevent the appearance of nonphysical effect in the solution. In the case of singularly perturbed problems such a natural requirement leads to complicated difference schemes (in the case when mixed derivatives are present in the equations, see, e.g., schemes in [112] for regular problems and in [138] for singularly perturbed boundary value problems). For the classes of boundary value problems studied here the problems that appear are to find appropriate classes of grid approximations and also to obtain sufficient conditions, close to necessary ones, for the ε-uniform convergence of the schemes constructed. The study of sufficiently wide classes of boundary value problems requires the development of new approaches to the construction of numerical methods and, in general, leads to more complicated schemes. Primarily, we need a priori estimates for the solutions that adequately reflect the behaviour of such singularities as a boundary layer in a neighborhood of different parts of the domain boundary. To construct special schemes, we use the simplest standard ε-uniformly monotone approximations to the boundary value problem. The


Main principles in the construction of special schemes

9

schemes are constructed on the simplest piecewise uniform meshes that ensure the ε-uniform convergence of the grid solution. The technique for constructing a priori estimates based on a decomposition of the solution first conceived in [118] and developed in [138] (see also [129, 131, 87] and the discussions, for example, in [105, 96, 95] for a two-dimensional problem) was called the Shishkin decomposition (see, e.g., [72, 73]). This technique uses the decomposition of the solution into regular and singular components and, in the case of problems in domains with piecewise smooth boundaries, the singular component, in turn, is represented in terms of its components as the sum of regular and corner layers of different dimension. The specific behaviour of the regular component, as well as of constituents of the singular components, is revealed using the first terms of their asymptotic expansions with respect to the parameter ε. Such a technique allows us to detect the distinctions in the behaviour of derivatives of the singular component in different directions, and to obtain estimates that are necessary to prove the ε-uniform convergence of the difference schemes being constructed. In the method of condensing grids conceived in [128], which uses piecewise uniform meshes, the distribution of mesh points in the direction across the boundary layer, normal to the boundary, is defined by one transition parameter, that is, by the point at which the mesh size changes. In the literature, such meshes are called Shishkin’s meshes (see, e.g., [83]). For difference schemes on these meshes, the difficulties in the development of ε-uniformly convergent grid methods are essentially reduced that allows the construction of ε-uniformly monotone difference schemes which converge ε-uniformly for representative classes of singularly perturbed boundary value problems. The first numerical results on piecewise uniform meshes were obtained in [88] for an ordinary differential equation, in [89] for a parabolic partial differential equation with a parabolic boundary layer, and in [40] for an elliptic convection-diffusion equation on a rectangle with a parabolic boundary layer that is generated on the characteristic parts of the boundary. An ε-uniformly convergent scheme for a problem in a two-dimensional domain with a curvilinear boundary and an equation with mixed derivatives was constructed in 1989 for the first time in [130] in the case of a linear parabolic reaction-diffusion equation. For some boundary layer problems in n-dimensional domains, such similar schemes were constructed in [131], and in [136, 137] for quasilinear equations in an n-dimensional slab (see also [139] (1992)). The ε-uniformly convergent schemes in all these cases were constructed on piecewise uniform meshes. The appearance of piecewise uniform meshes (see, e.g., [127, 128]) that simplified the construction and study of ε-uniformly convergent grid methods and also the progress in techniques for obtaining a priori estimates, as well as the identification of a large class of problems for which the approach based on a fitted operator method is restricted, contributed to the development of εuniformly convergent difference schemes for partial differential equations (see some results related to the development of ε-uniformly convergent schemes,


10

Introduction

for example, in the books [138, 87, 106, 33], in the survey [204], and in the references therein).

1.4 Modern trends in the development of special difference schemes In the book we point out some problems related to numerical methods for singularly perturbed problems that require further investigation. • As a rule, the order of ε-uniform convergence of difference schemes is low; therefore these schemes are too inefficient for practical use. Thus, the development of schemes with a high convergence order is an important problem.

• A priori estimates for solutions of boundary value problems largely “dictate the structure” of special robust schemes. Therefore, the development of a technique for obtaining a priori estimates for new classes of boundary value problems, in particular, problems that have several perturbation parameters, or problems for systems of equations is an important task.

• In a number of boundary value problems, for example, for parabolic equations with moving boundary and/or interior layers (see, e.g., [159]), the complicated solutions bring too complicated ε-uniformly convergent schemes under the construction. A problem of current interest is to find necessary and sufficient conditions for ε-uniform convergence of such special schemes.

• Even in regular problems, the conditioning number of classical difference schemes grows without bound when the mesh-size tends to zero. In the condensing grid method, the mesh-size in the layer for a fixed number of nodes can be arbitrarily small depending on the parameter ε. Thus, the study of conditioning, as well as ε-uniform conditioning of special schemes, is an important problem. These problems relate to the foundations of the theory of numerical methods (difference schemes) for singularly perturbed problems with complicated singularities.


The contents of the present book

1.5

11

The contents of the present book

The contents of the book can be divided into two parts. Part I is “translation” of certain parts of the book [138] published in 1992 in Russian, and Part II is a survey of some recent results in the development of robust methods for boundary value problems with boundary layers and other singularities. The work involved in the “translation” resulted in the development in detail of principles for constructing monotone difference schemes for some classes of singularly perturbed problems. In Part I (Chapters 2–7) boundary value problems are considered for elliptic and parabolic reaction-diffusion and convection-diffusion equations in n-dimensional domains with smooth and piecewise smooth boundaries in the case when the differential equations contain mixed derivatives. Here our goal is to develop a technique for constructing and justifying ε-uniformly convergent difference schemes, based on classical discrete approximations, for boundary value problems with less restrictions on the problem data. In this situation we are not too concerned about the rate of ε-uniform convergence of schemes constructed. Significant attention is given to the study of conditions that are necessary and sufficient for the ε-uniform convergence of the relevant schemes. When constructing special difference schemes that converge ε-uniformly, we use the techniques developed for canonical problems on a slab and a parallelepiped. Part II contains material published mainly in the last four years. These are problems with boundary layers, and additional singularities generated by nonsmooth data, unboundedness of the domain, and also by the presence of the perturbation vector parameter. Another aspect of the considerations in this part is that we find both the solution and its derivatives with errors that are independent of the perturbation parameters. In Chapters 8 and 13, boundary value problems for equations with perturbation vector parameters are studied. In Chapter 8, problems are considered for a scalar parabolic reaction-diffusion equation with three scalar parameters, and in Chapter 13, a problem is studied for a system of two elliptic reactiondiffusion equations with two scalar parameters. In these problems, layers of different types arise depending on the relation between the scalar parameters. For grid approximations of boundary value problems with a moving boundary layer, necessary and sufficient conditions for the schemes to be robust are obtained in Chapter 9. Taking into account these conditions, difference schemes are constructed for a parabolic reaction-diffusion equation in a domain with moving boundaries. Methods of improving the accuracy of grid solutions while preserving their robustness are examined in Chapter 10. Conditioning and correctness (or well-posedness) of a difference scheme that is constructed based on a classical grid approximation of a boundary value problem for a parabolic convection-


12

Introduction

diffusion equation on piecewise-uniform meshes are discussed in Chapter 12. In Chapter 11, difference schemes on a priori adapted meshes are developed. Promising approaches to the development of robust methods based on a posteriori adapted techniques are discussed in Chapter 14. In the case of singularly perturbed problems, the approaches known for regular problems improve the convergence only for finite values of the parameter ε, and do not provide for the robustness of the method. Here, difference schemes are discussed that converge “almost ε-uniformly”. In the survey Chapter 14 we give comments to some approaches in the construction of special difference schemes and research directions. In particular, the following problems are discussed: • some “applied” problems whose solutions require robust numerical methods; • approaches to the construction of robust numerical methods for parabolic equations with smooth and piecewise smooth boundary conditions; • approximation of solutions and derivatives; • specific boundary value problems for elliptic convection-diffusion equations such as a problem on an unbounded domain for an equation with a scalar parameter and a problem on a rectangle for an equation with a vector parameter; • in the case of an unbounded domain, the concept of the domain of essential dependence of the solution that allows robust schemes to be constructed whose solutions converge on bounded domains by increasing the number of nodes in the meshes used; • compatibility conditions for a problem on a rectangle that guarantee the smoothness of the solution and its regular and singular components that are required for the construction and justification of robust schemes.

1.6

The present book

We discuss the existing state of the art in the area of the development of difference schemes that converge ε-uniformly in the maximum norm. The books [138, 87, 33], as well as the present book, are focussed on the development of a technology for constructing finite difference schemes that converge ε-uniformly in the maximum norm for a wide class of singularly perturbed boundary value problems. In [106], ε-uniformly convergent numerical methods are considered that use various techniques and methods developed for


The present book

13

regular and singularly perturbed problems. The convergence of discrete methods is considered in norms corresponding to the applied numerical method. In [106] the basic results developed in [138] are given in a condensed form, and problems for equations with mixed derivatives and multi-dimensional problems are not discussed. The books [138, 87, 33] are devoted to the development of finite difference schemes for singularly perturbed elliptic and parabolic equations. Difference schemes are constructed based on standard monotone finite difference approximations of differential problems on piecewise uniform meshes. The books [138, 87] emphasize the theory, and [33] is strictly numerical. Appearing in 1988, piecewise-uniform meshes [127, 128] allow us to overcome a number of essential difficulties in the development and justification of ε-uniformly convergent finite difference schemes for representative classes of singularly perturbed equations with partial derivatives. The development of techniques to derive a priori estimates on the basis of the decomposition method applied to the solution of boundary value problems, and also techniques to construct monotone discrete approximations (of boundary value problems) constructed on piecewise uniform meshes, allow the development of special finite difference schemes that are convergent ε-uniformly. As a result of such a progress, the book [138] was written. In [138] rather wide classes of elliptic and parabolic equations as reactiondiffusion, and convection-diffusion, have been considered in n-dimensional domains with smooth and piecewise smooth boundaries, and also in composite domains in the presence of concentrated sources. For these problems, techniques to construct ε-uniformly convergent schemes have been developed, such schemes have been constructed and their convergence has been studied. The presentation of results derived in [138] has been given in a concentrated form as in a reference book, so that it was possible to cover large classes of boundary value problems. However, such a form of exposition makes it difficult to use them in the development of numerical methods for equations with partial derivatives. It could be that this circumstance is one of the reasons for the delay in using results from [138], as well as a rather long delay in the development of robust numerical methods for multidimensional problems. The book [87] is a thorough introduction to robust finite difference schemes. In [87], basic ideas and the techniques that are used to construct and justify ε-uniformly convergent difference schemes were considered using model problems as examples. Here, difference schemes based on the fitted operator method and the condensing mesh method have been constructed that converge ε-uniformly. Difficulties that arise when constructing robust numerical methods for parabolic equations with parabolic boundary layers have been considered. In the book [33], for a number of model problems with boundary layers, extensive numerical studies have been performed for robust difference schemes, and in particular, for those that have been considered in [138, 87]. For the


14

Introduction

quantitative analysis of numerical methods, an experimental technique has been developed which allows us to justify ε-uniform convergence, and also to find the parameters in the estimates of the convergence rate. A technique for experimental study of special difference schemes has been developed in [33]. This technique was tested by numerical experiments carried out for linear and nonlinear problems, in particular for elliptic convection-diffusion equations whose solutions have parabolic boundary layers that degenerate on the domain boundary. Details of the solvers that are used in the computations have also been given. In the concluding chapters of the book [33], the efficiency of the techniques that have been developed in the first ten chapters is demonstrated when solving the classical Prandtl problem for flow of an incompressible fluid past a semi-infinite flat plate [113]. For the nonlinear Prandtl problem, a difference scheme was constructed using a nonlinear solver; in this problem, ε = Re−1 is a perturbation parameter, and Re is the Reynolds number. The numerical method gives the problem solution (that is components of the flow velocity) and the normalized first-order derivatives with an accuracy close to order one. Computational expanses for the solution (up to a logarithmic multiplier, are proportional to the number of mesh points used) are independent of the parameter ε. In the book [106], numerical methods have been considered for singularly perturbed boundary value problems to ordinary differential equations and also to one-dimensional parabolic and two-dimensional elliptic equations (with the Laplace operator as the main part of the differential operator). Various ideas and techniques that are used in the numerical analysis for singularly perturbed differential equations, and also approaches developed for regular problems, are discussed. Finite difference methods and finite element methods have also been considered for singularly perturbed boundary value problems. Approximations of solutions to boundary value problems in various norms, in particular in the maximum and the energy norms, have been discussed. Significant attention has been devoted to ε-uniformly convergent numerical methods in which the fitted operator method and the fitted mesh method are applied using Bakhvalov meshes and piecewise-uniform meshes. A separate chapter is devoted to incompressible Navier-Stokes equations. The book contains an extensive bibliography on numerical methods for singularly perturbed boundary value problems. The present book is devoted to a systematic detailed development of approaches to the construction of ε-uniformly convergent finite difference schemes for some classes of singularly perturbed boundary value problems considered in [138]. Problems for multi-dimensional elliptic and parabolic equations that are reaction-diffusion and convection-diffusion are studied in the case when mixed derivatives are involved in the differential equations. The problems are considered in domains with piecewise smooth and curvilinear boundaries. Boundary value problems from [138] such as problems for equations with dis-


The present book

15

continuous coefficients in the presence of concentrated sources, and also problems for convection-diffusion equations with characteristic parts on the boundary, are not considered. The approach presented here includes: • a technique to construct a priori estimates for solutions of boundary value problems which are necessary to construct and justify finite difference schemes ε-uniformly convergent in the maximum norm, • a technique to construct finite difference schemes based on standard discrete approximations of boundary value problems, preserving monotonicity of differential problems, • a technique to construct special piecewise uniform meshes (grids) providing ε-uniform convergence of the schemes under construction. Furthermore, significant attention is devoted to the consideration of modern trends in the development of ε-uniformly convergent numerical methods for singularly perturbed problems. The investigations of a series of model problems can be considered as representative models of more general classes, for which approaches from the first part of the present book could be applied to the development of appropriate numerical methods. Approaches developed in the book, and results derived, should promote further development of efficient numerical methods for singularly perturbed problems with different types of singularities, and at the same time, preserve the same qualities already well-established in numerical methods for regular boundary value problems. In the present book, as well in the books [138, 87, 33], ε-uniformly convergent schemes are constructed on simplest piecewise uniform meshes, i.e., meshes having only one transition point in a neighborhood of the boundary layer. Schemes on Bakhvalov meshes, as well as on meshes with few transition points in a neighborhood of the boundary layer (see, e.g., [169]), are not considered here. The present book and the books [138, 87, 106, 33], considered as one package, cover a wide spectrum of problems that arise when constructing special numerical methods for problems with boundary layers. Nevertheless, there is a large gap between complicated systems of nonlinear singularly perturbed equations with partial derivatives which need to be solved, e.g., from mathematical modelling, and those singularly perturbed problems for which theoretically justified numerical methods exist. We hope that the results given here could be helpful in the development and analysis of new numerical methods for problems that have yet to be solved.


16

1.7

Introduction

The audience for this book

The present book, together with the books [138, 87, 106, 33], could be useful for scientists-researchers (from students up to professionals) in the development of numerical methods for singularly perturbed problems and also for specialist-researchers who are interesting in mathematical modelling and where problems with boundary and interior layers arise naturally. In the first part of the book devoted to multi-dimensional problems, techniques and ideas exploited to derive a priori estimates for the solutions and to construct the simplest discrete constructions, i.e., piecewise uniform meshes and standard monotone finite difference operators on them, are carefully discussed for each type of boundary value problem. Conditions imposed on discrete constructions are discussed, and it is shown that such conditions are close to necessary conditions. Techniques and principles given in the book allow the reader to construct independently robust finite difference schemes for new classes of boundary value problems, in particular, for problems from [138] that are not treated in the present book. In the second part of the book, a consideration of modern trends are carried out on model problems, which allow to check on ideas and techniques used in the construction. Here, we give references to original sources where details are given in the exposed bibliography. The discussion of modern trends allows the reader to be more confident in this fast growing area of research. The first and second parts of the book can be studied independently. The book can be also helpful for researchers studying singularly perturbed ordinary differential equations. If the reader wishes to develop new difference methods for particular singularly perturbed problems, it is only necessary to read Part I. If the reader wishes to apply difference methods already developed for singularly perturbed problems, the reader should pass to Part II.


Chapter 2 Boundary value problems for elliptic reaction-diffusion equations in domains with smooth boundaries

In this chapter boundary value problems are considered for elliptic reactiondiffusion equations in domains with smooth boundaries. The construction of finite difference schemes and the justification of their convergence is carried out using an analog of the well-known sufficient condition for convergence of schemes for regular boundary value problems that is a consequence of approximation by a stable finite difference scheme (its description see, e.g., [79, 108, 100]). When applied to singularly perturbed boundary value problems, this sufficient condition for ε-uniform convergence can be stated in the following way: ε-uniform convergence of a finite difference scheme follows from ε-uniform approximation of the boundary value problem by an ε-uniformly stable finite difference scheme. Our finite difference schemes are constructed based on classical difference approximations of differential equations. ε-uniform approximation of boundary value problems by finite difference schemes is obtained by a special distribution of mesh points. ε-uniform monotonicity (and, hence, stability) of the finite difference schemes is achieved by means of spatial meshes whose distributions of nodes in each coordinate direction are consistent with each other.

2.1

Problem formulation. The aim of the research

On an n-dimensional domain D with sufficiently smooth boundary Γ = Γ(D) and S D = D Γ, (2.1) where D is, in general, an unbounded domain, we consider the Dirichlet problem for elliptic reaction-diffusion equations L u(x) = f (x),

x ∈ D,

(2.2a)

u(x) = ϕ(x),

x ∈ Γ.

(2.2b)


18 Here

Elliptic reaction-diffusion equations 1

L ≡ ε2 L2 + L0 , n X

L2 = L2(2.2c) ≡

L0 ≡ −c0 (x), n

ask (x)

s, k=1

X ∂2 ∂ + bs (x) − c(x). ∂xs ∂xk s=1 ∂xs

(2.2c)

The coefficients in (2.2c) are bounded and satisfy the ellipticity condition a0

n X s=1

ξs2 ≤

n X

ask (x)ξs ξk ≤ a0

n X

ξs2 ,

x ∈ D,

a0 > 0.

(2.3a)

s=1

s,k=1

They also satisfy the condition c0 (x) ≥ c0 > 0,

c(x) ≥ 0,

x ∈ D.

(2.3b)

The perturbation parameter ε takes arbitrary values in the open-closed interval (0, 1]. The coefficients, the right-hand side f , and the boundary function ϕ in (2.2) are assumed to be sufficiently smooth. As well as the problem (2.2) on the arbitrary domain (2.1), we consider also the boundary value problem on the slab D

(0)

= D(0)

S (0)

Γ

,

(2.4)

D(0) = {x : d∗1 < x1 < d∗1 , | xs | < ∞, s = 2, . . . , n}. (0)

For a periodic boundary value problem on the slab D the coefficients, the right-hand side f of the differential equation, and the boundary function ϕ are assumed to be 2π-periodic functions in xs for s = 2, . . . , n. In the problems (2.2), (2.1) and (2.2), (2.4), a boundary layer appears in a neighborhood of the boundary as the parameter ε tends to zero. We give some definitions. A finite difference scheme 2 means a set of grids and systems of difference equations defined on these grids. The solution error of a difference scheme depends, in general, on the number of nodes in the meshes used and the value of the parameter ε. In the case when the solution error is independent of the value of the parameter ε, we say that the solution of the difference scheme (or, briefly, the scheme itself ) converges ε-uniformly. Convergence of a finite difference scheme is considered in the discrete maximum norm. For singularly perturbed boundary value problems, by classical finite difference schemes (classical difference approximations of boundary value problems) we mean schemes (approximations) such that have no special 1

The notation L(j.k) (M(j.k) , m(j.k) , Gh(j.k) , f(j.k) (x)) means that these operators (constants, grids, functions) are introduced in formula (j.k). 2 Further, briefly, we shall use terms “difference scheme” or “scheme” instead of “finite difference scheme”.


Boundary value problems in smooth domains

19

imposed restrictions (other than standard ones) on the choice of grids and difference equations that ensure ε-uniform convergence of the scheme. We say that a scheme is a fitted operator method when its ε-uniform convergence is ensured by a special choice of approximating difference equations. When constructing these schemes we use grids with arbitrary distribution of nodes. The condensing mesh method (or adapted mesh method) comprises schemes whose ε-uniform convergence is ensured by the choice of an appropriate distribution of mesh points. Such schemes are classical finite difference schemes on condensing/adapted meshes. When constructing such schemes we use systems of difference equations whose coefficients are, in general, functionals of the coefficients in the differential equation and the problem data (see, e.g., [79, 108]). Our aim for the boundary value problem (2.2), (2.1) is to construct a finite difference scheme that converges ε-uniformly on condensing meshes.

2.2

Estimates of solutions and derivatives

For the construction and investigation of difference schemes, we need estimates for the solution of the boundary value problem and its derivatives for ε ∈ (0, 1]. These bounds are derived by using a priori estimates for regular problems, i.e., Schauder-type interior estimates and estimates up to the boundary [69, 37]. Here an a priori estimate means an estimate of the solution under the assumption that this solution exists, that depends on the problem data, i.e., the coefficients in the equation, its right-hand side, and the boundary conditions. We denote by C l+α (D), where l ≥ 0 and α ∈ (0, 1), the Banach space whose elements are continuous functions in D that have in D continuous derivatives of order up to l which are in the sense of Hölder continuous with exponent α. For these elements a finite value is taken by the norm X

|u|l+α = |u|l +

Ã Hα

k1 +...+kn =k

! ∂k u . ∂xk11 . . . ∂xknn

k=l

Here |u|l =

l X

X

k=0 k1 +...+kn =k

¯ ¯ ∂k ¯ ¯ k1 ¯ ∂x1 . . . ∂xknn

¯ ¯ ¯u(x) − u(x0 )¯ ¯ ¯ H α (u) = sup , ¯x − x0 ¯α x, x0 ∈D


¯ ¯ ¯ u¯ , ¯ 0

|u|0 = max |u(x)|, D

20

Elliptic reaction-diffusion equations

and |x − x0 | is the distance between the points x and x0 , where x, x0 ∈ D, and H α (u) is Hölder coefficient of the function u in D. Solutions of singularly perturbed problems change sharply and their derivatives grow without boundary as ε → 0, however, in thin subdomains, i.e., in neighborhoods of boundary and interior layers. We are interested in estimates of the solutions and derivatives depending on the value of the parameter ε, and the behaviour of the derivatives in a neighborhood of the layer. When studying boundary value problems, a maximum principle is often used in the following form of a comparison lemma (see, e.g., [69, 37, 99, 205]). Lemma 2.2.1 Let v, w ∈ C 2 (D) |L v(x)| ≤ −L w(x),

T

C(D), and let the condition

x ∈ D,

|v(x)| ≤ w(x),

hold, where L = L(2.2a) . Then |v(x)| ≤ w(x),

x ∈ Γ,

x ∈ D.

Here w(x), x ∈ D is a majorant (barrier) function for v(x). We consider the boundary value problem (2.2) on the slab (2.4). Using the maximum principle, we deduce the inequality 3 £ ¤ |u(x)| ≤ M max max |f (x)|, max |ϕ(x)| , x ∈ D. (2.5) D

Γ

For ask , bs , c, c0 , f ∈ C l+α (D), ϕ ∈ C l+2+α (Γ), where l is an integer number, l ≥ k 0 , k 0 ≥ 0, α ∈ (0, 1), we have u ∈ C l+2+α (D) for each ε ∈ (0, 1] [69]. Next, we pass to the boundary value problem in the variables ξs = ξs (x) = ε−1 xs , where s = 1, . . . , n. In these variables (ξ = ξ(x)), the differential e = D e S Γ, e where D e becomes regular; here D e = {ξ = equation on the set D ξ(x) : x ∈ D}. Taking into account the a priori estimates up to the boundary [69], we find ¯ ¯ ¯ ¯ ∂k ¯ ¯ e k = k1 + . . . + kn , u e (ξ) ¯ k1 ¯ ≤ M, 0 ≤ k ≤ k 0 + 2, ξ ∈ D, k ¯ ∂ξ1 . . . ∂ξnn ¯ where u e(ξ) = u(x(ξ)). Returning to the variables x, we obtain the following estimate for the function u(x): ¯ ¯ ¯ ¯ ∂k ¯ ¯ u(x) (2.6) ¯ k1 ¯ ≤ M ε−k , x ∈ D, 0 ≤ k ≤ K, ¯ ∂x1 . . . ∂xknn ¯ where K = k 0 + 2. 3

Here and below, M, Mi , M i (or m, mi , mi ) denote sufficiently large (small) positive constants that are independent of the parameter ε and of the discretization parameters.



21

Lemma 2.2.2 Let ask , bs , c, c0 , f ∈ C l+α (D), s, k = 1, . . . , n, ϕ ∈ C l+2+α (Γ), l ≥ k 0 , k 0 ≥ 0, α ∈ (0, 1). Then the solution of the problem (2.2), (2.4) satisfies the estimate (2.6). Write the function u(x), i.e., the solution of the problem (2.2), as the decomposition, i.e., the sum of the functions: u(x) = U (x) + V (x),

(2.7)

where U (x) and V (x) are the regular and singular parts (components) of the solution of the problem. The function U (x) is the restriction to D of the function U e (x), x ∈ De , where U e (x) is the solution of a problem which is extended beyond the set Γ: Le U e (x) = f e (x),

x ∈ De ,

U e (x) = ϕ e (x),

x ∈ Γe .

(2.8)

Here the domain De involves the domain D together with its m-neighborhood; Γe = Γ(De ) is sufficiently smooth boundary of the domain De ; the coefficients of the operator Le and the right-hand side f e (x) are sufficiently e smooth extensions on D of corresponding data of the problem (2.2) and e e ϕ (x), x ∈ Γ , is a sufficiently smooth function. The functions f e (x), ϕ e (x), are assumed to be equal to zero outside an m1 -neighborhood of the set D, where m1 < m. The function V (x) is the solution of the problem L V (x) = 0,

x ∈ D,

V (x) = ϕ(x) − U (x),

x ∈ Γ.

(2.9)

e

Write the function U e (x), x ∈ D , as the sum of the functions U e (x) = U0e (x) + v1e (x),

e

x∈D .

(2.10a)

Here the function U0e (x) is the solution of a degenerated problem and is defined by the relation e −c0e (x) U0e (x) = f e (x), x ∈ D ; (2.10b) the function v1e (x) is the remainder term in the representation of the function U e (x) and is the solution of the problem Le v1e (x) = −ε2 Le2 U0e (x), x ∈ De ,

v1e (x) = ϕe (x) − U0e (x), x ∈ Γe , (2.10c)

where Le2 is an extension of the operator L2 = L2(2.2c) . Let the data of the problem (2.2), (2.4) satisfy the condition ask , bs , c, c0 , f ∈ C l+α (D), where l ≥ k 0 , k 0 ≥ 2, α ∈ (0, 1), and let the extensions of e e these data to the set D belong to the class C l+α (D ). Then for the function e v1e (x), one has v1e ∈ C l+α (D ), and also the estimate ¯ e ¯ ¯v1 (x)¯ ≤ M ε2 , x ∈ D e , that is established using the maximum principle. In the boundary value probe lem for the function v1e (x), x ∈ D , we pass to the variables ξs = ξs (x) =


22


ε−1 xs , where s = 1, . . . , n. In these variables, the differential equation on e e becomes regular, where D e = {ξ = ξ(x) : x ∈ D e }. Taking into the set D e account interior a priori estimates [69], we find the following estimates on D ¯ ¯ ¯ ¯ ∂k ¯ ¯ e e v e (ξ) ¯ k1 ¯ ≤ M ε2 , 0 ≤ k ≤ k 0 , ξ ∈ D, 1 ¯ ∂ξ1 . . . ∂ξnkn ¯ where ve1e (ξ) = v1e (x(ξ)). Returning to the variables x, by virtue of the explicit form of the function U0e (x), we obtain U ∈ C l+α (D); moreover, ¯ ¯ ¯ ¯ ∂k ¯ ¯ U (x) (2.11) ¯ k1 ¯ ≤ M [1 + ε2−k ], x ∈ D, 0 ≤ k ≤ K, ¯ ∂x1 . . . ∂xknn ¯ where K = k 0 . The regular component of the problem solution and its lowest derivatives (here they are no higher than second order) are bounded εuniformly on the set D. Using majorant functions of type ¡ ¢ ¡ ¢ £ ¤ exp − m ε−1 (d∗1 − x1 ) + exp − m ε−1 (x1 − d∗1 ) , x ∈ d∗1 , d∗1 , we establish the inequality ¯ ¯ ¡ ¡ ¢¢ ¯V (x)¯ ≤ M exp −m ε−1 r x, Γ ,

x ∈ D,

(2.12a)

where r(x, Γ) is the distance from the point x ∈ D to the boundary Γ and m is an arbitrary constant in the interval (0, m0 ), the constant m0 defined by the relation £ ¤1/2 0 . (2.12b) m0 = min a−1 11 (x) c (x) D

For the data of the problem (2.2), (2.4), as well as the conditions for smoothness given above, let the condition ϕ ∈ C l+α (Γ) be fulfilled also. Then V ∈ C l+α (D). With regard to the estimates up to the boundary, in the variables ξ = ξ(x) [69] one has ¯ ¯ ¯ ¯ ³ ´ ∂k ¯ e e (ξ)¯¯ ≤ M exp −m r(ξ, Γ) e , ξ ∈ D, V 0 ≤ k ≤ k0 , ¯ k1 k ¯ ∂ξ1 . . . ∂ξnn ¯ e and we obtain where Ve (ξ) = V (x(ξ)), ξ ∈ D, ¯ ¯ ¯ ¯ ¡ ¡ ¢¢ ∂k ¯ ¯ V (x) ¯ k1 ¯ ≤ M ε−k exp −m ε−1 r x, Γ , x ∈ D, k n ¯ ∂x1 . . . ∂xn ¯

(2.13)

0 ≤ k ≤ k0 . Let us refine the estimate (2.13) in the case of the problem (2.2), (2.4). The function V (x) is the solution of the problem (2.9). We differentiate



23

the equation and the boundary condition in xj , where j 6= 1. In the lefthand side of the equation obtained, we leave a term that involves the function V (1) (x) ≡ (∂/∂xj )V (x) and its derivatives. We transfer to the right-hand side of the equation those terms that contain the derivatives of the coefficients of (2.2a) as multipliers. We denote the right-hand side by f (1) (x), where f (1) ∈ C l−1+α (D)), and we have the inequality ¡ ¢ |f (1) (x)| ≤ M exp −mε−1 r(x, Γ) , x ∈ D. Taking into account the inequality ¯ ¯ ¯ ¯ ¯ (1) ¯ ¯¯ ∂ϕ(x) ∂U (x) ¯¯ − ≤ M, ¯ V (x)¯ = ¯ ∂xj ∂xj ¯

x ∈ Γ,

one can establish the estimate ¯ ¯ ¯ ¯ ∂k ¯ ¯ V (x) ¯ k ¯≤ ¯ ∂x 1 . . . ∂xkj−1 ∂xj ∂xkj+1 . . . ∂xknn ¯ 1 j−1 j+1 £ ¤ ¡ ¢ ≤ M ε−k1 + ε1−k exp −m ε−1 r(x, Γ) , 0 ≤ k ≤ k 0 , j > 1, k = k1 + . . . + kj−1 + 1 + kj+1 + . . . + kn . In a similar way, we obtain the estimate ¯ ¯ ¯ ¯ £ ¤ ¡ ¢ ∂k ¯ ¯ V (x) ¯ k1 ¯ ≤ M ε−k1 + ε2−k exp −mε−1 r(x, Γ) , ¯ ∂x1 . . . ∂xknn ¯ x ∈ D,

(2.14)

0 ≤ k ≤ K,

where m is an arbitrary constant in the interval (0, m0 ) with m0 = m0(2.12) in the case of the problem (2.2), (2.4), for K = k 0 . In the estimates (2.13), (2.14), one can choose the constant m as an arbitrary number in the interval (0, m0 ). The singular component of the problem solution decays rapidly when moving away from the set Γ (i.e., exponentially decreases as ε−1 r(x, Γ) grows); however, its derivatives along the normal to the boundary Γ in the neighborhood of Γ defined by ε−1 r(x, Γ) = O (1) grow without bounds as ε → 0 and ε−1 r(x, Γ) → ∞. Thus, the following theorem is established. Theorem 2.2.1 Let the condition ask , bs , c, c0 , f ∈ C l+α (D), for s, k = 1, . . . , n, ϕ ∈ C l+α (Γ), l ≥ k 0 , k 0 ≥ 2, α ∈ (0, 1) hold. Then for the components U (x) and V (x) of the solution u(x) to the problem (2.2), (2.4) in the representation (2.7), the estimates (2.11), (2.14) hold.


24


Let us discuss estimates of the problem (2.2), (2.1) in the case when the domain D has sufficiently smooth curvilinear boundary Γ. Here it is convenient to consider the problem solution in the space C l+α (D) (a definition can be found in Chapter 3, Section 3.2). Let Γ0 ⊆ Γ be a smooth part of the boundary Γ, i.e., Γ0 = Γg0 , and let Γ0 ∈ C l+α , where l ≥ k 0 , k 0 ≥ 2 and α ∈ (0, 1). Thus, in a neighborhood of each point of the boundary Γ, there exists a local representation of the boundary Γ0 in the form ¡ ¢ xi = ηi x1 , . . . , xi−1 , xi+1 , . . . , xn ; furthermore, the function ηi belongs to the class C l+α . In the case when ϕ ∈ C l+α (Γ0 ), then the function ϕ(x), x ∈ Γ0 (as a function of the local variables x1 , . . . , xi−1 , xi+1 , . . . , xn ), belongs to the regarded class C l+α . In the case when ask , bs , c, c0 , f ∈ C l0 +α (D), ϕ ∈ C l+α (Γ0 ) for l0 = l − 2, for the problem solution the estimate (similar to (2.6) and (3.4)) ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ u(x) ¯ k1 ¯ ≤ M ε−k + r−k (x, Γ \ Γ0 ) , x ∈ D, 0 ≤ k ≤ K, k n ¯ ∂x1 . . . ∂xn ¯ holds up to the smooth part of the boundary Γ0 , where K = l. Write the problem solution as the sum of the functions u(x) = U (x) + vu (x),

x ∈ D,

where U (x), x ∈ D, is the restriction to D of the solution to the problem (2.8). For l0 = l the function U (x) satisfies the estimate (2.11). For the function vu (x), one has vu ∈ C l+α (D), and also the estimate ( ¡ ¢ M exp − m ε−1 r(x, Γ) , r(x, Γ) ≤ M1 σ, |vu (x)| ≤ ¡ ¢ M ε2 , r x, Γ ≥ m1 σ, x ∈ D, holds. Here m is an arbitrary value in the interval (0, m0 ), σ is sufficiently ¡ ¢ 0 1/2 small value independent of ε; m0 = a−1 , a0 = a0(2.3) , c0 = c0(2.3) ; 0 c M1 σ is less than the radius of curvature of the boundary Γ. Taking into account interior a priori estimates of the derivatives, we find the estimate of the derivatives outside an σ-neighborhood of the boundary Γ: ¯ ¯ ¯ ¯ ¡ ¢ ∂k ¯ ¯ v (x) ¯ k1 ¯ ≤ M ε2−k , x ∈ D, r x, Γ ≥ σ. u ¯ ∂x1 . . . ∂xknn ¯ For the problem solution outside an σ-neighborhood of the boundary Γ, we obtain the estimate ¯ ¯ ¯ ¯ ∂k ¯ ¯ u(x) (2.15) ¯ k1 ¯ ≤ M [1 + ε2−k ], x ∈ D, ¯ ∂x1 . . . ∂xknn ¯


Boundary value problems in smooth domains ¡ ¢ r x, Γ ≥ σ,

25

0 ≤ k ≤ K,

where K = k 0 . In an σ-neighborhood of the boundary Γ, we write the problem solution as the sum ¡ ¢ u(x) = U (x) + V (x), x ∈ D, r x, Γ ≤ σ, (2.16) where U (x) and V (x) are the regular and singular ¡ ¢ components of the problem e solution, here U (x) = U(2.8) (x), x ∈ D, r x, Γ ≤ σ. For the function U (x) we have the estimate ¯ ¯ ¯ ¯ ∂k ¯ ¯ U (x) (2.17) ¯ k1 ¯ ≤ M [1 + ε2−k ], x ∈ D, ¯ ∂x1 . . . ∂xknn ¯ ¡ ¢ r x, Γ ≤ σ,

0 ≤ k ≤ K,

where K = k 0 . In an σ-neighborhood of the boundary Γ we pass to a new coordinate system x0s = x0s (x), for s = 1, . . . , n such that some part of the boundary Γ belongs to the plane x01 = const, where x01 = x01 (x); r(x, Γ) ≤ σ is the distance from the point x ∈ D to the boundary Γ. Similar to constructions on the slab, we estimate the problem solution in a neighborhood of this part of the boundary. Returning to the original coordinate system, we find that for the regular and singular components of the problem solution in an σneighborhood of the boundary Γ, the estimates similar to (2.11), (2.14) hold. Thus, assuming that the conditions of Theorem 2.2.1 are fulfilled and also the condition Γ ∈ C l+α holds, one obtains that outside some sufficiently small neighborhood of the boundary Γ (the size of the neighborhood is independent of the parameter), the solution of the boundary value problem satisfies the estimate (2.15). In a neighborhood of the boundary, the problem solution has the representation (2.16). For the regular part of the problem solution, the estimate (2.17) is valid, and for the singular part, one has the estimate ¯ ¯ ¯ ¯ £ ¤ ¡ ¢ ∂k ¯ 0 ¯ 0 V (x ) ≤ M ε−k1 + ε2−k exp −m ε−1 r(x0 , Γx0 ) , ¯ 0 k1 ¯ x k ¯ ∂x1 . . . ∂x0n n ¯ (2.18) T x0 ∈ Dx0 {r(x0 , Γx0 ) ≤ σ}, 0 ≤ k ≤ K, T

T

where K = k 0 , Dx0 {r(x0 , Γx0 ) ≤ σ} and Γx0 are the ¡ set ¢D {r(x, Γ) ≤ σ} and the boundary Γ in the variables x0 , Vx0 (x0 ) = V x(x0 ) ; the derivative in x01 is the derivative along the normal to the boundary Γ, and the derivatives in x0s , where s = 2, . . . , n, are the derivatives in the direction orthogonal to the normal to the boundary; m ∈ (0, m0 ), the constant m0 in the case of the estimate (2.18), is defined by the relation ¡ ¢ 0 1/2 , m0 = a−1 0 c


a0 = a0(2.3) ,

c0 = c0(2.3) .

26


Theorem 2.2.2 For the data of the boundary value problem (2.2), (2.1) in the domain D with the curvilinear boundary Γ, let the assumptions of Theorem 2.2.1 be fulfilled, and let Γ ∈ C l+α . Then for the problem solution u(x) and for the components U (x) and V (x) in the representation (2.16), the estimates (2.15), (2.17), (2.18) hold.

2.3 2.3.1

Conditions ensuring ε-uniform convergence of difference schemes for the problem on a slab Sufficient conditions for ε-uniform convergence of difference schemes

When deriving sufficient conditions for ε-uniform convergence of a difference scheme, we use a standard algorithm (principle) that is used for convergence proofs in the case of regular problems: “convergence of a finite difference scheme follows from an approximation of the boundary value problem by a stable finite difference scheme” (see, e.g., [80, 109, 106]). This principle allows us to establish convergence of the stable finite difference scheme that approximates the boundary value problem. We consider the boundary value problem (2.2) on a convex set D,

(2.19)

whose faces are orthogonal to coordinate axes. Let Dh

(2.20)

be a rectangular mesh on the set D, and let N = N (D) be the number of nodes in the mesh Dh , if the set D is bounded, and N = N (D) be the minimal number of nodes in the mesh Dh on the unit n-dimensional cube whose centre belongs to the set D, if the set D is unbounded. On the mesh Dh , consider a finite difference scheme that corresponds to the problem (2.2), (2.19) (see, e.g., schemes [79, 108], in particular, the scheme (2.47), (2.26) for the problem (2.2), (2.4)): Λ z(x) = f h (x), x ∈ Dh ,

z(x) = ϕh (x), x ∈ Γh .

(2.21)

Here the functions f h (x), x ∈ Dh , and ϕh (x), x ∈ Γh , as well as the coefficients multiplied by the difference derivatives in the operator Λ, are functionals of the data of the problem (2.2) and of the coefficients in the differential equation. We say that:



27

1) the finite difference scheme (2.21),¡ (2.20) approximates the boundary ¢ value problem (2.2), (2.19) ε-uniformly in the discrete maximum norm if the following inequalities are fulfilled: |(L − Λ) u(x)| ≤ M λ(N (D)),

x ∈ Dh ; ) |f (x) − f h (x)| ≤ M λ(N (D)), x ∈ Dh

|ϕ(x) − ϕh (x)| ≤ M λ(N (D)), x ∈ Γh

(2.22) ,

(2.23)

where λ(N (D)) − → 0 as N → 0 (the notation f (ξ) − → 0 as ξ → ξ0 means ε

ε

that f (ξ) → 0 as ξ → ξ0 ε-uniformly), and M λ(N (D)) is the ε-uniform error bound of the approximation of the boundary value problem by the finite difference scheme (2.21), (2.20); 2) the finite difference scheme (2.21), (2.20) is ε-uniformly stable (in the discrete maximum norm) if the following inequality is valid: · ¸ h h |z(x)| ≤ M max max |f (x)| , max |ϕ (x)| , x ∈ Dh ; Dh

Γh

3) the finite difference scheme (2.21), (2.20) converges ε-uniformly at the ¡ ¢ rate of O λ(N (D)) if the following inequality holds: |u(x) − z(x)| ≤ M λ(N (D)),

x ∈ Dh .

Here it is convenient to give a formulation of the maximum principle for the grid problem (2.21), (2.20) [108] (see also [214]). The difference equation (2.21) in the nodes x ∈ Dh can be written in the canonical form X Λ z(x) ≡ −A(x) z(x) + B(x, y) z(y) = f h (x), x ∈ Dh . (2.24a) y∈S(x)

Here A(x) and B(x, y) are coefficients of the discrete equation. The stencil of the finite difference scheme, i.e., the grid equation (2.24a), is the set of S nodes x S(x) on which the grid functions are involved in the grid equation written at the point x. The node x is the stencil centre and the set S(x) is a stencil neighborhood of the node x. Let the coefficients A(x) and B(x, y) of the difference operator Λ(2.21) on the grid (2.20) satisfy the conditions [108] A(x) > 0, B(x, y) > 0, C(x) = A(x) −

X

B(x, y) ≥ 0,

(2.24b)

y∈S(x)

x ∈ Dh , y ∈ S(x), ε ∈ (0, 1]. In this case, we say that the scheme (2.21), (2.20) (the operator Λ(2.21) ) is ε-uniformly positive.


28


In the case when the inequality z(x) ≥ 0, x ∈ Dh , for ε ∈ (0, 1], follows from the condition Λ z(x) ≤ 0,

x ∈ Dh ,

z(x) ≥ 0,

x ∈ Γh ,

we say that the finite difference scheme (2.21), (2.20) (the operator Λ(2.21) ) is ε-uniformly monotone. The condition (2.24b) implies the monotonicity of the finite difference scheme (2.21), (2.20) (the operator Λ(2.21) ), but the converse is false. We call this the monotonicity condition of the finite difference scheme (the operator). We say that the condition (2.24b) is the condition of strong ε-uniform monotonicity of the scheme (2.21), (2.20) (the operator Λ(2.21) ) if in (2.24b) one has C(x) > 0 for x ∈ Dh , and ε ∈ (0, 1]. The maximum principle for the discrete Dirichlet problem is stated in the following way. Lemma 2.3.1 Let the coefficients of the operator Λ satisfy the condition (2.24). Then from the condition Λ z(x) ≥ 0 x ∈ Dh , z(x) 6≡ const, x ∈ Dh , it follows that z(x) can not take its largest positive value on the nodes of Dh . The maximum principle is often applied in the following form of a comparison lemma. Lemma 2.3.2 Let the coefficients of the operator Λ satisfy the condition (2.24), and for the functions v(x) and w(x), x ∈ Dh , let the condition |Λ v(x)| ≤ −Λ w(x) x ∈ Dh ,

|v(x)| ≤ w(x), x ∈ Γh ,

be satisfied. Then |v(x)| ≤ w(x), x ∈ Dh . Here w(x), x ∈ Dh is a majorant (barrier) function for v(x), x ∈ Dh . The following convergence theorem holds. Theorem 2.3.1 Let the finite difference scheme (2.21), (2.20) approximate the boundary value problem (2.2), (2.19) ε-uniformly with the estimates (2.22), (2.23), and let this scheme be ε-uniformly stable. ¡ ¢ Then the difference scheme (2.21), (2.20) converges at the rate O λ(N (D)) ε-uniformly as N (D) → ∞: |u(x) − z(x)| ≤ M λ(N (D)),

x ∈ Dh .

In the hypotheses of Theorem 2.3.1, the ε-uniform monotonicity of the scheme (2.21), (2.20) is not assumed. The solution of the boundary value problem (2.2), (2.19) is ε-uniformly stable, and it satisfies the estimate (2.5).



29

The difference scheme (2.21), (2.20) is also ε-uniformly stable if the operator Λ is ε-uniformly monotone and approximates the operator L ε-uniformly (in the maximum norm) on the smooth functions: sup |(L − Λ) v(x)| ≤ M λ1 (N (D)).

(2.25a)

ε, Dh

Here λ1 (N (D)) − → 0 as N → ∞ and v(x) is a sufficiently smooth function ε satisfying the condition ¯ ¯ ¯ ¯ ∂k ¯ ¯ v(x) ¯ k1 ¯ ≤ M, x ∈ D, 0 ≤ k ≤ 3, k = k1 + . . . + kn . (2.25b) ¯ ∂x1 . . . ∂xknn ¯ Inequality (2.25) is established using the maximum principle, where one takes into account the strong L on the majorant func£ negativity of the operator ¤ tion v(x) = M max max |f h (x)|, max |ϕh (x)| , x ∈ D. Thus, the following statement is valid.

Dh

Γh

Lemma 2.3.3 Let the operator Λ(2.21) be ε-uniformly monotone and approximate the operator L ε-uniformly on sufficiently smooth functions satisfying the condition (2.25). Then the difference scheme (2.21), (2.20) is ε-uniformly stable: · ¸ h h |z(x)| ≤ M max max |f (x)|, max |ϕ (x)| , x ∈ Dh . Dh

Γh

A corollary of Theorem 2.3.1 and Lemma 2.3.3 is the following result. Theorem 2.3.2 The difference scheme (2.21), (2.20) converges ε-uniformly at the rate O(λ(N (D))) if it approximates the boundary value problem (2.2), (2.19) ε-uniformly with the estimates (2.22), (2.23) and the operator Λ is εuniformly monotone.

2.3.2

Sufficient conditions for ε-uniform approximation of the boundary value problem

We now discuss sufficient conditions for ε-uniform approximation of the boundary value problem (2.2) on the slab (2.4) by a difference scheme. We consider classical difference approximations on rectangular grids. Assume that the estimates of Theorem 2.2.1 hold for the solution of the boundary value problem (2.2). The simplest classical difference schemes are obtained by changing derivatives in the differential equations into appropriate difference derivatives. In certain difference schemes, the coefficients in the differential and the difference equations (2.2) and (2.21), in general, do not coincide, but this discrepancy tends to zero as the maximal mesh step-size tends to zero; furthermore,


30


the coefficients in the difference equations are independent of the solutions of the differential equations; such schemes are also classical. For boundary value problems, the construction of grid approximations including difference schemes can be found in, e.g., [79, 108]; for the approximation of mixed derivatives see also Subsections 2.4.1–2.4.3. Note that we do not consider schemes of fitted operator type. (0)

On the set D = D(2.4) we introduce the grid Dh = ω 1 × ω2 × . . . × ωn .

(2.26)

Here ω 1 , ω2 , . . ., ωn are meshes with an arbitrary distribution of nodes, n o ∗ 1 ω 1 = xi1 : d∗1 ≤ xi1 ≤ d∗1 , i = 0, 1, . . . , N1 , x01 = d∗1 , xN 1 = d1 , ωs = {xis : | xis | < ∞, i = 0, ±1, ±2, . . .},

s = 2, . . . , n.

Set h = maxs hs , where hs = maxi his with his = xi+1 − xis for s = 1, . . . , n. s Let N1 + 1 be the number of nodes in the mesh ω 1 and let Ns + 1 be the minimal number of nodes in the mesh ωs , s = 2, . . . , n, per unit length; N = mins Ns , s = 1, . . . , n. Assume that the condition h ≤ M N −1 is fulfilled. T T Set Dh = D Dh , Γh = Γ Dh . Write the operator L(2.2) as the sum of operators L(2.2) = L∗ + L∗∗ ,

(2.27a)

where ∂2 , ∂x21

L∗ ≡ ε2 a11 (x) ( L

∗∗

2

≡ε

n X s, k=1 s+k>2

(2.27b)

) n X ∂2 ∂ ask (x) + bs (x) − c(x) − c0 (x). ∂xs ∂xk s=1 ∂xs

(2.27c)

It is not difficult to show that in the class of functions v(x), x ∈ D, that satisfy the condition ¯ ¯ ¯ ¯ ∂k ε2 ¯¯ ks kr v(x)¯¯ ≤ M, x ∈ D, 0 ≤ k ≤ K, 0 ≤ k1 ≤ 2, ∂xs ∂xr where k = ks + kr , s, r = 1, . . . , n, K ≥ 3, the operator L∗∗ is approximated ε-uniformly by the classical finite difference operators Λ∗∗

(2.28)

on the grids Dh(2.24) with an arbitrary distribution of the nodes xis in the meshes ωs for s = 1, . . . , n: |(L∗∗ − Λ∗∗ ) v(x)| ≤ M λ2 (N ),


x ∈ Dh ,

(2.29)


31

λ2 (N ) − → 0 as N → ∞. The same operators Λ∗∗ , by virtue of the bounds of ε Theorem 2.2.1, approximate ε-uniformly the operator L∗∗ (2.27) on the solution of the boundary value problem: ∗∗ |(L∗∗ (2.27) − Λ ) u(x)| ≤ M λ2 (N ),

x ∈ Dh .

However, already the difference operator ε2 δx1 x1 c does not approximate ε-uniformly the operator ε2 ∂ 2 /∂x21 (on the problem solution) on the grid Dh(2.26) , where ω 1 is a mesh with an arbitrary distribution of nodes. Here δx1 x1 c is the second-order difference derivative on a nonuniform mesh [108]: ¡ i ¢ £ ¤ i−1 −1 i δx1 x1 δx1 u(xi ) − δx1 u(xi ) , c u(x ) = 2 h1 + h1 δx1 u(xi ) and δx1 u(xi ) are the first-order (forward and backward) difference derivatives ¡ ¢−1 £ ¤ δx1 u(xi ) = hi1 u(xi+1 ) − u(xi ) , ¡ ¢−1 £ ¤ δx1 u(xi ) = hi−1 u(xi ) − u(xi−1 ) , 1 where xi , xi−1 , xi+1 ∈ Dh , xi = (xi1 , x2 , . . . , xn ). Let us consider an error of an approximation of the operator L∗ by the operator Λ∗ ≡ ε2 ah11 (x) δx1 x1 (2.30) c on the problem solution. The local error of the approximation of the operator ε2 (∂ 2 /∂x21 ) by the operator ε2 δx1 x1 c on the problem solution is ¯µ 2 ¯ ¶ ¯ ∂ ¯ 2 ¯ i ¯ ε ¯ u(x )¯ ≡ ψ11 (xi ; u(·)) = ψ11 (xi ), xi ∈ Dh . − δx1 x1 (2.31) c ∂x21 This quantity is specified by local a priori estimates of derivatives of the problem solution and the stencil of the scheme: ¯ 3 ¯ ½ ¯ ∂ ¯ ¡ i ¢ i−1 i 2 ¯ ψ11 (x ) ≤ M ε min h1 + h1 u(ξ)¯¯ ; £ max ¤ 3 ¯ ∂x 1 ξ1 ∈ xi−1 , xi+1 1

1

ξs =xs , s=2,...,n

¯ 2 ¯¾ ¯ ∂ ¯ ¯ u(ξ)¯¯ , ¤ £ max 2 ¯ ∂x i−1 i+1

ξ 1 ∈ x1 , x 1 ξs =xs , s=2,...,n

1

xi = (xi1 , x2 , . . . , xn ) ∈ Dh .

Taking into account the representation (2.7) of the solution to the boundary value problem and also the estimates (2.11), (2.14) for its components, we obtain the following estimate of the value ψ11(2.31) (xi ): £ ¤ ψ11(2.31) (xi ) ≤ M β(xi ) + λ3 (N ) , (2.32)


32


where β(xi ) is the main term (up to a constant multiplier) in the estimate of the local error of the approximation ψ11(2.31) (xi ): £ ¤ −1 β(xi ) = β(xi1 ) = β(xi1 ; ε, ω 1 ) ≡ min η(xi1 ), (hi1 + hi−1 η(xi1 ) ; (2.33) 1 )ε ¡ ¢ ¡ ¢ ∗ −1 ∗ η(xi1 ) = exp − m ε−1 r(xi+1 r(xi−1 1 , {d∗1 , d1 }) + exp − m ε 1 , {d∗1 , d1 }) ; λ3 (N ) = N1−1 ;

xi = (xi1 , x2 , . . . , xn ) ∈ Dh ,

m < m0(2.12) .

Let the grid function ah11 (x), x ∈ Dh , from (2.30) approximate the function a11 (x) ε-uniformly: |a11 (x) − ah11 (x)| ≤ M λ4 (N ),

x ∈ Dh ,

(2.34)

where λ4 (N ) − → 0 as N → ∞. The products ε

ε2

∂2 u(x), x ∈ D, ∂x21

ε2 δx1 x1 c u(x), x ∈ D h

(we call them normalized derivatives) are bounded ε-uniformly (see the estimates (2.11), (2.14)). Hence for the error of the approximation of the operator L∗ by the operator Λ∗ on u(x), we obtain the estimate ¸ · |(L∗ − Λ∗ ) u(x)| ≤ M max β(xi1 ) + λ3 (N ) + λ4 (N ) , x ∈ Dh , xi1 ∈ω1

and for the error of the approximation of the operator L by the operator Λ, one has the estimate · ¸ 4 X i |(L − Λ) u(x)| ≤ M max β(x ) + λ (N ) , x ∈ Dh . n 1 i x1 ∈ω1

Here

n=2

Λ = Λ∗(2.30) + Λ∗∗ (2.28) .

(2.35)

According to the statement of Theorem 2.3.2, for the boundary value problem (2.2), (2.4) the construction of the ε-uniformly convergent difference scheme {(2.21), (2.35), (2.26)} in the case when the difference scheme is monotone amounts to the construction of a distribution of nodes in the mesh ω 1 (on the interval [d∗1 , d∗1 ]), under which the condition max β(xi1 ) − →0 i

ε

as N1 → ∞

(2.36)

holds. This guarantees the ε-uniform approximation of the operator ε2 ∂ 2 /∂x21 by the operator ε2 δx1 x1 c (in the case of the conditions (2.29), (2.34), one also has ε-uniform approximation of the operator L(2.2) by the operator Λ(2.35) ) on the solution of the boundary value problem.



33

Theorem 2.3.3 For the solution of the boundary value problem (2.2), (2.4), let the estimates (2.11), (2.14) hold for K = 3. Then the condition (2.36) is necessary for the ε-uniform approximation of the boundary value problem by the difference scheme {(2.21), (2.35), (2.26)} satisfying the conditions (2.29), (2.30), (2.32), (2.34). Theorem 2.3.4 For the solution of the boundary value problem (2.2), (2.4), let the estimates (2.11), (2.14) hold for K = 3. In order that the scheme {(2.21), (2.35), (2.26)} satisfying the conditions (2.29), (2.30), (2.32), (2.34) 4 ³ ´ X be ε-uniformly convergent at the rate O maxi β(xi1 ) + λn (N ) , it is suffin=2

cient for the operator Λ to be ε-uniformly monotone and for the condition (2.36) to hold. There exist meshes ω 1∗ = ω 1∗ (d∗1 , d∗1 ),

(2.37)

that ensure the fulfillment of the condition (2.36) (and the ε-uniform convergence of the scheme {(2.21), (2.35), (2.26)}. For their construction, one can use, e.g., the technique of [15]; see also the following subsections.

2.3.3

Necessary conditions for distribution of mesh points for ε-uniform convergence of difference schemes. Construction of condensing meshes

For classical difference schemes, we now impose conditions on the distribution of grid nodes that are necessary for the ε-uniform approximation of the operator L(2.2) by the operator Λ(2.35) on the problem solution and that are sufficient for the ε-uniform convergence of the difference schemes. We construct piecewise-uniform meshes along the normal to the domain boundary that ensure the ε-uniform approximation of the operator L by the operator Λ. Note that, by virtue of the estimate (2.14), the k-th derivative along the normal ¡ ¢to the boundary in an M ε-neighborhood of the boundary is of order O ε−k . The following theorem defines a condition on the distribution of grid nodes in a boundary layer that is necessary for the ε-uniform approximation of the boundary value problem by the difference scheme. Theorem 2.3.5 Let a difference scheme be constructed by a classical approximation of the boundary value problem (2.2), (2.4) on the grid Dh(2.26) . Then the condition  ε−1 max hi−1  1 i d∗1 <x1 ≤d∗1 +M ε − → 0 as N1 → ∞, xi1 ∈ ω1 , (2.38) i ε ε−1 ∗ max h  1 i ∗ d1 −M ε≤x1 0, ρ∗s ≤ ρ∗s , ρ∗s and ρ∗s depend on ask for s, k = 1, . . . , n, such that for any hs satisfying the condition hs ∈ [ρ∗s h∗ , ρ∗s h∗ ],

s = 1, . . . , n,

(2.54a)

where h∗ > 0 is an arbitrary number, the inequality akk h−1 k −

n X

|ask | h−1 s > 0,

k = 1, . . . , n,

(2.54b)

s=1 s6=k

holds. This is equivalent to the inequality n X

|ask | ξs ξk < akk ξk2 ,

k = 1, . . . , n, ξk = h−1 k(2.54a) .

s=1 s6=k

At the point x0 where the operator L∗(2) is canonical, such inequality is valid for all nonnegative values ξs and also in a small neighborhood of x0 in the case when the values ξs ξk−1 are bounded. In the case of the condition (2.54a), one has ρ∗s ρ∗k −1 ≤ ξs ξk−1 ≤ ρ∗s ρ−1 ∗k .


42

Elliptic reaction-diffusion equations Recall that ³ ´ ρ∗s = ρ∗s L∗(2) ,

³ ´ ρ∗s = ρ∗s L∗(2) ,

s = 1, . . . , n.

(2.54c)

The values ρ∗s and ρ∗s can be chosen to satisfy the relations ρ∗s → 0,

ρ∗s → ∞

as

max |ask | → 0;

s, k, s6=k

the values ρ∗s and ρ∗s , as well as the coefficients ask , depend in general on x, i.e., ρ∗s = ρ∗s (x), ρ∗s = ρ∗s (x), and ask = ask (x). We say that the operator L∗(2) at the point x has almost canonical form in the variables x1 , . . . , xn with the condition of almost canonicity (2.54) (or, briefly, has almost canonical form) if its coefficients satisfy the relations (2.54). In the relations (2.54), the values ρ∗s , ρ∗s , hs = ξs−1 for s = 1, . . . , n and h∗ appear. The value hs in the definition of almost canonicity of the operator L∗(2) in (2.54a), (2.54b) and the value hs in the definition of the uniform grid (2.52) are, in general, not related to each other. Note that the fulfillment of the condition (2.54b) at the point x guarantees the monotonicity of the operator Λ∗(2)(2.51) at this point in the case of the uniform grid Dh(2.52) , where hs = hs(2.54a) . Thus, in the case of the difference scheme {(2.47), (2.46), (2.52)} the condition (2.54) is the condition of strong ε-uniform monotonicity of the operators Λ∗(2)(2.51) , Λ(2.46) and the scheme at the point x itself. In general, there exist no values hs for s = 1, . . . , n, satisfying the condition (2.54a) such that the inequality (2.54b) is valid for all x in D. But if it turns out that the relation max ρ∗s (x) < min ρ∗s (x), D

s = 1, . . . , n,

D

holds, then it is possible to find values hs under which the inequality (2.54b) is valid for all x in D. We now generalize the relations (2.54). We shall assume that for an arbitrary point x∗ ∈ D, one can find (i) a sufficiently small neighborhood Q(x∗ ) (we shall say, a neighborhood of the point x∗ ) with x∗ ∈ Q(x∗ ), Q(x∗ ) ⊆ D and (ii) functions ρ∗s (x) and ρ∗s (x), x ∈ Q(x∗ ) such that the relation max ρ∗s (x) ≤ min ρ∗s (x),

Q(x∗ )

Q(x∗ )

s = 1, . . . , n,

(2.55a)

holds and also the following relation (similar to (2.54b)): min akk (x)h−1 k −

Q(x∗ )


n X s=1 s6=k

max |ask (x)| h−1 s > 0,

Q(x∗ )

k = 1, . . . , n.

(2.55b)

Boundary value problems in smooth domains Here the values hs satisfy the condition · ¸ ∗ ∗ ∗ hs ∈ max ρ∗s (x) h , min ρs (x) h , Q(x∗ )

Q(x∗ )

43

s = 1, . . . , n,

(2.55c)

and h∗ > 0 is an arbitrary number. We emphasize that the functions ρ∗s (x) and ρ∗s (x) are defined by the coefficients of the operator L∗(2)(2.50) : ¡ ¢ ρ∗s (x) = ρ∗s x; L∗(2) ,

¡ ¢ ρ∗s (x) = ρ∗s x; L∗(2) ,

s = 1, . . . , n.

(2.55d)

The values ρ∗s (x) and ρ∗s (x) can be chosen to satisfy the relations max ρ∗s (x) → 0,

Q(x∗ )

min ρ∗s (x) → ∞

Q(x∗ )

for

max

s,k,s6=k,Q(x∗ )

|ask (x)| → 0. (2.55e)

In the general case, one can ensure fulfillment of the relations (2.55) if one brings the operator L∗(2)(2.50) to the canonical form at the point x∗ (or close to the canonical form) and chooses the neighborhood Q(x∗ ) to be sufficiently small. We say that the operator L∗(2) on the set D has local almost canonical form (in the variables x1 , . . . , xn ), if for an arbitrary point x∗ ∈ D, one can find a neighborhood Q(x∗ ) and n pairs of the functions ρ∗s (x) and ρ∗s (x), x ∈ Q(x∗ ), such that the relations (2.55) hold for the coefficients of the operator L∗(2) . 1

1

If on a set D ⊆ D, the relations (2.55), where Q(x∗ ) = D , hold for the coefficients of the operator L∗(2) , we say that the operator L∗(2) has al1

most canonical form (or, briefly, is almost canonical) on the set D . In the case when the coefficients of the operator L∗(2) allow us to construct the functions ρ∗s (x), and ρ∗s (x) for s = 1, . . . , n, such that the relations (2.55), where 1 Q(x∗ ) = D , are valid, we say that the operator L∗(2) permits the transforma1

tion to the almost canonical form on the set D . Note that when defining the local almost canonical operator, in the relations (2.55) we use the values hs for s = 1, . . . , n, which, in general, have no relations to grids on the set Q(x∗ ); grids on the set Q(x∗ ) will be considered below. T For the set Q Dh and the operator Λ we consider the node x0 as strongly interior, or, briefly, as interior, if all nodes of the stencil (for which the point T x0 is centre) of the operator Λ belong to the set Q Dh . The fulfillment of the condition (2.55b) under the condition (2.55c) ensures the monotonicity of the operator Λ∗(2)(2.51) on the interior nodes of the uniform grid Dh(2.53) (Q(x∗ )). In the case of the difference scheme {(2.47), (2.46), (2.52)} the condition (2.55) is the condition of strong ε-uniform monotonicity of the operators Λ∗(2) , Λ and of the difference scheme on the interior nodes of the set Dh(2.53) (Q(x∗ )).


44


We say that the grids Dh , with a special distribution of nodes in the meshes ωr , . . . , ωk that ensure the ε-uniform monotonicity of the operator Λ on the T interior nodes of the set Q Dh , are consistent grids on the set Q in the variables xr , . . . , xk with the monotonicity condition for the operator Λ; restrictions on the meshes ωs in the other variables are not imposed. For brevity, we shall call such meshes the consistent meshes (without indicating the variables) T guaranteeing the monotonicity of the operator Λ on the set Q Dh . Theorem 2.4.2 Let the grid Dh be uniform on the set Q(x∗ ) in each of the variables, i.e., ¢ ¡ ¢ u¡ Dh = Dh Q(x∗ ), L∗(2) = Dh∗ Q(x∗ ), L∗(2) , x1 , . . . , xn = (2.56) n© o ª = Q(x∗ ) h(2.53) under the condition (2.55c) while assuming (2.55) . ¡ ¢ Then the operators Λ(2.46) (L) and Λ∗(2)(2.51) L∗(2) are ε-uniformly monotone on the set Dh(2.56) (in the case when the set of interior nodes is not empty). Thus, the uniform grid Dh(2.56) is consistent on Q(x∗ ) in x1 , . . . , xn with respect to the monotonicity condition for the operators Λ∗(2)(2.51) and Λ(2.46) . The lower index h∗ in the notation of the grid Dh∗ is used to show that the distribution of nodes in the grid satisfies a condition (the condition (2.55c)) controlled by the parameter h∗ . In the list of “arguments” of the grid Dh in (2.56), the variables x1 , . . . , xn in which the grid is consistent are written. The conditions (2.55) that are imposed on the coefficients ask (x) and on the relations between the values h1 , . . . , hn allow us to construct uniform consistent grids on the set Q(x∗ ). The difference scheme (2.47), (2.46) on the uniform grid ¡ ¢ Dh(2.56) = Dh∗ (2.56) Q(x∗ ), L∗(2) with Q(x∗ ) = D is monotone, but this scheme does not satisfy the condition (2.36) (sufficient for ¡ ε-uniform convergence of a difference scheme) nor the condition (2.40b) necessary for ε-uniform convergence of a difference scheme on the family of ¢ the grids Dh(2.41) when the grid Dh(2.56) ⊂ Dh(2.41) .

2.4.2

Problems on piecewise-uniform meshes

We are interested in monotone difference approximations of the boundary value problem (2.2), (2.4). The following statements are valid. Lemma 2.4.2 For the mesh ω 1 that defines the grid Dh(2.26) , let the condition (2.38) be violated. Then the operator Λ(2.46) (L) does not approximate the operator L(2.2) ε-uniformly on the problem solution.



45

This lemma follows from Theorem 2.3.5. The condition (2.38) is necessary for the ε-uniform approximation of the operator L by the operator Λ(2.46) on the problem solution. Theorem 2.4.3 Suppose that the coefficients a1s (x) for s = 2, . . . , n, where a1s (x) = as1 (x), do not satisfy the condition a1s (x) = 0,

x ∈ Γ,

s = 2, . . . , n.

(2.57)

Then the difference scheme (2.47), (2.46) on the grid Dh(2.26) under the condition (2.38) is not ε-uniformly monotone for any distribution of nodes in the meshes ωs , for s = 1, . . . , n. In particular, one does not obtain ε-uniform monotonicity on the grid Dh = Dh (ω 1(2.48) ) = ω 1 × ω2 × . . . × ωn ,

(2.58)

where ω 1 = ω 1(2.48) is the piecewise-uniform mesh; and ωs , for s = 2, . . . , n, are uniform meshes (with step-size hs ); h1 = max hi1 . i

The condition (2.57) is necessary for the ε-uniform monotonicity of the operator Λ(2.46) on the grid Dh(2.58) satisfying the condition (2.38). Thus, the condition (2.57) is the necessary condition for ε-uniform monotonicity of the operator Λ(2.46) under the condition that the difference scheme approximates the boundary value problem. We shall consider the difference scheme (2.47), (2.46) on the grid (2.58). The case a1s (x) ≡ 0 for x ∈ D We consider the boundary value problem in the case when the equation (2.2a) does not contain mixed derivatives involving differentiation in x1 , i.e., under the condition a1s (x) ≡ 0,

x ∈ D,

s = 2, . . . , n.

(2.59)

In the case of the condition ask (x) = 0, where

1

D ⊆ D,

1

x∈D ,

s = k + 1, . . . , n,

1 ≤ q ≤ n − 1,

k = 1, . . . , q,

s, k = 1, . . . , n, 1

s 6= k,

we say that the operator L∗(2)(2.50) on the set D is canonical in the variables x1 , . . . , xq for q < n − 1 and canonical in the variables x1 , . . . , xn for q = n − 1. Note that ask (x) = aks (x), x ∈ D for s, k = 1, . . . , n. In a similar way, one can define the canonical form of the operator L∗(2) in the variables xq , . . . , xn , where 1 ≤ q ≤ n.


46


In the case of the conditions (2.57) and (2.59), the operator L∗(2)(2.50) has canonical form in the variable x1 on Γ and on D, respectively. We say that the condition (2.57) (the condition (2.59)) is the canonicity condition in the variable x1 of the elliptic part of the differential equation (of the elliptic part of the operator L) on the boundary Γ (respectively, on the set D), or, briefly, the canonicity condition of the boundary value problem in x1 on the boundary Γ (respectively, on the set D). On the set Q ⊆ D we introduce the grid T

Dh (Q) = {Q }h = Q Dh(2.58)

(2.60)

that is piecewise-uniform in x1 and give conditions that guarantee the ε-uniform monotonicity of the operator Λ(2.46) on this grid. In the case of the condition (2.59), using an appropriate linear transformation of the variables x2 , . . . , xn (the transformation X = X(x), X1 = x1 , x ∈ D), the operator L∗(2)(2.50) can be brought to canonical form at the point X0 ∈ DX , X0 = X(x0 ). In the canonical form, the coefficients of the mixed derivatives not involving differentiation in X1 (here X1 = x1 ) are sufficiently small in a small neighborhood of the point X0 . In the new variables, the coefficients of the mixed derivatives involving X1 are equal to zero on the whole domain. Next, we shall assume that the coefficients of the operator L∗(2)(2.50) already satisfy both the condition (2.59) and the following condition: For an arbitrary point x∗ ∈ D, one can find a sufficiently small neighborhood Q(x∗ ), where Q(x∗ ) ⊆ D, and n−1 pairs of functions ρ∗s (x) and ρ∗s (x), for s = 2, . . . , n, such that the following relations are fulfilled: max ρ∗s (x) ≤ min ρ∗s (x),

Q(x∗ )

Q(x∗ )

min akk (x)h−1 k −

Q(x∗ )

where hs ∈

n X s=2 s6=k

h

s = 2, . . . , n;

max |aks (x)| h−1 s > 0,

Q(x∗ )

i max ρ∗s (x) h∗ , min ρ∗s (x) h∗ ,

Q(x∗ )

(2.61a)

Q(x∗ )

k = 2, . . . , n,

s = 2, . . . , n,

(2.61b)

(2.61c)

h∗ > 0 is an arbitrary number. In the case of the conditions (2.59) and (2.61), we say that the operator L∗(2) ¡ ¢ on the set D is canonical in x1 ¡because the condition (2.59) holds and locally almost canonical in x2 , . . . , xn because, besides (2.59), for an arbitrary point x∗ ∈ D, one can find a neighborhood Q(x∗ ) and functions ρ∗s (x) and ρ∗s (x),



47

¢ for s = 2, . . . , n, such that the relations (2.61) hold . Note that the conditions (2.61) involve only the coefficients ask (x), for s, k = 2, . . . , n, of the truncated operator ¢ ∗[1] ∗[1] ¡ L(2) = L(2) L∗(2) , (2.62) i.e., the operator involving differentiation in the variables x2 , . . . , xn ; thus, the ∗[1] truncated operator L(2) is local almost canonical. In the case of the condition (2.59), the locally almost canonicity of the ∗[1] operator L(2) allows us to construct piecewise-uniform meshes in x1 on which the operator Λ is monotone. ¡ ¢ Theorem 2.4.4 Consider the grid Dh constructed on the set Q(x∗ ) , which is piecewise-uniform in x1 and uniform in x2 , . . . , xn : ³ ´ Dh = Dh∗ Q(x∗ ), L∗(2) =

(2.63)

³ ´ = Dh∗ Q(x∗ ), L∗(2) , ω 1 = ω 1(2.48) , ωs = ωsu , s = 2, . . . , n, x2 , . . . , xn = n o = {Q(x∗ )}h(2.60) under the condition (2.61c) subject to (2.61) . In the case of the conditions (2.59), (2.61), the operator Λ(2.46) = Λ(2.46) (L) is ε-uniformly monotone. ¡ ¢ Thus, the grid Dh = Dh∗ (2.63) Q(x∗ ), L∗(2) is consistent in the variables ¡ ¢ ¡ ¢ x2 , . . . , xn on Q(x∗ ) ⊆ D for the operators Λ∗(2)(2.51) L∗(2) and Λ(2.46) L under the condition (2.59). The case a1s (x) = 0 for x ∈ Γ We consider the boundary value problem in the case when the coefficients a1s (x) vanish on the boundary Γ (i.e., the condition (2.57) holds) and, furthermore, for these coefficients one has a1s ∈ C 1 (D),

s = 2, . . . , n.

(2.64)

We construct a grid that is consistent in x1 , . . . , xn , uniform in x2 , . . . , xn , and condenses in the boundary layer in x1 . In the case of condition (2.57), it is appropriate to use the piecewise-uniform meshes (2.48). We shall assume that the coefficients of the operator L∗(2)(2.50) , as well as ¡ ¢ the condition (2.57) i.e., the operator L∗(2) on the set Γ is canonical in x1 , satisfy the following condition. For an arbitrary point x∗ in D, one can find sufficiently small values m∗ and ν > 0, a neighborhood Q(x∗ ) and n pairs of


48


functions ρ∗s (x) and ρ∗s (x), for s = 1, . . . , n, such that the relations (2.55)¡ are valid for the coefficients ask (x), for s, k = 1, . . . , n, of the operator L∗(2) i.e., ¢ the operator L∗(2) is locally almost canonical in x1 , . . . , xn on the set D ; the values m∗ and ν > 0 do not appear in the relations (2.55). Furthermore, in a neighborhood of Γ, the coefficients ask (x), for s, k = 2, . . . , n, of the operator ∗[1] L(2)(2.62) satisfy the relations (2.55), where s, k = 2, . . . , n, as well as the ¡ ¢ additional relation similar to (2.61b) h

n i X akk (x) − ν h−1 − k

min

Q(x∗ )

max

Q(x∗ )

|ask (x)| h−1 s > 0,

(2.65)

s=2 r(x,Γ)≤m∗ s6=k

r(x,Γ)≤m∗

k = 2, . . . , n. In the case when the relations (2.55) and (2.65) are both fulfilled with the same “arguments”, we shall use the notation {(2.55), (2.65)}. The fulfillment of {(2.55), (2.65)} in the case of the condition (2.57) can be ensured by a linear transformation of the variables x2 , . . . , xn that brings the operator L∗(2) to the canonical form at the point x∗ ∈ Γ (or to a form close to canonical), and by choosing the neighborhood Q(x∗ ) and the values m∗ and ν to be sufficiently small. Note that the relations (2.65), in general, follow from the relation (2.55b). Because the values h2 , . . . , hn are commensurable, one can subtract from the left part of the inequality (2.55b), without violating it, the value ν h−1 k , choosing the value ν and the neighborhood Q(x∗ ) sufficiently small. This way of obtaining the relations (2.65) is less preferable because the smallness of the values ν and m∗ leads to strong restrictions imposed on the parameters of the piecewise-uniform mesh ω 1 constructed later (see the condition (2.67) below). In the case when the relations {(2.55), (2.65)} hold for the coefficients of ∗[1] ∗[1] the operator L(2) for s, k ≥ 2, we say that the operator L(2) is strongly locally almost canonical in x2 , . . . , xn in a neighborhood of the set Γ. 1

If on a set D ⊆ D the relations {(2.55), (2.65)} hold for the coefficients 1 ∗[1] of the operator L(2)(2.62) for s, k = 2, . . . , n and Q(x∗ ) = D , we say that the ∗[1]

1

operator L(2) is strongly almost canonical in x2 , . . . , xn on the subset D in a neighborhood of the set Γ. In an m∗ -neighborhood of Γ, the condition (2.65) is more restrictive than the condition (2.61b); note that the conditions (2.55a), (2.55c) for s, k = 2, . . . , n and the conditions (2.61a), (2.61c) are similar. The condition (2.57) and the condition the coefficients ask (x) of the operator L∗(2) on the set Q(x∗ ) ⊆ D satisfy the relations {(2.55), (2.65)}


(2.66)


49

together allow us to construct on the set Q(x∗ ) piecewise-uniform (in the x1 axis) consistent grids and thereby ensure the monotonicity of the operators Λ∗(2)(2.51) and Λ(2.46) . Let us construct such a mesh. Let the conditions (2.57) and (2.66) hold. We now impose conditions on the parameters of the grid (2.60) under which the operator Λ∗(2)(2.51) is monotone. We consider the operator Λ∗(2)(2.51) on the grid Dh(2.58) that is uniform in xs , for s = 2, . . . , n, and piecewise-uniform in x1 . Let the step-sizes hs in the meshes ωs , for s = 2, . . . , n, and the step-size h1 (i.e., the maximal step-size in the mesh ω 1(2.48) ) satisfy the condition (2.55c). Note that h1 = h(2) in the mesh (2.48). Let the value σ = σ(2.48) satisfy the condition σ ≤ m∗(2.65) ,

(2.67a)

and let the condition (2.64) hold. By virtue of the conditions (2.57), (2.64), one has |a1s (x)| ≤ M1 σ for r(x, Γ) ≤ σ, where M1 = max |(∂/∂x1 ) a1s (x)|. s,s6=1,D

−1 Therefore |a1k (x)| h−1 (1) ≤ M1 h(1) σ, for k 6= 1. The inequality

ν h−1 k −

max

Q(x∗ ) x1 ∈(d / ∗1 +σ, d∗ 1 −σ)

−1 holds if M1 h(1) σ 0,

k = 2, . . . , n

min [h−1 k ], i.e., under the condition

k=2,...,n

−1 N∗1 ≡ σ h−1 (1) < M1 ν

min [h−1 k ],

k=2,...,n

N1 ≤ M N∗1 ,

(2.67b)

where N∗1 + 1 is the number of nodes in the mesh ω 1(2.48) in the interval [d∗1 , d∗1 + σ] or on [d∗1 − σ, d∗1 ] . Under the above conditions on the values σ and N∗1 , the operators Λ∗(2)(2.51) and Λ(2.46) are ε-uniformly monotone on the grid Dh(2.60) (Q), where Q is Q(x∗ ) from {(2.55), (2.65)}. Thus in the case of (2.57), (2.66), (2.64), the condition (2.67) is a sufficient condition for the local ε-uniform monotonicity of the operators Λ∗(2) and Λ on the piecewise-uniform mesh Dh(2.60) (Q). In the case of the condition (2.66), when constructing consistent meshes based on the grid n T Dh = Dh∗ (Q) = Dh∗ (Q(x∗ )) = Dh(2.58) Q(x∗ ), o under the condition (2.55c) subject to {(2.55), (2.65)} ,

(2.68)

we use the meshes n o ω 1 = ω 1(2.48) under the condition (2.67) .


(2.69)

50


Under the conditions (2.57), (2.66), (2.64), (2.67) the grid ³ ´ ³ ´ Dh = Dh∗ Q(x∗ ), L∗(2) = Dh∗ Q(x∗ ), L∗(2) , ω 1(2.69) , x1 , . . . , xn = © ª = Dh∗ (2.68) , ω 1 = ω 1(2.69) , ωs = ωsu , s = 2, . . . , n = (2.70) n T ∗ = Dh(2.58) Q(x ), ω 1 = ω 1(2.48) subject to (2.55c) and (2.67) o in the case of the relations (2.57), {(2.55), (2.65)} is consistent on the set Q(x∗ ) in the variables x1 , . . . , xn with the monotonicity condition for the operators Λ∗(2)(2.51) and Λ(2.46) . Thus, when constructing consistent meshes, the passage from the uniform grids Dh(2.56) to the piecewise-uniform grids Dh(2.60) requires us to impose ¡ besides the condition (2.55) for the coefficients ask (x) and the parameters hs ¢ of the meshes ω s additional conditions (2.65) and (2.67) on the coefficients ask (x) and on the mesh ω 1(2.48) . Note that the grid Dh(2.70) (the grid on the set Q(x∗ )) is generated by the meshes ω 1(2.58) defined on [d∗1 , d∗1 ], and by the meshes ωs defined on the xs -axis for s = 2, . . . , n. However, any change of the distribution of nodes in the meshes ω 1 , or ω2 , . . ., ωn outside the projection of the set Q(x∗ ) on the xs -axis ¡ does not¢ affect the distribution of nodes in the grid Dh(2.70) = Dh(2.70) Q(x∗ ), L∗(2) . Theorem 2.4.5 Under the conditions (2.57), (2.66), (2.64), (2.67)¢the opera¡ tor Λ(2.46) = Λ(2.46) (L) on the grid Dh = Dh∗ (2.70) Q(x∗ ), L∗(2) is ε-uniformly monotone. Remark 2.4.1 Let the coefficients of the mixed derivatives satisfy the condition £ ¤ max |ask (x)| < (n − 1)−1 min αs αk−1 ass (x), αk αs−1 akk (x) , (2.71) D

D

s, k = 1, . . . , n,

s 6= k,

where α1 , . . . , αn are some positive numbers, i.e., we have dominance of the diagonal elements in the matrix of the coefficients of the operator L∗(2)(2.50) on the whole set D. Then one can choose the value ν and the functions ρ∗s (x) and ρ∗s (x), for s = 1, . . . , n, such that the condition (2.66) is valid for Q(x∗ ) = D.

(2.72)

The fulfillment of the condition (2.71) ensures existence of a transformation of the operator L∗(2) to the strongly almost canonical form (in the variables



51

x1 , . . . , xn ) on the set D. Indeed, under the condition (2.71), one can choose the value ν to be sufficiently small such that the same inequality (2.71) will be satisfied if the coefficients app (x) in its right-hand side are changed to app (x) − ν, for p = s, k. Setting ρ∗s (x) = ρ∗s (x) = ρs = α1 αs−1 , for s = 1, . . . , n, we come to the condition (2.55) and also to the condition (similar to (2.65)) £

n X ¤ min akk (x) − ν h−1 − max |ask (x)| h−1 s > 0, k D

s=1 s6=k

k = 1, . . . , n.

D

In the case of the condition (2.71), under the additional condition (2.57), ∗ ¡using the grid Dh(2.58) , where ω 1 = ω 1(2.69) for ¢ hs = ρs h , with s = 1, . . . , n for appropriate m1(2.40) , ν, N∗1 = N∗1 (N1 ) , one can satisfy the conditions (2.66), (2.67) that ensure ε-uniform monotonicity of the operator Λ(2.46) for Q(x∗ ) = D and any h∗ . In the case when the coefficients ask (x), for s, k > 1, satisfy the condition £ ¤ max |ask (x)| < (n − 2)−1 min αs αk−1 ass (x), αk αs−1 akk (x) , (2.73) D

D

s, k = 2, . . . , n,

s 6= k,

where α2 , . . . , αn are some positive numbers (i.e., we have dominance of the ∗[1] diagonal elements in the matrix of the coefficients of the operator L(2)(2.62) ), under the condition ρ∗s (x) = ρ∗s (x) = ρs = α2 αs−1 , for s = 2, . . . , n, the relations (2.61) hold for Q(x∗ ) = D. (2.74) ∗[1]

Conditions (2.73) guarantee existence of a transformation of the operator L(2) to almost canonical form (in the variables x2 , . . . , xn ) on the set D. In the case of conditions (2.59) and (2.73) on the grid Dh(2.63) , which is uniform in x2 , . . . , xn for hs = ρs h∗ , with s = 2, . . . , n, condition (2.61b) holds and ensures the ε-uniform monotonicity of the operator Λ(2.46) for Q(x∗ ) = D and any h∗ specifying the meshes ω s , for s = 2, . . . , n; here ω 1 = ω 1(2.48) .

2.4.3

Consistent grids on subdomains

For the construction of difference schemes of the domain decomposition method on overlapping subdomains, in particular, for boundary value problems in domains with curvilinear boundaries, we need piecewise-uniform consistent meshes that condense in a neighborhood of only one of the side-faces adjoining to the boundary. Let us discuss the construction of such consistent meshes. ¡ ¢ ¡ ¢ In the mesh ω 1(2.69) = ω 1(2.69) ω 1(2.48) from Dh(2.70) Q(x∗ ), L∗(2) , the ¡ ¢ use of the uniform mesh ω 1u d∗1 , d∗1 whose step-size h1 equals the step-size


52


h(2) in the mesh ω 1(2.48) instead of the piecewise-uniform mesh ω 1(2.48) = ¡ ¢ ω 1(2.48) d∗1 , d∗1 leads to the uniform grid ³ ´ ³ ´ u Dh = Dh Q(x∗ ), L∗(2) = Dh(2.70) Q(x∗ ), L∗(2) , ω 1(2.69) (ω 1u ) = (2.75a) ³ = Dh(2.70) Q(x∗ ), L∗(2) , ω 1 = ω 1u under the condition (2.55c) ´ in the case of the relations {(2.55), (2.65)} , which is consistent (in the¡ variables x¢1 , . . . , xn ) on the set Q(x∗ ). The consistency of the grid Dh(2.70) Q(x∗ ), L∗(2) on Q(x∗ ) is preserving also in the case when instead of the mesh£ ω 1(2.48) ¤ that condenses in neighborhoods of both endpoints of the interval d∗1 , d∗1 , we use the piecewise-uniform mesh either ¡ ¢ ¡ ¢ ω 1l d∗1 , d∗1 or ω 1r d∗1 , d∗1 that condenses in ¤a neighborhood either of the left £ or the right endpoint of the interval d∗1 , d∗1 , i.e., in the case of the grids ³ ´ ³ ´ l Dh = Dh Q(x∗ ), L∗(2) = Dh(2.70) Q(x∗ ), L∗(2) , ω 1(2.69) (ω 1l ) , ³ r

´

Dh = Dh Q(x∗ ), L∗(2) = Dh(2.70) l

³

´ Q(x∗ ), L∗(2) , ω 1(2.69) (ω 1r ) .

(2.75b) (2.75c)

r

For the consistency of the grids Dh and Dh on Q(x∗ ),¡ it is required that ¢ ¡ the pa¢ l r ∗ ∗ rameters σ, h(1) , and h(2) specifying the meshes ω d , and d ω d , d ∗1 ∗1 1 1 1 1 ¡ ¢ would be the same as in the mesh ω 1(2.48) d∗1 , d∗1 . ¡ ¢ u Note that the consistent uniform grid Dh(2.75a) Q(x∗ ), L∗(2) is, in general, ¡ ¢ different from the consistent uniform grid Dh(2.56) Q (x∗ ), L∗(2) . Produced requirements to the coefficients ask (x) and generating meshes ω 1 and ωs , for u s = 2, . . . , n, in the case of the grid Dh(2.75a) , turn out to be more restricted compared with the grid Dh(2.56) . We now give sufficiently natural conditions imposed on the coefficients ask (x) and the meshes ω 1 , ω2 , . . ., ωn that guarantee the consistency of the grids Dh constructed based on the elementary meshes in the x1 -axis, i.e., ω 1u , ω 1l , ω 1r , or ω 1b , where ω 1b £is a mesh ¤ that condenses in neighborhoods of both endpoints of the interval d∗1 , d∗1 . Let S Γ = j Γj , j = 1, 2, (2.76) where Γ1 and Γ2 are the left and right parts of the boundary Γ. Let the piecewise-uniform (in x1 ) grid Dh condense in a neighborhood either of the side Γ1 and Γ2 , or the grid Dh is uniform. We denote by Γe those parts of the boundary Γ in whose neighborhoods it is required to condense the mesh ω 1 . Let J ∗ J ∗ = J ∗ (D) = J ∗ (D; Γe ) (2.77)



53

be the set of indexes j related to Γj in whose neighborhood the mesh ω 1 condenses; here Γj ⊆ Γe . For example, J ∗ = ∅ in the case of the uniform mesh ω 1 ; but if the mesh ω 1 is ω l1 then J ∗ = {j = 1}; here J ∗ (D; Γe = ∅) = ∅. Thus, the set of the indexes J ∗ is defined by the set Γe of the parts of the boundary Γ. Suppose that on the boundary Γ the coefficients ask (x) of the operator L∗(2) satisfy the condition ask (x) = 0, x ∈ Γj , j ∈ J ∗ , s = 1 or k = 1, s 6= k, if J ∗ 6= ∅;

(2.78)

restrictions on ask (x) on the set Γ are not imposed if J ∗ = ∅, i.e., the operator L∗(2) is canonical on the side Γj for j ∈ J ∗ , where J ∗ 6= ∅. Assume that on the set D the coefficients ask (x) satisfy the conditions (2.55). In the case when J ∗ 6= ∅, we assume that the following additional condition similar to (2.65) holds: h min Q(x∗ ) r(x,Γj )≤m∗ j∈J ∗


max

Q(x∗ ) s=2 ∗ s6=k r(x,Γj )≤m j∈J ∗

k = 2, . . . , n,

|ask (x)| h−1 s > 0,

(2.79)

J ∗ 6= ∅.

In the case of condition (2.78) together with the relations {(2.55), (2.79)}, the operator L∗(2) on the set D has local almost canonical form in x1 , . . . , xn ∗[1]

and the operator L(2)(2.62) has strongly local almost canonical form in x2 , . . ., xn in a neighborhood of the set Γj , where j ∈ J ∗ . The condition the coefficients ask (x) of the operator L∗(2) on the set D

(2.80)

satisfy the relations (2.78), {(2.55), (2.79)} if J ∗ 6= ∅, ∗ and the relations (2.55) if J ∗ = ∅, where Q(x∗ ) ⊆ D, J ∗ = J(2.77) (D),

allows us on the set Q(x∗ ) to construct piecewise-uniform (controlled by the set J ∗ (D)) consistent grids that ensure the monotonicity of the operators Λ∗(2)(2.51) and Λ(2.46) . We now construct such meshes. For J ∗ = ∅ the uniform grid ¡ Dh = Dh(2.81) Q(x∗ ), L∗(2) , ω 1 = ω 1u , ωs = ωsu , s = 2, . . . , n ¢ (2.81) under the condition (2.55c) in the case of the relations (2.55) ¡ ¡ ¢¢ this is the grid Dh = Dh(2.56) Q(x∗ ), L∗(2) is consistent on the set Q(x∗ ). Let the condition (2.80) hold. In the case of the condition J ∗ 6= ∅ (i.e., when the mesh ω 1 is piecewise uniform), we assume that the step-sizes h1 = h(2)1


54


and hs in the meshes ω 1 and ωs , for s = 2, . . . , n, satisfy the condition (2.55c) (on the set Q(x∗ )) and, moreover, for the mesh ω 1 , one has the condition (similar to (2.67a)): σ = σ l ≤ m∗(2.79) , for j = 1, j ∈ J ∗ , σ = σ r ≤ m∗(2.79) , for j = 2, j ∈ J ∗ ;

(2.82a)

and also the following condition (similar to (2.67b)): −1 N∗1 ≡ σ h−1 (1) < M1 ν

∗ min [h−1 k ], j ∈ J ,

k=2,...,n

(2.82b)

where N∗1 + 1 is the number of nodes on the interval [d∗1 , d∗1 + σ] for {j = 1} ∈ J ∗ and on the interval [d∗1¡− σ, d∗1 ] for {j = 2} ∈ J ∗ ; M1 = M1(2.67) . In the case when J ∗ = {j = 1, 2} i.e., the mesh ω 1 condenses in neighborhoods ¢ of both endpoints of the interval [d∗1 , d∗1 ] , set σ l = σ r . Furthermore, we assume that the condition (2.80) is fulfilled. Along the x1 -axis, we use the piecewise-uniform or uniform meshes  l ω1     r ω1 ω1 =   ωb   1u ω1

   under the conditions ∗ if J = {j = 2} (2.55c), (2.82)   ∗ if J ∗ 6= ∅, if J = {j = 1, 2} under the condition (2.55c) if J ∗ = ∅;

if J ∗ = {j = 1}

(2.83a)

the mesh ω 1(2.83) condenses only in a neighborhood of the sets Γj for which j ∈ J ∗ , where J ∗ 6= ∅. We expose the mesh ¡ ¡ ¢ ¢ ω 1(2.83) d∗1 , d∗1 = ω 1 d∗1 , d∗1 ; ω 1(2.48) , (2.83b) ¡ ¢ that is constructed based on the mesh ω 1(2.48) d∗1 , d∗1 . Set ¡ ¡ ¢ ¢ ω 1 = ω b1 d∗1 , d∗1 = ω 1(2.48) d∗1 , d∗1 if J ∗ = {j = 1, 2}.

(2.83c)

For J ∗ = {j = 1} we define the mesh ω 1 by the relation ¡ ¡ ¢ ¤ (∗) ¢ T £ ω 1 = ω 1l d∗1 , d∗1 = ω 1(2.48) d∗1 , d1 d∗1 , d∗1 , J ∗ = {j = 1}.

(2.83d)

¡ £ (∗) ¢ (∗) ¤ Here ω 1(2.48) d∗1 , d1 is the mesh on the interval d∗1 , d1 with 2 N1 + 1 £ (∗) ¤ nodes; moreover, x1 = d∗1 is the midpoint of the interval d∗1 , d1 ; d∗1 = ¡ ¡ ¢ (∗) ¢ 2−1 d∗1 + d1 , N1 + 1 is the number of nodes in the¡ mesh ω¢1(2.83) d∗1 , d∗1 . In a similar way, for J ∗ = {j = 2} the mesh ω 1 = ω 1r d∗1 , d∗1 is constructed. ¡ (∗) ¢ The parameters σ, h(1)1 , and h(2)1 of the mesh ω 1(2.48) d∗1 , d1 are the same as those in the meshes ω 1l , ω 1r .



55

Theorem 2.4.6 In the case of the conditions (2.80), (2.64), (2.82), the grid ¡ ¢ Dh = Dh Q(x∗ ), L∗(2) ; J ∗ = ³ = Dh Q(x∗ ), L∗(2) , ω 1 = ω 1(2.83) (ω 1(2.48) ),

(2.84)

ωs = ωsu , s = 2, . . . , n under the conditions (2.55c) and (2.82) in the case of the relations (2.78), {(2.55), (2.79)} if J ∗ 6= ∅, and under the condition (2.55c) in the case of the relations (2.55) if J ∗ = ∅;

´ ∗ J ∗ = J(2.77) (D) ,

is consistent on the set Q(x∗ ). i

i

But if in (2.84) Q(x∗ ) = D , where D is a subset from D, the grid Dh(2.84) i

is consistent on the set D . In the case when i

D = Di

© Γ , Di = x : di∗s < xs < di∗ s , s = 1, . . . , n}

S i

(2.85)

is a rectangular parallelepiped whose faces Γij belong to coordinate planes ¢ ¡ i S Γ = j Γij , for j = 1, . . . , 2n , it is convenient to construct a consistent grid i

directly on D based on the meshes ω is , for s = 1, . . . , n, introduced on the intervals [di∗s , di∗ s ]. We show such grids. By j ∗ we denote the index of the face Γj ⊆ Γe , for which the condition i T Γi Γj 6= ∅ holds (i.e., the set D adjoins to the face Γj∗ ); Γj = Γj(2.76) , Γe = Γe(2.77) , Γi = Γ(Di ). We denote by J i∗ , where J i∗ ¡ ¢ ¡ ¢ J i∗ = J i∗ Di = J i∗ Di ; Γe ,

(2.86)

the set of all indexes j ∗ . We assume that the following condition (similar to (2.80)) holds: i

the coefficients ask (x) of the operator L∗(2) on the set D ⊆ D

(2.87)

satisfy the relations (2.78), {(2.55), (2.79)} if J i∗ 6= ∅, i

i∗ and the relations (2.55) if J i∗ = ∅, where Q(x∗ ) = D , J i∗ = J(2.86) (Di ). i On the intervals [di∗s , di∗ s ], we construct the meshes ω s , for s = 1, . . . , n. i iu i∗ The mesh ω 1 is uniform, i.e., ω 1 if J = ∅, or is piecewise uniform, i.e., ω il 1, ib i∗ ω ir , ω if J is or the corresponding set {j = 1}, {j = 2}, or {j = 1, 2}. Let 1 1


56


ω 1i =

 il ∗  ω 1 if J = {j = 1}   ω ir if J ∗ = {j = 2} 1

   under the conditions (2.55c), (2.82)   ∗ if J ∗ 6= ∅, if J = {j = 1, 2}

 ω 1ib     iu ω 1 under the condition (2.55c) ¡ i e¢ i∗ J ∗ = J i∗ = J(2.86) D;Γ ,

(2.88a)

if J ∗ = ∅;

and let the meshes ω is , for s = 2, . . . , n, be uniform. The mesh ¡ i ¢ ¡ ¢ i ω 1i = ω 1(2.88) d∗1 , di∗ = ω 1i di∗1 , di∗ (2.88b) 1 1 ; ω 1(2.48) , £ ¤ ¤ £ i constructed on d∗1 , di∗ (from the set d∗1 , d∗1 ) based on the mesh ω 1(2.48) , 1 is defined by the relation ¡ i ¢ ¡ i ¢ i i∗ ω 1i = ω 1(2.88) d∗1 , di∗ 1 ; ω 1(2.48) = ω 1(2.83) d∗1 , d1 ; ω 1(2.48) . (2.88c) ¡ ¢ Assume that the parameters of the meshes ω si = ω si di∗s , di∗ s , for s = 1, . . . , n, i

i∗ satisfy the conditions (2.55c), (2.82), where Q(x∗ ) = D and J ∗ = J(2.86) 6= ∅; i ∗ i∗ for J = J = ∅ the parameters of the meshes ω s satisfy the conditions (2.55c).

Theorem 2.4.7 In the case of the conditions (2.64), (2.87), (2.82) under the i∗ condition that in (2.78), (2.79), (2.82) J ∗ is J(2.86) , the grid (similar to (2.84)): ¢ i i¡ i¡ i Dh = Dh Di , L∗(2) ; J i∗ = Dh Q(x∗ ) = D(2.85) , L∗(2) , (2.89) i ω 1i = ω 1(2.88) (ω 1(2.48) ), ω si = ω siu , for s = 2, . . . , n,

under the conditions (2.55c) and (2.82) subject to (2.78), {(2.55), (2.79)} if J i∗ 6= ∅, and under the condition (2.55c) subject to (2.55) if J i∗ = ∅, ¢ i i∗ where Q(x∗ ) = D , J ∗ = J i∗ = J(2.86) (Di ) , i

is consistent on the set D . ¡ ¢ iT For J i∗ = ∅ i.e., for either D Γe = ∅ and Γe 6= ∅ or Γe = ∅ the grid i

Dh(2.89) is uniform. i

Note that the grid Dh(2.89) is constructed directly on the set D at that i

i

time as the grid Dh = Dh(2.84) (Q(x∗ ) is constructed on D for Q(x∗ ) = D based on meshes introduced on the set D. Grids similar to (2.84), (2.89) are used for the construction of domaindecomposition-based difference schemes, in particular, of a difference scheme for boundary value problems in domains with curvilinear boundaries; see Subsections 2.5.1, 2.5.2 in this section.



2.4.4

57

ε-uniformly convergent difference schemes

We now consider difference schemes for the boundary value problem (2.2) (0)

on the slab D(2.4) in the case when the operator L∗(2) is almost canonical on the set D. This property of the operator L∗(2) allows us by means of a (0)

choice of a consistent mesh on D to construct a monotone difference scheme that approximates the boundary value problem ε-uniformly for different type restrictions imposed on coefficients of the mixed derivatives, in particular, on the coefficients a1s (x). A corollary of Theorems 2.3.7, 2.4.4, and 2.4.5 is the following theorems on the convergence of the scheme (2.47), (2.46 ). Theorem 2.4.8 Let the conditions ask ∈ C 1 (D), for s 6= k, and ass , bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, hold and the coefficients ask (x) satisfy the conditions (2.57), (2.71), and also (2.66), (2.67), where Q = D. For the solution of the problem (2.2), (2.4), let the estimates of Theorem 2.2.1 be satisfied for K = 3. Then the difference scheme (2.47), (2.46) on the grid Dh(2.70) , where Q = D, ω 1(2.48) = ω 1(2.44) , converges ε-uniformly at the rate O(N −1 ln N ) as N → ∞: |u(x) − z(x)| ≤ M N −1 ln N,

x ∈ Dh .

Theorem 2.4.9 Let ask , bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, the coefficients ask (x) satisfy the conditions (2.59), (2.73), and also (2.61), where Q = D, and let for the solution of the problem (2.2), (2.4), the estimates of Theorem 2.2.1 be satisfied for K = 3. Then the difference scheme (2.47), (2.46) on the grid Dh(2.63) , where Q = D, ω 1(2.48) = ω 1(2.44) , converges ε-uniformly at the rate O(N −1 ln N ) as N → ∞. Theorem 2.4.10 Let ass , bs , c, c0 , f ∈ C(D), for s = 1, . . . , n, the coefficients ask (x) satisfy the condition (2.49), and let for the solution of the problem (2.2), (2.4), the estimates of Theorem 2.2.1 be satisfied for K = 3. Then the difference scheme (2.47), (2.46) on the grid Dh(2.58) , where ω 1(2.48) = ω 1(2.44) , converges ε-uniformly at the rate O(N −1 ln N ) as N → ∞. Under the hypotheses of Theorems 2.4.8, 2.4.9, and 2.4.10, for the interpolant z(x) = z (2.42) (x; z(·), Dh ), x ∈ D, where z(x), x ∈ Dh is the solution of the difference scheme (2.47), (2.46), respectively, on the grids (2.70), (2.63), where Q = D, and (2.58), one has the estimate |u(x) − z(x)| ≤ M N −1 ln N,


x ∈ D.

58


2.5

Boundary value problems in domains with curvilinear boundaries

2.5.1

A domain-decomposition-based difference scheme for the boundary value problem on a slab

Theorem 2.4.8 establishes ε-uniform convergence of the difference scheme {(2.47), (2.46), (2.70), (2.72)} constructed in Subsections 2.4.1 and 2.4.2 for the problem on the slab (2.4) in the case of the conditions (2.57), (2.71). When considering the boundary value problem in a domain with a curvilinear boundary, it is appropriate to pass to a new coordinate system in which the new boundary of the domain belongs to the coordinate plane and for the transformed problem to construct ε-uniformly convergent scheme. Keeping in mind such an idea, we construct a finite difference scheme for the problem (0) (2.2) on the slab D(2.4) in the case when the coefficients a1s (x) satisfy only the condition (2.57) and fulfillment of the condition (2.71) is not assumed. Emphasize that the condition (2.72), where Q(x∗ ) is a neighborhood from the relations {(2.55), (2.65)}, in general, is violated if the condition (2.71) is not satisfied. In this case, to approximate the boundary value problem we use a scheme of the domain decomposition method. Accuracy of the solution of an iterative scheme to the domain decomposition method depends on the (0) width of overlapping to subdomains covering D , the number of nodes in the grid domain and the number of iterations used for the solution of the difference scheme and also on the value of the parameter ε. We are interested in methods whose errors of solutions are independent of the value of the parameter ε and are defined only by the number of nodes in the mesh and the number of iterations used for the solution of the boundary value problem. For simplicity, when constructing the scheme, we assume that the boundary value problem is periodic. An approach that we use in this subsection for the construction of the difference scheme, we shall apply in Subsection 2.5.2 for a problem in a domain with a curvilinear boundary. In the case of a periodic problem in the domain D(0) , one can cover this domain by a finite system of overlapping bounded simply connected sets Q(x∗i ) = Di , for i = 1, . . . , I (neighborhoods of the points x∗i ), such that D

(1)

⊂

S

iD

i

and x∗i ∈ D(1) , for i = 1, . . . , I, where

D(1) = {x : d∗1 < x1 < d∗1 , 0 < xs < 2 π, s = 2, . . . , n},

(2.90)

is a domain of periodicity to the boundary value problem. We assume that the sets Di ,

i = 1, . . . , I,

are 2π-periodic in xs , for s = 2, . . . , n.


(2.91a)


59

To solve the boundary value problem (2.2), (2.4), we use a discrete analogue of the Schwartz method. Let us first describe this method for the differential problem. (0) On the set D(2.4) we construct a sequence of functions u[k] (x), where k = 1, 2, 3, . . . is the iteration number, in the following way. We assume that u[0] (x) = 0,

x∈D

(0)

.

(2.91b)

Let the function u[k−1] (x) be already constructed for k ≥ 1. We construct u[k] (x). For this we consider the auxiliary functions [k]

[k]

(k)

[k]

ui (x) = ui (x; ui (·), ui−1 (·)) ≡   i  u(k) (x),  x∈D i ≡ ,  u[k] (x), x ∈ D (1) \ D i  i−1 [k]

u0 (x) = u[k−1] (x),

x∈D

(0)

,

x∈D

(0)

;

(2.91c)

i = 1, . . . , I;

the values i and k define the number of interior and exterior iterations. Here i i (k) ui (x), x ∈ D , is the solution of the boundary value problem on the set D (k)

Lui (x) = f (x), x ∈ Di , ( T ) ϕ(x), x ∈ Γi Γ (k) ui (x) = , [k] ui−1 (x), x ∈ Γi \ Γ

(2.91d) x ∈ Γi ,

i

where Γi = D \ Di . Thus, [k]

[k] ¡

ui (x) = ui

¢ (k) x; {ul (·)}, 1 ≤ l ≤ i, u[k−1] (·) ,

x∈D

(0)

,

(2.91e)

i = 1, . . . , I. (k)

i

[k]

The functions ui (x), x ∈ D , and ui (x), x ∈ D [k]

for s = 2, . . . , n. The function u (x), x ∈ D [k]

u[k] (x) = uI (x), [k]

(0)

(0)

, are 2π-periodic in xs ,

, is determined by the relation

x∈D (0)

(0)

.

(2.91f)

We say that the function u(2.91f) (x), x ∈ D , is the solution on the k-th iteration of the continual Schwartz method (2.91), where k ≥ 1. We assume that the following condition for the width of overlapping to subdomains holds. Let D i be the set from (2.91a), where i = 1, . . . , I, and let Γ i,(1) be the part of the boundary to the set D i that belongs to D(1) . S Let D {i} = i1 D i1 , where i1 = 1, . . . , I and i1 6= i, be an union of the sets from (2.91a) which do not contain D i , and let Γ {i},(1) be the part of the


60


boundary to the set D {i} that belongs to D(1) . The minimum of the minimal distance between the sets Γ i,(1) and Γ {i},(1) for all i, where i = 1, . . . , I (their T infimum), is the minimal overlapping of the subdomains {D i D(1) }, covering D(1) . Assume that the minimal overlapping of subdomains, covering D(1) , is independent of ε. This condition ensures a convergence rate of the iterative Schwartz method (as the number of iteration grows) which is independent of the value of the parameter ε. About the iterative method with such convergence rate, we say that the method (with growth of the iteration number) converges ε-uniformly. [k]

Theorem 2.5.1 The function u(2.91) (x), i.e., the solution of the continual Schwartz method (2.91) for the boundary value problem (2.2), (2.4), converges to the solution of the boundary value problem ε-uniformly at the rate of a geometric progression as k → ∞: |u(x) − u[k] (x)| ≤ M q k ,

x∈D

(0)

,

q ≤ 1 − m.

(2.92)

In (2.92) and in what follows, the constant M is independent of k, and the value q is independent of ε. For the Schwartz method (2.91) we construct a difference scheme and formulate conditions which ensure ε-uniform convergence of this scheme. For simplicity, we assume that the sets Di are chosen in the form of rectangular parallelepipeds, whose faces are formed by the coordinate planes xs = iT const, for s = 1, . . . , n. Under the condition that D Γ = ∅, the distance between the sets Di and the boundary Γ is independent of ε. May happen that the inequality (2.71) is violated on D, but a similar inequality is satisfied pointwisely: £ ¤ |ask (x)| < (n − 1)−1 αs (x) αk−1 (x) ass (x), αk (x) αs−1 (x) akk (x) , x ∈ D,

s, k = 1, . . . , n,

s 6= k,

(2.93)

moreover, £ ¤ max αs (x) αk−1 (x) ≤ M,

s, k = 1, . . . , n,

s 6= k,

s,k,D

i.e., we have pointwise dominance of the diagonal elements in the matrix of the coefficients of the operator L∗(2) (or, the diagonal elements of the elliptic part of the differential equation). i In this case, one can choose the diameter of the sets D sufficiently small i such that the following inequality takes place on D £ ¤ maxD i |ask (x)| < (n − 1)−1 minD i αsi (αki )−1 ass (x), αki (αsi )−1 akk (x) , s, k = 1, . . . , n,


s 6= k,

(2.94)


61

that ensures a transformation of the operator L∗(2) to almost canonical form on i

∗[1]

D and also a transformation of the operator L(2) to strongly almost canonical i

iT

form on D if D Γ 6= ∅ and the coefficients a1s (x) on the boundary Γ satisfy the condition (2.57). i

Otherwise, for sufficiently small diameter (independent of ε) of the set D , it is possible to transfer to new variables X i = X i (x), in which the elliptic operator containing the second-order derivatives (we denote it by L∗X i (2) ) will i

i

be almost canonical form on the set DX i , i.e., the set D in the variables X i . iT But if D Γ 6= ∅, then new variables can be chosen such that the truncated T ∗[1] ∗[1] iL iL operator LX i (2) = LX i (2) (L∗X i (2) ) on the face ΓX = Γ Γ i, Γ i = i , where Γ i

iL D \Di , is canonical with respect to a variable orthogonal to the face ΓX i and, i

besides, on the set DX i is strongly almost canonical with respect to variables ∗ iL orthogonal to the face ΓX i . The coefficients of the operator LX i (2) on the set iT

i

DX i satisfy relations similar to (2.55) for D Γ = ∅ and satisfy the relations iT {(2.55), (2.65)} for D Γ 6= ∅; moreover, in (2.55) and {(2.55), (2.65)} the i set Q(x∗ ) is DX i . Such approach is considered in Subsection 2.5.2. Here we assume that the coefficients a1s (x) satisfy the condition (2.57). We consider that the operator L∗(2) is locally almost canonical on D and almost i

∗[1]

canonical on the subdomains D , and the operator L(2) under the condition iT

D Γ 6= ∅ is strongly almost canonical. We also suppose that the condition (2.64), (2.67) holds. Furthermore, we assume to be fulfilled the condition the coefficients ask (x) of the operator L∗(2) on the set D

i

(2.95)

satisfy the relations (2.94), (2.57), {(2.55), (2.65)}, (2.64), (2.67) if D and the relations (2.55) if D

iT

iT

Γ 6= ∅,

Γ = ∅,

i

where Q(x∗ ) = D , i = 1, . . . , I, that admits the construction of consistent grids in the variables x1 , . . . , xn with the monotonicity condition for the difference operator Λ∗(2)(2.51) (L∗(2) ) i

i

j

on D . Note that the grids defined on the subdomains D and D , on the iT j intersection of the subdomains D D , for i 6= j, in general, do not coincide. That requires a special organization of interaction to the grid solutions on the boundaries of subdomains. i Taking account of the condition (2.95) on the set D we construct the


62


consistent grid

i Dh

=

 D D

¡ h(2.56)

¡

h(2.70)

Q(x∗ ) = D i

Q(x∗ ) = D

¢

¢ i

if D if D

iT iT

Γ = ∅, Γ 6= ∅,

(2.96) i = 1, . . . , I,

that ensures the monotonicity of the operators Λ∗(2) = Λ∗(2)(2.51) (L∗(2) ) and i

i

Λ = Λ(2.46) (L) on Dh . The nodes of the grid Dh belong to D i

is not empty. We assume that the sets D and the grids in xs , for s = 2, . . . , n. i

i Dh

iT

Γ if this set

are 2π-periodic

i

The grid Dh(2.96) on the set D is constructed based on the grid Dh(2.56) ¡ (0) ¢ or Dh(2.70) introduced on the set D(2.4) and on the basis, respectively, the i

relations (2.55) or {(2.55), (2.65)}, (2.67) for Q(x∗ ) = D . Note that any ¡ ¢ i change of the distribution of nodes in the meshes ω s generating Dh(2.96) i

outside the projection of the set D on the xs -axis does not affect the distribui i T© i T jª j T© i T jª tion of nodes in the grid Dh . In general, Dh D D 6= Dh D D for D

iT

j

D = ∅, where i 6= j. (0)

i

i

On the slab D = D(2.4) on the basis of the grids Dh = Dh(2.96) , for i = 1, . . . , I, we construct the grid {0}

Dh = Dh ,

(2.97a)

where we set iS

1

D(1)h = Dh , {0}

Dh

D(i)h = Dh

n o i D(i−1)h \ D ,

i = 2, . . . , I,

(2.97b)

= D(I)h .

On the slab D i.e.,

(0)

{i}

we also introduce a set of the grids Dh , for i = 1, . . . , I, {i}

Dh

iS

= Dh

{i−1}

{Dh {0}

i

\ D }, {I}

i = 1, . . . , I.

(2.97c)

It is not difficult to see that Dh = Dh . On the grid Dh(2.97a) we construct a sequence of grid functions z [k] (x), for k = 1, 2, 3, . . ., in the following way. Set z [0] (x) ≡ 0, x ∈ Dh . We construct z [k] (x) assuming that the function z [k−1] (x), x ∈ Dh , for k ≥ 1, has already [k] been constructed. Let us consider the auxiliary functions zi (x) defined on



63

{i}

the grids Dh(2.97c) : ¢ [k] [k] ¡ (k) [k] zi (x) = zi x; zi (·), zi−1 (·) ≡    z (k) (x), x ∈ D i  h i ≡ ,  z [k] (x), x ∈ D {i−1} \ D i  h i−1 [k]

z0 (x) = z [k−1] (x),

x ∈ Dh ,

{i}

x ∈ Dh ,

(2.98a)

i = 1, . . . , I;

here the values i and k define the number of interior and exterior iterations. i (k) In these relations zi (x), x ∈ Dh , is the solution of the discrete boundary i value problem on the set Dh (k)

Λi zi (x) = f (x), x ∈ Dhi ; ( T ) ϕ(x), x ∈ Γih Γ (k) , zi (x) = [k] zˇi−1 (x), x ∈ Γih \ Γ

(2.98b) x ∈ Γih , i = 1, . . . , I.

i

Here Λi is Λ = Λ(2.46) (L) on the grid Dh , [k]

zˇ0 (x) = zˇ[k−1] (x),

x ∈ Γ1h \ Γ;

(2.98c)

[k]

the functions zˇi−1 (x), x ∈ D, and zˇ[k−1] (x), x ∈ D, are constructed as the {i−1}

[k]

interpolation of the functions zi−1 (x), x ∈ Dh i = 1, . . . , I. Thus, [k]

[k] ¡

zi (x) = zi

, and z [k−1] (x), x ∈ Dh , for

¢ (k) x; {zl (·)}, 1 ≤ l ≤ i, z [k−1] (·) ,

(i)

x ∈ Dh ,

(2.98d)

i = 1, . . . , I. The function z [k] (x), x ∈ Dh , is determined by the relation [k]

z [k] (x) = zI (x),

x ∈ Dh .

(2.98e)

We say that the function z [k] (x), x ∈ Dh , where Dh = Dh(2.97) , is kth iteration of the discrete Schwartz method {(2.98), (2.97), (2.96)}, where k ≥ 1. The difference scheme {(2.98), (2.97), (2.96)} is the iterative scheme of the domain decomposition method on the overlapping subdomains. [k] The functions zˇi (x) and zˇ [k] (x), x ∈ D, i.e., the interpolants used in (2.98b), (2.98c), are defined by the relations    z (k) (x), x ∈ D i  i [k] zˇi (x) = , x ∈ D, i = 1, . . . , I, (2.98f)  zˇ[k] (x), x ∈ D \ D i  i−1


64

Elliptic reaction-diffusion equations [k]

[k]

zˇ0 (x) = zˇ [k−1] (x), zˇ [k] (x) = zˇI (x), x ∈ D, k ≥ 1; zˇ [0] (x) = 0, x ∈ D. i

(k)

Here the interpolant z i (x), x ∈ D is defined by the relation (k)

(k)

(k)

z i (x) = z i(2.42) (x; zi

i

(·), Dh ),

i

x∈D .

Such interpolation preserves accuracy of the solution of the discrete problem (2.98b) that approximates the problem (2.91d). The iterative difference scheme {(2.98), (2.97), (2.96)} is defined by the canonical elements of the difference scheme of the domain decomposition method, i.e., the operator Λ(2.46) and the grids Dh(2.56) (ω u1 ) and Dh(2.70) (ω 1(2.48) ) on the basis of which the difference scheme is constructed. Set N i = min Nsi and N = min N i , where Nsi + 1 is the number of nodes in the i

s

i

grid Dh along the xs -axis, for s = 1, . . . , n. We now show a reduced problem for the iterative difference scheme {(2.98), (2.97), (2.96)}. For k → ∞ the discrete problem (2.98b) transforms into the following i problem on the set Dh Λi zi0 (x) = f (x), x ∈ Dhi ; ( T ) ϕ(x), x ∈ Γih Γ 0 , zi (x) = zˇi−1 (x), x ∈ Γih \ Γ

(2.99a) x ∈ Γih , i = 1, . . . , I.

{i}

[k]

The function zi (x), x ∈ Dh that corresponds to the function zi (x) from (2.98a) is defined by the relation ¡ ¢ zi (x) = zi x; zi0 (·), zi−1 (·) ≡   i z 0 (x),  D x ∈ {i} h i ≡ , x ∈ Dh , i = 1, . . . , I, (2.99b) z (x), x ∈ D {i−1} \ D i i−1 h z0 (x) = zI (x),

x ∈ Dh .

The interpolant zˇi (x), x ∈ D, in (2.99a) is defined by the relation   i  z 0 (x),  D x ∈ i zˇi (x) = , x ∈ D, i = 1, . . . , I,  zˇ (x), x ∈ D \ D i  i−1

zˇ0 (x) = zˇI (x),


x ∈ D.

(2.99c)

Boundary value problems in smooth domains i

65

i

0 Here z i0 (x) = z i(2.42) (x; zi0 (·), Dh ), x ∈ D , is an interpolant of the function i

zi0 (x), x ∈ Dh . Thus, ¡ ¢ zi (x) = zi x; {zl (·)}, l = 1, . . . , n, l 6= i ,

{i}

x ∈ Dh ,

(2.99d)

i = 1, . . . , I. We define the functions z(x), x ∈ Dh , by the relation z(x) = zI (x),

x ∈ Dh ;

(2.99e)

the interpolant of the discrete function z(x), x ∈ Dh , is defined by the relation zˇ(x) = zˇI (x),

x ∈ D.

(2.99f)

The function z(x), x ∈ Dh , is the solution of the difference scheme {(2.99), (2.97), (2.96)}, i.e., the noniterative difference scheme of the domain decomposition method which is reduced for the scheme {(2.98), (2.97), (2.96)}. The noniterative difference scheme {(2.99), (2.97), (2.96)} is defined by canonical elements, i.e., the operator Λ(2.46) and the grids Dh(2.56) (ω u1 ) and Dh(2.70) (ω 1(2.48) ). The schemes (2.98), (2.97), (2.96) and (2.99), (2.97), (2.96) are ε-uniformly monotone and approximate the boundary value problem (2.2), (2.4) ε-uniformly. T We assume that the minimal overlapping of the subdomains {Di D(1) } is independent of ε. For k → ∞ the solution of the iterative scheme {(2.98), (2.97), (2.96)} converges to the solution of the noniterative scheme {(2.99), (2.97), (2.96)} ε-uniformly with the estimate |z(x) − z [k] (x)| ≤ M q k ,

x ∈ Dh ,

q ≤ 1 − m.

(2.100)

Here and below the value q is independent of ε and N , and the constant M is independent of q. For the solution of the boundary value problem (2.2), (2.4), let the estimates of Theorem 2.2.1 be fulfilled for K = 3. Then the difference scheme {(2.99), (2.97), (2.96)}, where Dh(2.70) = Dh(2.70) (ω 1(2.48) = ω 1(2.44) ) converges ε-uniformly with the estimate |u(x) − z(x)| ≤ M N −1 ln N,

x ∈ Dh ,

(2.101)

as N → ∞. We have the following theorems on the convergence of the noniterative and iterative difference schemes of the domain decomposition method. Theorem 2.5.2 Let ask , bs , c, c0 , f ∈ C(D), s, k = 1, . . . , n, the coefficients ask (x) satisfy the conditions (2.93), (2.95), and let for the solution of the problem (2.2), (2.4), the estimates of Theorem 2.2.1 be satisfied for


66


K = 3. Then the iterative difference scheme {(2.98), (2.97), (2.96)} on the grid Dh(2.97a,2.96) , where ω 1(2.48) is ω 1(2.44) , converges ε-uniformly at the ¡ ¢ rate O N −1 ln N + q k as N, k → ∞: |u(x) − z [k] (x)| ≤ M [N −1 ln N + q k ],

x ∈ Dh .

Theorem 2.5.3 The solution of the iterative difference scheme {(2.98), (2.97), (2.96)}, for k → ∞, converges to the solution of the noniterative difference scheme {(2.99), (2.97), (2.96)} ε-uniformly. In the case when the hypotheses of Theorem 2.5.2 hold, the solution of the noniterative difference scheme {(2.99), (2.97), (2.96)} on the grid Dh(2.97a,2.96) , where ω 1(2.48) is ω 1(2.44) converges ε-uniformly to the solution of the problem (2.2), (2.4) as N → ∞. The discrete solutions satisfy the estimates (2.100), (2.101). [k]

The interpolants zˇ(2.99f) (x) and zˇ(2.98f) (x) of the solutions of the noniterative and iterative difference schemes converge ε-uniformly with the estimates |u(x) − zˇ(x)| ≤ M N −1 ln N,

x ∈ D,

|u(x) − zˇ [k] (x)| ≤ M [N −1 ln N + q k ],

x ∈ D,

q ≤ 1 − m.

Remark 2.5.1 Let k f be the number of iterations required for the solution of the iterative difference scheme of the decomposition method {(2.98), (2.97), (2.96)} (where ω 1(2.48) is ω 1(2.44) ), i.e., the number of iterations on which the estimate f |u(x) − z [k ] (x)| ≤ M N −1 ln N, x ∈ Dh , holds. The number of iterations k f is ε-uniformly bounded. The value k f satisfies the estimate k f ≤ M ln N. Remark 2.5.2 When constructing a difference scheme for the problem (2.2), (2.4), along with the grids (2.96) (constructed based on the grid Dh(2.56) , Dh(2.70) introduced on the set D) it is convenient to apply also consistent i

grids (that use the grids Dh(2.89) , constructed on D ) ¢ i i¡ Dh = Dh Di , L∗(2) ; J i∗ = (2.102) ¡ ¢ i∗ = Dh(2.89) Di , L∗(2) ; ω i1 = ω i1(2.88) (ω 1(2.48) ), J(2.86) , i = 1, . . . , I, where ω 1(2.48) is ω 1(2.44) . The statement of Theorem 2.5.2 is preserved in the i

i

case when, instead of the grid Dh(2.96) , we use the grid Dh(2.102) .



2.5.2

67

A difference scheme for the boundary value problem in a domain with curvilinear boundary

We construct a difference scheme for the boundary value problem (2.2) in the case when D is a domain with a curvilinear boundary. The fulfillment of the condition of a dominance of the diagonal elements of the elliptic operator on whole set D, i.e., the condition (2.71), as well as restrictions on the coefficients of the mixed derivatives on the domain boundary similar to the condition (2.57), is not assumed. We consider the problem (2.2) in a doubly-connected bounded domain D=D

S

Γ,

Γ = Γ1

S

Γ2 ,

Γ1

T

Γ2 = ∅;

(2.103)

the boundaries Γ1 and Γ2 are assumed to be sufficiently smooth. The problem (2.2), (2.103) is equivalent to a periodical boundary value problem on the slab with a curvilinear boundary. We construct a difference scheme using the domain decomposition method. First, we made some preliminary constructions. Let x∗ ∈ Γ, and let on the closure Q(x∗ ), i.e., a neighborhood Q(x∗ ) of the point x∗ , the boundary Γ be given by the equation x1 = x1 (x2 , . . . , xn ), x ∈ Q(x∗ ). By transforming to the variables X = X(x): X1 = x1 − x1 (x2 , . . . , xn ),

Xs = xs ,

s = 2, . . . , n,

(2.104)

we make the boundary Γ “plane” in a neighborhood of the point x∗ (in a neighborhood of the part to the boundary Γ, the tangent plane for which is not parallel to x1 -axis). In the case when the neighborhood Q(x∗ ) is sufficiently small, the system (2.104) can be uniquely solved with respect to x1 , . . . , xn for x ∈ Q(x∗ ). ∗ We denote by X −1 (X) the map inverse to X(x). We assume that D = ∗ T ∗ D Q(x ). For the functions v(x), W (X) and subdomains D0 ⊆ D we will use the notation v(x(X)) = vX (X) = {v(x)}X ,

W (X(x)) = WX −1 (x),

0 DX = {D0 }X = X(D0 ) = {X : X −1 (X) ∈ D0 }.

We assume that e X −1 = X −1 (D) e = {x : X(x) ∈ D}, e D ∗

∗

e is a subset of the set DX = {D }X . where D T ∗ Let the face {Γ D }X be belong to the plane X1 = 0, and let the set ∗ DX belong to the half-space X1 ≥ 0. We choose the neighborhood Q(x∗ ) i i ∗ such that the set DX = {D }X (which belongs to DX ) is a rectangular parallelepiped whose faces are formed by the coordinate planes Xs = const,


68

Elliptic reaction-diffusion equations iT

T

i

for s = 1, . . . , n; here D Γ 6= ∅. We denote the face {Γ D }X by ΓiL X; i T ΓiL = Γ Γi , Γi = D \ Di . i S iL In the new variables, the problem (2.2), (2.103) on the set DX ΓX is transformed into the subproblem

Here

LX U (X) = F (X),

i , X ∈ DX

(2.105a)

U (X) = Φ(X),

ΓiL X.

(2.105b)

LX ≡ ε2 L2X + L0X , n X

L2X ≡

X∈

L0X = −C 0 (X), n

Ask (X)

s,k=1

X ∂2 ∂ + Bs (X) − C(X); ∂Xs ∂Xk s=1 ∂Xs

set L∗X(2) ≡

n X

Ask (X)

s,k=1

∂2 . ∂Xs ∂Xk

(2.105c)

The coefficients Ask and Bs are determined from the formulae ( n ) X ∂ ∂ arp (x) Ask (X) = Xs Xk , s, k = 1, . . . , n; ∂xr ∂xp r,p=1 Bs (X) =

X

( n X

n X

∂ ∂2 br (x) Xs + arp (x) Xs ∂xr ∂ xr ∂xp r=1 r,p=1

) , s = 1, . . . , n. X

0

The functions U , C, C , F , and Φ are defined by the relation V (X) = vX (X), where v(x) is one of the functions u(x), c(x), c0 (x), f (x), ϕ(x). Note that i i i i Ask ∈ C 1 (DX ), Bs ∈ C(DX ) under the condition ask ∈ C 1 (D ), bs ∈ C(D ), Γ ∈ C 2. The coefficients Ask of the operator LX(2.105) satisfy the condition of strong ellipticity A0

n X s=1

ξs2 ≤

n X

Ask (X) ξs ξk ≤ A0

s,k=1

n X

i

ξs2 , X ∈ DX , A0 > 0.

(2.106)

s=1

We assume the fulfillment of the canonicity condition on the boundaries ΓiL X A1s (X) = 0, X ∈ ΓiL s = 2, . . . , n. (2.107) X, If the condition (2.107) is not fulfilled we pass to the new variables X 1 , X 1 = X 1 (X): ¡ ¢ ¡ ¢ X11 (X) = X1 , Xs1 (X) = Xs − A−1 11 0, X2 , . . . , Xn A1s 0, X2 , . . . , Xn X1 , s = 2, . . . , n



69

S

and then consider the problem (2.2) on the set {Di ΓiL }XX 1 in these new variables. For the coefficients A11s (X 1 ) of the operator LXX 1 , one has the condition A11s (X 1 ) = 0, X 1 ∈ ΓiL s = 2, . . . , n. XX 1 , In this case we choose the neighborhood Q(x∗ ) such that the set {Di }XX 1 is a rectangular parallelepiped whose faces are formed by the coordinate planes Xs1 = const, where s = 1, . . . , n. T

Assume that under the condition Γ D

i

6= ∅ the operator L∗X(2)(2.105)

i DX

i

defined on the set has almost canonical form in X1 , . . . , Xn on DX ¡ i¢ ∗[1] for sufficiently small diameter of the set D , and the operator LX(2) = ¡ ∗[1] LX(2) (L∗X(2) ) i.e., the truncated operator involving differentiation in the vari¢ ables X2 , . . . , Xn ; see (2.62) has strongly almost canonical form in X2 , . . ., ∗ Xn in a neighborhood of ΓiL X . Let the coefficients of the operator LX(2) defined © ª i on DX = X : di∗s ≤ Xs ≤ di∗ s , s = 1, . . . , n satisfy the condition (similar to (2.87)): i

the coefficients Ask (X) of the operator L∗X(2) on the set DX satisfy the relations (2.78), {(2.55), (2.79)} if J i∗ 6= ∅, and the relations (2.55) if J i∗ = ∅, ∗

(2.108) ∗

where ask (x), . . . , Q(x ) is Ask (X), . . . , {Q(x )}X , i

i under the condition {Q(x∗ )}X = DX , J i∗ = J i∗ (DX ) = {j = 1}. i Here the condition J i∗ (DX ) = {j = 1} means that when constructing piecei i wise-uniform grids DXh on the set DX , the piecewise-uniform mesh ω X1 in the iL X1 -axis (orthogonal to the face ΓX )£ condenses ¤ (for small ε) in a neighborhood of the left endpoint of the interval di∗1 , di∗ 1 . iL Note that on the face ΓX the condition (2.107) holds (i.e., the condition (2.78), where the coefficients a1k (x) on the set Γj , for j ∈ J ∗ , are the coefficiiL ents A1k (X) on the set ΓX ). In the case of the operator L∗X(2) , the condition ¡ ¢ i i similar to (2.55c) for Q(x∗ ) = D , where D ⊆ D imposed on the parameters hs takes the form

¤ £ ρ∗s (X) h∗ , min ρ∗s (X) h∗ , hs ∈ max i i DX

s = 1, . . . , n.

(2.109)

DX

The fulfillment of the condition (2.108) allows us to construct on the set i DX a piecewise-uniform grid that of ¢the operators ¡ ¢ ensures the monotonicity ¡ Λ∗X(2) = Λ∗X(2)(2.51) L∗X(2)(2.105c) and ΛX = ΛX(2.46) LX(2.105a) . The operators Λ∗X(2)(2.51) and ΛX(2.46) are constructed using the operators L∗X(2) and LX , respectively.


70


Let the conditions ask ∈ C 1 (D) and Γ ∈ C 2 hold, and let also the conditions i (2.108) and (2.82), where J ∗ = J i∗ (DX ) = {j = 1}, imposed on the parameters of the piecewise-uniform mesh ω X1 = ω lX1 and on the parameters ¡ ¢ of uniform meshes ω Xs = ω uXs , for s = 2, . . . , n, where ω Xs = ω iXs di∗s ,¡di∗ s , for s = 1, . . . , n, be fulfilled. Then ¢ the following piecewise-uniform grid the grid (2.89), where J ∗ = {j = 1} , ¢ i i ¡ i DXh = DXh {Q(x∗ )}X = DX , L∗X(2) ; J i∗ =

(2.110)

³ i i i l i u = DXh(2.89) {Q(x∗ )}X = DX , L∗X(2) ; ω X1 = ω X1 , ω Xs = ω Xs , for s = 2, . . . , n, under the conditions (2.109), (2.82), ´ J i∗ = {j = 1} in the case of the condition (2.108) ¡ ¢ i i is consistent in the variables X1 , . . . , Xn on the set DX . The grid DXh(2.110) l u is generated by the meshes ω X1 and ω Xs , for s = 2, . . . , n, i

l u u DXh(2.110) = ω X1 × ω X2 × . . . , ω Xn .

For the subproblem (2.105) we construct a grid approximation, i.e., the monotone difference scheme i ΛX Z(X) = F (X), X ∈ DXh ; i

iL Z(X) = Φ(X), X ∈ ΓXh .

(2.111)

i

Here ΛX = ΛX(2.46) (LX ), and DXh = DXh(2.110) . Returning to the variables x = (x1 , . . . , xn ), we obtain the monotone difference scheme i Λ∗ z ∗ (x) = f (x), x ∈ DXh X −1 ;

Here

i i DXhX −1 = D i

T i

i

DXh X −1 ,

DXh X −1 = {DXh }X −1 ; i

z ∗ (x) = ϕ(x), x ∈ ΓiL XhX −1 . T

(2.112)

i

ΓiL Xh X −1 = Γ D Xh X −1 , z ∗ (x) = Z(X(x)),

i

x ∈ DXh X −1 .

Note that the grid DXh X −1 is, in general, not rectangular. We now give a finite difference scheme for the problem (2.2), (2.103). i Let the subdomains D , for i = 1, . . . , I , form a covering of the set D; i moreover, the subdomains D , for i = 1, . . . , I1 , adjoin to the boundary Γ, 0 i S and the set D formed by the union i D , for i = I1 + 1, . . . , I, has no common points with the boundary Γ. 0 Let the distance between the sets D and Γ be independent of ε. We assume that the condition for the width of overlapping to subdomains holds, i.e., for all i, where i = 1, . . . , I, the minimum of the minimal distance



71

between the boundary of the set D i and the boundary of the union of all sets which do not contain D i (the minimal overlapping of subdomains that cover D) is independent of the value of the parameter ε. i The sets D , for i = I1 + 1, . . . , I, which have no common points with the boundary Γ, are chosen as rectangular parallelepipeds formed by coordinate i planes in the variables x1 , . . . , xn . The sets D , for i = 1, . . . , I1 , adjoining i to the boundary Γ, in new variables X i = X i (x) are the sets DX i which are rectangular parallelepipeds formed by coordinate planes in the variables X1i , . . . , Xni . We assume that the coefficients of the operator LX i , for i = 1, . . . , I1 , i on the set DX i satisfy the condition (2.108). We also consider that on the i subdomains D , for i = I1 + 1, . . . , I, the operator L is almost canonical; on these subdomains, let the following condition (similar to (2.87), where J ∗ = ∅) hold: the coefficients ask (x) of the operator L∗(2) on the set D satisfy the relations (2.55),

i

(2.113)

i

where Q(x∗ ) = D , for i = I1 + 1, . . . , I. i

But if the operator L is not almost canonical on D , for i = I1 + 1, . . . , I, then i on the set D we transfer to new variables X = X(x), in which the operator i ¡ i L∗X(2) has almost canonical form on DX for sufficiently small diameter DX , the coefficients of the operator L∗X(2) satisfy relations similar to (2.55) under ¢ 1 the condition {Q(x∗ )}X = DX in the new variables . i

i

These assumptions admit the construction of the grids DX i h and Dh that are consistent in the variables X1i , . . . , Xni and x1 , . . . , xn with the monotonicity condition for the corresponding grid operators Λ∗X i (2)(2.51) (L∗X i (2) ) and Λ∗(2)(2.51) (L∗(2) ). i

On the sets D , for i = 1, . . . , I1 , on the basis of the consistent grids i DX i h(2.110) , we construct the grids i

i

Dh = DX i h{X i }−1 ,

i = 1, . . . , I1 ,

(2.114a)

and the monotone difference operators Λi = Λ∗(2.112) for

i

x ∈ DX i h{X i }−1 ,

i = 1, . . . , I1 .

(2.114b)

i

Here DX i h are grids that are piecewise-uniform in the variable orthogonal to T i iL D }X i and uniform in other variables. On the sets the boundary ΓX i = {Γ i

D , for i = I1 + 1, . . . , I, we introduce consistent grids that are uniform in all


72


variables (similar to the grid (2.89), where J ∗ = ∅): ³ ´ i i Dh = Dh Q(x∗ ) = Di , L∗(2) ; J i∗ (Di ) =

(2.114c)

³ i = Dh(2.89) Q(x∗ ) = Di , L∗(2) , ω s = ω su , for s = 1, . . . , n, ´ under the condition (2.55c); J i∗ = ∅ ,

i = I1 + 1, . . . , I,

and the monotone difference operators Λi = Λ(2.46) (L(2.2a) ) for x ∈ Dhi ,

i = I1 + 1, . . . , I.

(2.114d)

i

By Nsi +1 we denote the number of nodes in the grid DX i h , for i = 1, . . . , I1 i (the grid Dh , for i = I1 + 1, . . . , I), on the Xsi -axis (respectively, on the xs axis), for s = 1, . . . , n. Set N i = mins Nsi and N = mini N i . To solve the problem (2.2), (2.103) in the domain with curvilinear boundary, we use the difference scheme (2.98), (2.97), where  Di i h(2.114a) for i = 1, . . . , I1 , Dh = (2.114e) Di h(2.114c) for i = I1 + 1, . . . , I; ( Λi(2.114b) for i = 1, . . . , I1 , Λi = (2.114f) Λi(2.114d) for i = I1 + 1, . . . , I. In this difference scheme unlike the scheme {(2.98), (2.97), (2.96)} for the [k] problem (2.2), (2.103), the functions zˇi (x), x ∈ D for k ≥ 1, used in (2.98b), now are defined by relations (different from (2.98f))  ©  ª (k)  i     Z i (X ) {X i }−1 , i = 1, . . . , I1 , x ∈ D i   [k]  zˇi (x) = z (k) (x), , (2.114g) i = I + 1, . . . , I 1 i      [k]  i zˇi−1 (x), x∈D\D x ∈ D, i = 1, . . . , I, [k] zˇ0 (x)

= zˇ

[k−1]

[k]

(x), zˇ [k] (x) = zˇI (x), x ∈ D, k ≥ 1; zˇ [0] (x) = 0, x ∈ D.

(k)

i

Here the interpolant Z i (X i ), X i ∈ DX i , is defined by the relation (k)

(k)

(k)

Z i (X i ) = Z i(2.42) (X i ; Zi

i

(·), DX i h ),

i

X i ∈ DX i ;

this interpolant is piecewise-linear in the variables Xsi , for s = 1, . . . , n, at the (n+1-vertex) polyhedral elements, i.e., partitions of elementary n-dimensional i rectangular parallelepipeds formed by the nodes of the grid DX i h .



73

The difference scheme of the domain decomposition method {(2.98), (2.97), (2.114)} is determined by canonical elements, i.e., the operators Λ(2.46) (L) and ΛX i (2.46) (LX i ) and the consistent uniform and piecewise-uniform grids i

i

Dh(2.114c) and DX i h(2.114a) . The difference scheme {(2.98), (2.97), (2.114)} is ε-uniformly monotone and approximates the boundary value problem (2.2), (2.103) ε-uniformly. The following convergence theorem holds. Theorem 2.5.4 Let ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, i i and Γ ∈ C 2 , the coefficients ask (x), x ∈ D , and Ask (X i ), X i ∈ DX i , satisfy, respectively, the conditions (2.93), (2.113) for i = I1 + 1, . . . , I and (2.107), (2.108), where X = X i , for i = 1, . . . , I1 , and let for the solution of the problem (2.2), (2.103), the estimates of Theorem 2.2.2 be satisfied for K = i l 3. Then, on the grid Dh(2.97a,2.114e) , in which the mesh ω X i 1 in D X i h , for i i = 1, . . . , I1 , is defined by the mesh ω 1(2.44) , the difference scheme {(2.98), (2.97), (2.114)} converges ε-uniformly as N, k → ∞: ¡ ¢ |u(x) − z [k] (x)| ≤ M N −1 ln N + q k , x ∈ Dh , q ≤ 1 − m. In the case of the scheme {(2.98), (2.97), (2.114)}, the interpolant zˇ[k] (x), x ∈ D, of the solution z [k] (x), x ∈ Dh , to the difference scheme of the decomposition method is defined by the relation [k]

zˇ[k] (x) = zˇI(2.114g) (x),

x ∈ D,

k ≥ 1.

For the interpolant zˇ[k] (x) we have the estimate ¡ ¢ |u(x) − zˇ[k] (x)| ≤ M N −1 ln N + q k ,

x ∈ D.

For the solution of the difference scheme {(2.98), (2.97), (2.114)}, the statement of Remark 2.5.1 to Theorem 2.5.2 holds. i

Remark 2.5.3 When solving differential subproblems on the sets D , for i = 1, . . . , I1 , in the decomposition method it is convenient to use the difference equations (k)

(X i ) = F (X i ), X i ∈ DXi i h ,  T i Φ(X i ), X i ∈ ΓX ΓX i , ih   ) (k) i = 2, . . . , I1 Zi (X i ) = Zˇ [k] (X i ), i  , X i ∈ ΓX i h \ ΓX i ,  © [k−1] ª zˇ (x) X i , i = 1

Λi Zi

i = 1, . . . , I1 not transferring to the original variables x.


Chapter 3 Boundary value problems for elliptic reaction-diffusion equations in domains with piecewise-smooth boundaries

In this chapter the boundary value problem is considered for an elliptic reaction-diffusion equation in a domain with a piecewise-smooth boundary. In this problem in a neighborhood of the corner points, edges, and smooth parts of the boundary, the derivatives of the problem solution in each direction orthogonal to the boundary grow without boundary as the parameter ε tends to zero. To construct finite difference schemes, classical difference approximations of the corresponding differential operators are used. Sufficient conditions for ε-uniform convergence of the finite difference schemes are derived. The fulfillment of these conditions is ensured by choosing special grids that condense near the boundary and have a consistent distribution of nodes.

3.1

Problem formulation. The aim of the research

On an n-dimensional domain D with a piecewise-smooth boundary Γ, we consider the Dirichlet problem for the elliptic reaction-diffusion equation

Here

L u(x) = f (x),

x ∈ D;

(3.1a)

u(x) = ϕ(x),

x ∈ Γ.

(3.1b)

L ≡ ε2 L2 + L0 ,

the operators L2 and L0 are defined by the relations (2.2c). The coefficients satisfying the conditions (2.3) and the right-hand side f in the differential equation are assumed to be sufficiently smooth. Let Γc be the set of vertices and edges of the domain D and Γj , for j = 1, . . . , J, be its curvilinear S faces; here Γ = j Γj and Γj = Γj . Assume that the neighboring faces are intersected at nonzero angles and the domain D has not entering corners and edges. The function ϕ(x) is sufficiently smooth on each of the sides Γj , for j = 1, . . . , J and continuous on Γ. The ¡ fulfillment of other compatibility conditions on the set Γc is not assumed under these assumptions, in general, one ¢ has u 6∈ C 2 (D) .


76


We shall consider the boundary value problem either on a rectangular parallelepiped D = {x : d∗s < xs < d∗s , s = 1, . . . , n}, (3.2) or in a domain with a piecewise-smooth curvilinear boundary. Our aim for the boundary value problem (3.1) in a domain with a piecewisesmooth boundary is to construct a finite difference scheme that converges ε-uniformly.

3.2


We estimate the solution of the problem (3.1) on the rectangular parallelepiped D(3.2) . The estimates are established using a priori estimates [69, 37]. Let the functions u(x), x ∈ D on the boundary Γ take values of ϕ(x), moreover, on the set Γg the function ϕ(x) is “good”, i.e., ϕ ∈ C l+α (Γg ) and g Γg ∈ C l+α , Γg = Γ . The set Γg can coincide with Γ, or can be empty. We denote by C l+α (D) the Banach space whose elements are continuous funcS tions in D Γg taking “good” boundary values on the set Γg and having in S g D Γ continuous derivatives of order up to l which are in the sense of Hölder continuous with exponent α. For these elements a finite value is taken by the norm Ã ! X ∂k g g g k |u|l+α = |u|l + Hα d u . ∂xk11 . . . ∂xknn k +...+k =k 1

k=l

n

Here ¯ ¯ ∂k ¯ k |u|gl = ¯d ¯ ∂xk11 . . . ∂xnkn k=0 k1 +...+kn =k l X

X

p+α Hαg (dp u) = sup dx, x0 x, x0 ∈D

dx = dgx = r(x, Γ \ Γg ),

¯g ¯ ¯ u¯ , ¯

|u(x) − u(x0 )| , |x − x0 |α dx, x0 =

dgx, x0

|u|g0 = sup |u(x)|, D

0

|dp v|g0 = sup |dpx v(x)|, D

= min(dx , dx0 ).

We assume that the coefficients in the equation (3.1a) and the functions f (x) and ϕ(x) are sufficiently smooth D and Γj , and ϕ(x) is continuous on Γ. Under these conditions, the problem solution is sufficiently smooth on D, however, u ∈ / C 2 (D) (see, e.g., [212, 69, 38]). Using a maximum principle we establish the ε-uniform stability of the problem solution ¸ · |u(x)| ≤ M max max |f (x)|, max |ϕ(x)| , x ∈ D. (3.3) D


Γ

Boundary value problems in piecewise-smooth domains

77

Under the condition ask , bs , c, c0 , f ∈ C l+α (D), ϕ ∈ C 1 (Γj ), for j = T 1, . . . , 2n, we have u ∈ C α1 (D) C l+2+α (D), where l ≥ 0 is an integer number and α, α1 ∈ (0, 1). From interior ¡a priori estimates and estimates up to the smooth parts of the boundary Γ derived in the variables ξ for the function u e(ξ), see [69], where u e(ξ) = u(x(ξ)) is the solution of a¢ regular equation in the variables ξ, ξ = ξ(x) and ξs = ε−1 xs , for s = 1, . . . , n , it follows the estimate ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ (3.4) u(x) ¯ k1 ¯ ≤ M ε−k + r−k (x, Γ) , ¯ ∂x1 . . . ∂xknn ¯ x ∈ D,

x∈ / Γ for k > 0, 0 ≤ k ≤ K,

where K = l + 2. Furthermore, under the condition ϕ ∈ C l1 (Γ), for l1 ≥ 2, using a maximum principle, we establish the following estimate for the function u(x): |u(x) − ϕ(x∗ (x))| ≤ M ε−1 r(x, Γ),

x ∈ D,

(3.5)

where x∗ (x) is a point on the boundary Γ nearest to the point x ∈ D. We shall study the behaviour of the solution in a neighborhood of the boundary layer and outside it. For convenience of presentation and formulation of the results obtained, we introduce some sets on which we shall consider auxiliary problems. We assume that the faces Γj = Γj (D),

j = 1, . . . , 2n

(3.6a)

of the parallelepiped D for j = s and j = n + s, where s = 1, . . . , n, are orthogonal to the xs -axis and the face Γs contains the vertex (d∗1 , . . . , d∗n ); here Γj = Γj , for j = 1, . . . , 2n. By D(j) we denote the half-space {x : d∗j < xj < ∞} for j ≤ n and the half-space {x : −∞ < xj−n < d∗j−n } for j > n. We set D(j... r) =

T q

D(q) ,

q = j, . . . , r,

1 ≤ j, . . . , r ≤ 2n.

(3.6b)

The set Γ(j... r) = Γ(D(j... r) ), i.e., the boundary of the set D(j... r) , is formed S by the faces Γq (D(j... r) ), where Γ(D(j... r) ) = q Γq (D(j... r) ), for q = j, . . . , r; the faces Γq (D(j... r) ) are defined similar to those Γq = Γq (D), moreover, Γq ⊆ Γq (D(j... r) ), for q = j, . . . , r. Note that D = D(1) and Γq = Γq (D(1) ), for q = 1, . . . , 2n, in the case when D(1) = D(1... 2n) . The problem solution on the set D is decomposed into the sum u(x) = U (x) + V (x),

x ∈ D,

(3.7)

where U (x) and V (x) are the regular and singular components of the solution. The function U (x), x ∈ D, is the restriction of the function U e (x), x ∈ D e ,


78


to the set D, i.e., U (x) = U e (x), x ∈ D; here, for simplicity, as the set D e we choose an n-dimensional space IRn . The function U e (x) is the bounded solution of the following problem on the unbounded domain D e : Le U e (x) = f e (x),

x ∈ De .

(3.8)

The domain De is an extension of the domain D beyond the boundary Γ; the coefficients and right-hand side f e of the equation (3.8) are smooth continuations of those in the equation (3.1a) with preserving their properties. Let L e = ε2 4 − c, and let the function f e (x) be vanished beyond an m-neighborhood of the set D. The function V (x), x ∈ D, is the solution of the problem L V (x) = 0, x ∈ D;

V (x) = ϕV (x), x ∈ Γ,

where ϕV (x) = ϕ(x) − U (x), x ∈ Γ. Let us estimate the functions U (x), V (x) and their derivatives. For the data of the problem (3.1), (3.2), let the condition ask , bs , c, c0 , f ∈ C l+α (D) be fulfilled for l ≥ 2 and α ∈ (0, 1). Then U ∈ C l+α (D). Furthermore, let the condition ϕ ∈ C l+α (Γj ) hold for j = 1, . . . , 2n. Then T V ∈ C α1 (D) C l+α (D), α1 ∈ (0, 1), moreover, ϕV ∈ C l+α (Γj ), i.e., Γj = Γgj , for j = 1, . . . , 2n, in the case of the function V (x), x ∈ D. For the function U (x), the estimate (2.11) holds: ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ U (x) (3.9) ¯ k1 ¯ ≤ M 1 + ε2−k , x ∈ D, 0 ≤ k ≤ K. k n ¯ ∂x1 . . . ∂xn ¯ We write the singular part V (x) as the sum of functions V (x) = V0 (x) + v(x),

x ∈ D.

(3.10)

Here the functions V0 (x) and v(x), i.e., the main term of the singular part of the solution and the remainder term, are solutions of homogeneous equations. We represent the function V0 (x) as the sum of functions of type to regular and corner boundary layers: X X V0 (x) = V(j) (x) + V(j... r) (x), x ∈ D, (3.11) j=1,...,2n

j,..., r=1,...,2n j 1. The functions ϕe(j... r) (x) are q {Γq (D sufficiently smooth on the faces D(1) , and on the faces Γq ⊂ Γ they satisfy the condition U (x) + ϕe(j) (x) = ϕ(x), x ∈ Γq , q = j, |j . . . r| = 1; X X U (x) + V(q) (x) + V(q... k) (x) + ϕe(j... r) (x) = ϕ(x), q=j,..., r

x∈

S

q Γq ,

q,..., k=j,..., r q 0, k ≤ l.

Here the values k p = k p (ks )P for p = j, . . . , r, s = 1, . . . , n, where j, . . . , r = n 1, . . . , 2n, |j . . . r| ≥ 1, k = s=1 ks , are defined by the following relations. In the case when ks 6= 0 and either p = s or p = n + s, we set k p (ks ) = ks ; otherwise, we set k p (ks ) = 0. The value k p = k p (ks ) is the order of the derivative along the xs -axis orthogonal to the face Γp , p = j, . . . , r for |j . . . r| ≥ 1. For the function vV(j...r) (x), using majorant functions, we find ¯ ¯v V

³ ¯ ¡ ¢´ ¯ ≤ M ε exp − mε−1 r x, T (x) Γ (D ) , q (j... r) q=j,...,r (j...r)

x ∈ D(j... r) ;

for the derivatives we have the estimate ¯ ¯ h ¯ ¯ ¡ ¢i ∂k ¯ ¯ ≤ M ε ε−k + r−k x, Γ(D(j... r) ) × v (x) V ¯ ∂xk1 . . . ∂xkn (j...r) ¯ n 1 ³ ¡ T ¢´ × exp − mε−1 r x, q=j,...,r Γq (D(j... r) ) , x ∈ D(j... r) , x ∈ / Γ(D(j... r) ) for k > 0, 0 ≤ k ≤ K, K = l.



81

Taking into account the estimates of the derivatives of the functions e V(j... r)0 (x) and vV(j...r) (x), we obtain ¯ ¯ n ¯ ¯ ¡ ¢ ∂k e −(kj +...+kr ) ¯ ¯ V (x) + r−(kj +...+kr ) x, Γ(D(j... r) ) + ¯ ∂xk1 . . . ∂xkn (j... r) ¯ ≤ M ε 1

n

³ £ ¤o ¡ T ¢´ + ε ε−k + r−k (x, Γ(D(j... r) )) exp − mε−1 r x, q=j,...,r Γq (D(j... r) ) , x ∈ D(j... r) , x ∈ / Γ(D(j... r) ) for k > 0, 0 ≤ k ≤ K, where K = l and k p = k p(3.16) (ks ) with p = j, . . . , r. For the function v(x) from (3.10) one has the estimate ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ ≤ M ε2 ε−k + r−k (x, Γ) , v(x) ¯ ∂xk1 . . . ∂xkn ¯ n 1 x ∈ D, x ∈ / Γ for k > 0,

(3.17)

(3.18)

0 ≤ k ≤ K.

Next, we consider the problem solution u(x) as the decomposition u(x) = U(0) (x) + V0 (x),

x ∈ D,

(3.19)

where U(0) (x) = U (x) + v(x). For the function u(x) and its components U(0) (x) and V(j... r) (x) in the representations (3.19), (3.11) we have the estimates ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ u(x) (3.20) ¯ k1 ¯ ≤ M ε−k + r−k (x, Γ) , k n ¯ ∂x1 . . . ∂xn ¯ ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ U (x) ¯ k1 ¯ ≤ M 1 + ε2−k + ε2 r−k (x, Γ) , (0) ¯ ∂x1 . . . ∂xknn ¯ ¯ ¯ ¯ ¯ n ∂k ¯ ¯ V (x) ¯ k1 ¯ ≤ M ε−(kj +...+kr ) + r−(kj +...+kr ) (x, Γ) + (j... r) k n ¯ ∂x1 . . . ∂xn ¯ ³ ´ £ ¤o T + ε ε−k + r−k (x, Γ) exp − mε−1 r(x, q=j,...,r Γq ) , x ∈ D,

x∈ / Γ for k > 0,

|u(x) − ϕ(x∗ (x))| ≤ M ε−1 r(x, Γ), j, . . . , r = 1, . . . , 2n,

x ∈ D;

1 ≤ |j . . . r| ≤ n,

0 ≤ k ≤ K,

where x∗ (x) = x∗(3.5) (x). In (3.20), k p = k p(3.16) , p = j, . . . , r, m = m(3.13) . The following theorem holds.


82


Theorem 3.2.1 Let ask , bs , c, c0 , f ∈ C l+α (D), for s, k = 1, . . . , n, ϕ ∈ C(Γ), ϕ ∈ C l+α (Γj ) for j = 1, . . . , 2n, l ≥ K, K ≥ 2, α ∈ (0, 1). Then for the function u(x) and its components U(0) (x) and V(j... r) (x) in the representations (3.19), (3.11) of the solution to the boundary value problem (3.1), (3.2), the estimates (3.20) hold. In some problems for the function u(x) one has the inclusion u ∈ C K+α (D),

α ∈ (0, 1),

(3.21)

if the data of the problem (3.1), (3.2) on the set Γc (D) satisfy special conditions. Compatibility conditions on the set Γc (D) are special conditions imposed on the coefficients in the equation, its right-hand side and the boundary conditions in a neighborhood of the set Γc (D) that guarantee membership of the solution to a class of functions having required smoothness in the neighborhood of the set Γc (D), (see, e.g., [212, 213, 66]). The membership (3.21) allows us to obtain estimates of derivatives for the components in (3.19), (3.11) on the set D. The smoothness of the data of the problem on the parallelepiped does not, in general, imply the smoothness of the solution on the set D (see, e.g., [38]). In [212, 213] for Poisson equation ∆u = f (x) at n = 2, 3,¡imposed compatibility conditions are given for the functions f (x) and ϕ(x) here u(x) = ϕ(x), ¢ x ∈ Γ on the set Γc that ensure the smoothness u ∈ C 4+α (D). For the problem (3.1), where L ≡ ε2 4 − c0 (x),

x ∈ D,

r(x, Γc ) ≤ m ε,

(3.22)

on the basis of results from [212, 213], it is not difficult to write down compatibility conditions on the set Γc (D) under which one has (3.21) (in the case n = 2 see, e.g., [180]). Applying estimates of up to the ¡ a priori ¡ ¢ the solution ¢ smooth parts of the boundary for r x, Γc (D(j... r) ) ¡ ≥ m¡1 ε and estimates ¢ of [212, 213] in a neighborhood of the set Γc (D(j... r) ) for r x, Γc (D(j... r) ) ≤ ¢ e m2 ε, for m2 > m1 , we justify that for the functions V(j...r) (x) and v(x) for n = 2, 3, the following estimates are valid: ¯ ¯ ¯ ¯ h i ∂k ¯ ¯ e −(kj +...+kr ) 2−k V (x) ≤ M ε + ε × ¯ k1 ¯ ¯ ∂x1 . . . ∂xknn (j... r) ¯ ¡ ¢ T × exp − m ε−1 r(x, q=j,...,r Γq ) , x ∈ D(j... r) ; ¯ ¯ ¯ ¯ ∂k ¯ ¯ v(x) ¯ k1 ¯ ≤ M ε2−k , x ∈ D, 0 ≤ k ≤ K. ¯ ∂x1 . . . ∂xknn ¯

(3.23)

(3.24)

For n ≤ 3 in the case of the condition (3.22) if (3.21) holds, then the


Boundary value problems in piecewise-smooth domains functions u(x), U(0) (x) , and V(j...r) (x) satisfy the estimates ¯ ¯ ¯ ¯ ∂k ¯ ¯ u(x) ¯ k1 ¯ ≤ M ε−k , ¯ ∂x1 . . . ∂xknn ¯ ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ U (x) ¯ k1 ¯ ≤ M 1 + ε2−k , (0) ¯ ∂x1 . . . ∂xknn ¯ ¯ ¯ ¯ ¯ h i ∂k ¯ ¯ −(kj +...+kr ) 2−k ≤ M ε V (x) + ε × ¯ k1 ¯ (j... r) k ¯ ∂x1 . . . ∂xnn ¯ ¡ ¢ T × exp − m ε−1 r(x, q=j,...,r Γq ) , j, . . . , r = 1, . . . , 2n,

83

(3.25)

x ∈ D;

1 ≤ |j . . . r| ≤ n,

0 ≤ k ≤ K.

In (3.25) one has k p = k p(3.16) , where p = j, . . . , r, and m = m(3.13) . The following theorem holds. Theorem 3.2.2 Let n ≤ 3, the hypotheses of Theorem 3.2.1 hold, and also the condition (3.22) and the membership (3.21) be satisfied. Then for the function u(x) and its components U(0) (x) and V(j... r) (x) in the representations (3.19), (3.11) of the solution to the boundary value problem (3.1), (3.2), the estimates (3.25) hold. Let us discuss estimates in the case of the problem (3.1) in a domain with a piecewise-smooth boundary having curvilinear faces. It is convenient to write the estimates of the solution in the case of the problem (3.1), (3.2) in a form different from the estimates (3.20), (3.25). We write the solution u(x) of the problem (3.1), (3.2) in an σ-neighborhood of the boundary Γ, where σ is sufficiently smooth, as the sum u(x) = U(0) (x) + V0 (x),

x ∈ D,

r(x, Γ) ≤ σ,

(3.26)

where U(0) (x) = U0 (x) + v(x). For the function u(x) considered outside the σ-neighborhood of the boundary Γ and its components U(0) (x) and V(j... r) (x) in the representations (3.19), (3.11) considered only in the σ-neighborhood of the boundary Γ, by virtue of (3.20), we have the estimates ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ u(x) (3.27a) ¯ k1 ¯ ≤ M 1 + ε2−k + ε2 σ −k , r(x, Γ) ≥ σ; ¯ ∂x1 . . . ∂xknn ¯ ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ 2−k 2 −k U (x) ≤ M 1 + ε + ε r (x, Γ) , (3.27b) ¯ k1 ¯ (0) k ¯ ∂x1 . . . ∂xnn ¯


84

Elliptic reaction-diffusion equations ¯ ¯ ¯ ¯ n ∂k ¯ ¯ V (x) ≤ M ε−(kj +...+kr ) + ¯ k1 ¯ ¯ ∂x1 . . . ∂xknn (j... r) ¯ £ ¤o + r−(kj +...+kr ) (x, Γ) + ε ε−k + r−k (x, Γ) × ³ ¡ T ¢´ × exp − mε−1 r x, q=j,...,r Γq , |u(x) − ϕ(x∗ (x))| ≤ M ε−1 r(x, Γ), x ∈ D,

j, . . . , r = 1, . . . , 2n,

(3.27c)

x∈ / Γ for k > 0,

r(x, Γ) ≤ σ;

(3.27d)

1 ≤ |j . . . r| ≤ n,

0 ≤ k ≤ K.

For n ≤ 3 in the case of the condition (3.22), by virtue of (3.25), function u(x), U(0) (x) and V(j...r) (x) the estimates ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ u(x) ¯ k1 ¯ ≤ M 1 + ε2−k , r(x, Γ) ≥ σ; k n ¯ ∂x1 . . . ∂xn ¯ ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ 2−k U (x) ≤ M 1 + ε , ¯ k1 ¯ (0) k ¯ ∂x1 . . . ∂xnn ¯ ¯ ¯ ¯ ¯ h i ∂k ¯ ¯ −(kj +...+kr ) 1−k V (x) ≤ M ε + ε × ¯ k1 ¯ (j... r) ¯ ∂x1 . . . ∂xknn ¯ h ³ ¡ T ¢í × exp − mε−1 r x, q=j,...,r Γq , r(x, Γ) ≤ σ; x ∈ D,

j, . . . , r = 1, . . . , 2n,

1 ≤ |j . . . r| ≤ n,

for the

(3.28a)

(3.28b)

(3.28c)

0≤k≤K

are valid. In (3.27), (3.28) one has k p = k p(3.16) , where p = j, . . . , r, and m = m(3.13) . Estimates of type (3.27), (3.28) take place also in the case of the problem (3.1) in a domain with a piecewise-smooth boundary, in particular, in the domain D, which is an n-dimensional parallelepiped with curvilinear faces, when the hypotheses of Theorem 3.2.1, 3.2.2 are fulfilled if Γj ∈ C l+α , for j = 1, . . . , 2n. Outside the σ-neighborhood of the boundary Γ, the estimates (3.27a) and (3.28a) hold for the solutions of the boundary value problem. We cover the σ-neighborhood of the boundary Γ by finite set of subdomains. On each such subdomain we introduce a new coordinate system x0s = x0s (x), for s = 1, . . . , n, in which the curvilinear faces become pieces of planes orthogonal to the coordinate axes. In a neighborhood of the set Γ(j...r) , the variables x0s = x0s (x), for s = j, . . . , r are the distances from the point x ∈ D, with r(x, Γ) ≤ σ, to the faces Γj0 , . . . , Γr0 generating Γ(j...r) . Similar to the constructions on the rectangular parallelepiped, we shall derive an estimate of the problem solution in variables x0 , x0 = (x01 , . . . , x0n ) on subdomains from the σ-neighborhood of the boundary Γ. In a neighborhood of the boundary, the problem solution has a representation similar to (3.22). Let in the original



85

variables the membership (3.21) hold and in the new variables the condition (3.22) be satisfied. Now we shall preserve previous notations on the partition subdomains from the σ-neighborhood of the domain boundary for the new variables and also for the domain and its boundary, for the solution of the boundary value problem and its regular and singular components. Let the functions U(0) (x) and V(j...r) (x), which are the regular and singular parts of the solution, be determined as above. For these functions, the estimates (3.28b) and (3.28c), respectively, are valid in the case of additional conditions (3.21), (3.22), or the estimates (3.27b) and (3.27c) when these conditions are violated, moreover, the estimate (3.27d) is fulfilled for the solution of the boundary value problem. In (3.27c), (3.28c) in the case of the function V(j... r) (x) the value k p = k p (ks ) is the order of the derivative along the xs -axis orthogonal to the face Γp , for p = j, . . . , r, from the σ-neighborhood of the domain boundary. The constant m in these estimates is chosen to satisfy the condition m < m00 , where £ 0 ¤1/2 , m00 = (a00 )−1 c0 )

(3.29)

0

a00 and c0 are the constant from the ellipticity condition and the constant from the estimate of the reaction term in the differential operator written in the new variables in the σ-neighborhood of the boundary Γ. Theorem 3.2.3 For the data of the boundary value problem (3.1) in the domain D, i.e., an n-dimensional parallelepiped with curvilinear faces, let the hypotheses of Theorem 3.2.1 be fulfilled and also the condition Γj ∈ C l+α , for j = 1, . . . 2n. Then for the solution u(x) and its components in the representation (3.26), the estimates (3.27), (3.29) hold; in the representation (3.26) and in the estimates (3.27b, c, d) the variable x is new. But if for n ≤ 3 under the additional condition (3.21), the condition (3.22) holds, then the estimates (3.28), (3.29) are satisfied; in the relations (3.26), (3.22) and in the estimates (3.28b, c) the variable x is new.

3.3

Sufficient conditions for ε-uniform convergence of a difference scheme for the problem on a parallelepiped

The solution of the boundary value problem (3.1), (3.2) does not, in general, belong to C 2 (D) and that does not allow us immediately to use results of Chapter 2 for the construction and justification of ε-uniformly convergent finite difference schemes. We shall study finite difference schemes on grids condensing in a neighborhood of the boundary layers and find sufficient conditions for ε-uniform convergence of the difference schemes. Assume that the


86


solution of the problem (3.1) satisfies the condition T

u ∈ C(D) C 3 (D)

(3.30)

and the estimates (3.20) of Theorem 3.2.1. On the parallelepiped D(3.2) we introduce the rectangular grid Dh = ω 1 × . . . × ω n ,

(3.31)

where ω s , for s = 1, . . . , n is a mesh on the interval [d∗s , d∗s ] at the xs -axis. We set h = max hs , hs = max his , and N = min Ns , where Ns + 1 is the s

s

i

number of nodes in the mesh ω s , for s = 1, . . . , n; assume that the condition h ≤ M N −1 holds. On the grid Dh(3.31) for the problem (3.1) we consider a difference scheme Λz(x) = f h (x), x ∈ Dh ,

z(x) = ϕh (x), x ∈ Γh ,

(3.32)

and find sufficient conditions for its ε-uniform convergence. In the case when the problem solution is sufficiently smooth on D for each fixed value of the parameter, the conditions sufficient for the ε-uniform convergence of the scheme (3.32), (3.31) (and for its stability) are given by Theorem 2.3.2 that follows from Theorem 2.3.1 and Lemma 2.3.3 (Lemma 2.3.3 provides the stability conditions). If, however, u ∈ C 2+α (D), but u 6∈ C 2 (D) (ε-uniformly), then the difference operator Λ from the scheme (3.32), (3.31) approximating the operator L on the problem solution at each point x ∈ Dh |(L − Λ)u(x)| ≤ M µ (h(x), x) ,

x ∈ Dh ,

in general, does not approximate the operator L uniformly on D. Here µ(h(x), x) → 0 as h(x) → 0; h(x) is the maximal mesh step in the stencil of the operator Λ centered at the point x (ε is fixed). Therefore, Theorems 2.3.1 and 2.3.2 and Lemma 2.3.3 that establish the ε-uniform convergence of the scheme turn out to be directly inapplicable for study of the ε-uniform convergence of the scheme (3.32), (3.31). Thus, a problem arises to justify ε-uniform convergence of the schemes in the case of boundary value problems whose solutions belong to the class C 2+α (D) but not belong to C 2 (D). Let us discuss some concepts on the basis of which we shall formulate conditions that allow us to justify ε-uniform convergence of the schemes constructed. Let u 6∈ C 2 (D), and let for the solution of the problem the estimates (3.20) hold. In that case, the estimates of the problem solution considered on the set D(ρ) from D, where © ª S D(ρ) = D(ρ) Γ(D(ρ)), D(ρ) = x : r(x, Γ) > ρ , (3.33)



87

are the same as the estimates (3.25) in the case when u ∈ C 3 (D) up to a multiplying factor (dependent on ρ). For example, by virtue of (3.20), for the derivative (∂ 3 /∂x31 ) u(x) on D(ρ), we have ¯ 3 ¯ ¯ ∂ ¯ ¯ ¯ ρ) ε−3 , x ∈ D(ρ), ¯ ∂x3 u(x)¯ ≤ M (e 1 where M (e ρ) = M(3.20) (1 + ρe 3 ) and ρe = ε−1 ρ is the normalized distance to the boundary. If on the boundary Γ(D(ρ)) for not too small ρ the solution of the difference problem (3.32), (3.31) for N → ∞ approximates the solution of the problem (3.1) ε-uniformly, one would use the conditions of Theorems 2.3.1 and 2.3.2 and Lemma 2.3.3 as sufficient conditions for ε-uniform convergence of the difference scheme (3.32), (3.31) on D(ρ). To estimate an error of the solution on the set D \ D(ρ), we shall use the following. Assume that the solution of the problem (3.32), (3.31) satisfies the estimate similar to (3.5): |z(x) − z(x∗ (x))| ≤ M ε−1 r(x, Γ), x ∈ Dh , ¡ where x∗ (x) = x∗(3.5) (x) in the case of monotone schemes the estimate is ¢ derived similar to (3.5) . Then for the error in the problem solution we have |u(x) − z(x)| ≤ M ε−1 r(x, Γ),

x ∈ Dh ,

and, on the set D \ D(ρ) that is the ρ-neighborhood of the boundary Γ and also on the boundary Γ(D(ρ)), we obtain the estimate |u(x) − z(x)| ≤ M ρe, x ∈ D \ D(ρ), x ∈ Dh . (3.34) ¡ ¢ By choosing the value ρe the value ρ that satisfies the condition ρe = ρe(N ), where ρe(N ) → 0 as N → ∞, it is possible to satisfy the conditions of convergence of the scheme on the set D(ρ) that follow from Theorems 2.3.1 and 2.3.2 and Lemma 2.3.3 and, furthermore, to ensure the ε-uniform convergence of the scheme on D \ D(ρ) by virtue of the estimate (3.34) that leads to the ε-uniform convergence on Dh . Thus, the ε-uniform convergence principle: ε-uniform convergence of a finite difference scheme follows from ε-uniform approximation of the boundary value problem by an ε-uniformly stable finite difference scheme transforms into the principle: ε-uniform convergence of a finite difference scheme follows from (i) continuity in the variable ρe of the solutions to the boundary value problem and to the difference scheme on the domain boundary


88

Elliptic reaction-diffusion equations and (ii) ε-uniform approximation and ε-uniform stability of the difference scheme outside an ρ-neighborhood of the boundary with constants in the error bounds of the approximation and stability, which depend on ρe.

Keeping in mind such an approach, we obtain conditions under which the finite difference scheme converges ε-uniformly. Let the functions f h (x), x ∈ Dh , and ϕh (x), x ∈ Γh , approximate the functions f (x), x ∈ D, and ϕ(x), x ∈ Γ, ε-uniformly on the sets Dh and Γh , respectively: |f (x) − f h (x)| ≤ M λ(N ),

x ∈ Dh ;

(3.35)

|ϕ(x) − ϕh (x)| ≤ M λ(N ),

x ∈ Γh ,

(3.36)

where λ(N ) − → 0 as N → ∞. ε

We say that the operator Λ from the scheme (3.32), (3.31) acting on the solution of the ¡ problem (3.1) approximates the ¢ operator L(3.1a) ε-uniformly on the set D(ρ) or ε-uniformly inside the set D if the following inequality holds: |(L − Λ) u(x)| ≤ M ψ(e ρ) λ(N ),

x ∈ Dh

T

D(ρ),

(3.37)

where M ψ(e ρ) is a constant in the error bound of the approximation; in general, ψ(e ρ) → ∞ as ρe → 0, and ψ is independent of ε. Let us define stability of the finite difference scheme (3.32), (3.31) inside the set D. Let Dh1 be a subset of Dh . We call a point x ∈ Dh the nearest node to the set Dh1 if x 6∈ Dh1 and this point x is a mesh node in the stencil of the difference scheme (3.32), (3.31) with the centre at any point ξ belonging to Dh1 . We denote the set of these nodes by S(Dh1 ),

(3.38)

and we call it the boundary of the subset Dh1 . We say that the difference scheme or, the¡scheme (3.32), (3.31) ε-uniformly stable ¢ on the set D(3.33) (ρ) uniformly in ρ or,ε-uniformly stable inside the set D if for the function w(x), T S T x ∈ {Dh D(ρ)} S(Dh D(ρ)) that satisfies the relations Λw(x) = fwh (x), x ∈ Dh

T

D(ρ),

w(x) = ϕhw (x), x ∈ S(Dh the estimate holds |w(x)| ≤ M max

h max T

Dh

D(ρ)

|fwh (x)|,

T

max T

S(Dh

n o ³ ´ T S T x ∈ Dh D(ρ) S Dh D(ρ) ,


D(ρ)),

D(ρ))

i |ϕhw (x)| ,

(3.39)


89

where M is independent of ρ. We assume that the error bound of the difference scheme satisfies the following condition of convergence in the nearest neighborhood of the boundary ¡ ¢ |u(x) − z(x)| ≤ M λ0 ε−1 r(x, Γ) ,

x ∈ Dh ,

(3.40)

where λ0 (ξ) − → 0 as ξ → 0, i.e., the error bound of the difference scheme for ε

ε−1 r(x, Γ) → 0 converges to zero ε-uniformly with the estimate (3.40). The following statements are valid. Theorem 3.3.1 For the difference scheme (3.32), (3.31) let the following conditions be fulfilled: a) the functions f h (x) and ϕh (x) approximate the functions f (x) and ϕ(x) ε-uniformly (with the estimates (3.35), (3.36)); the operator Λ(3.32) acting on the solution of the boundary value problem approximates the operator L(3.1a) ε-uniformly inside the set D (with the estimate (3.37)); b) the difference scheme (3.32), (3.31) is ε-uniformly stable inside the set D (with the estimate (3.39)); c)the error bound of the solution to the difference scheme satisfies the condition (3.40). Then the difference scheme (3.32), (3.31) converges ε-uniformly at the rate O (λ0 (ρ1 (λ(N )))) as N → ∞: ³ ¡ ¢´ |u(x) − z(x)| ≤ M λ0 ρ1 λ(N ) , x ∈ Dh , where ρ1 (λ) is the solution of the equation λ ψ(ρ) = λ0 (ρ). Lemma 3.3.1 Let the operator Λ from the difference scheme (3.32), (3.31) be monotone and approximate the operator L ε-uniformly on smooth functions (satisfying the condition (2.25)). Then the scheme (3.32), (3.31) is εuniformly stable inside the set D (and on the set D). Theorem 3.3.2 The scheme (3.32), (3.31) converges ε-uniformly at the rate O (λ0 (ρ1 (λ(N )))) if the conditions a), c) of Theorem 3.3.1 and the hypotheses of Lemma 3.3.1 are fulfilled.

3.4

A difference scheme for the boundary value problem on a parallelepiped

We construct a difference scheme for the problem (3.1) on the parallelepiped D(3.2) . We assume that the membership (3.30) and either the estimates (3.25) or (3.20) are fulfilled. Unlike problem (2.2), classical difference approximations to each of the terms of the equation (3.1a) that contain the second-order derivatives do not


90


approximate these terms ε-uniformly everywhere in the domain if at least one of the meshes ω s , for s = 1, . . . , n (generating the grid Dh(3.29) ), is a mesh with an arbitrary distribution of its nodes. The behaviour of the solution of the boundary value problem (3.1) in neighborhoods of the smooth parts of the boundary Γ is similar to the behaviour of the solution of (2.2). Therefore, when constructing difference schemes for (3.1), we will use special meshes condensing in a neighborhood of the boundary layer. Let us consider the error of approximation of the operators L(sk) ≡ ε2

∂2 , ∂xs ∂xk

s, k = 1, . . . , n

by difference operators on the problem solution. We assume that the estimates (3.25) hold for the problem solution and its components in the representations (3.19), (3.11). In the case of the operator L(ss) , we write the problem solution on the parallelepiped as the sum of two functions (s)

(s)

u(x) = u1 (x) + u2 (x),

x ∈ D,

s = 1, . . . , n,

which satisfy the estimates ¯ k ¯ ¯ ∂ s (s) ¯ ¯ ¯ ≤ M [1 + ε1−ks ], (x) u ¯ ∂xks 1 ¯ s ¯ k ¯ ¯ ∂ s (s) ¯ £ ¡ ¢¤ ¯ ¯ ≤ M ε−ks exp −m ε−1 r(x, Γs S Γn+s ) , u (x) ¯ ∂xks 2 ¯ s x ∈ D,

0 ≤ ks ≤ 3,

similar to the estimates (2.11), (2.14). According to the results of Section 2.1, for the ε-uniform approximation of the operator L(ss) by the operator Λ(ss) ≡ Λ(ss) (L(ss) ) ≡ ε2 δxs xcs ,

s = 1, . . . , n,

(3.41)

on the problem solution, it suffices to use the mesh ω s that ensures the fulfillment of the condition max β (s) (xis ) − →0 i

Here

ε

for

Ns → ∞,

β (s) (xis ) = β(2.33) (xis ; ε, ω s ),

s = 1, . . . , n.

(3.42)

xis ∈ ωs , s = 1, . . . , n.

To approximate the operator L(sr) , for s, r = 1, . . . , n, s 6= r, by the operator ( ) either 2−1 ε2 (δxs xr + δxs xr ) (sr) (sr) (sr) Λ = Λ (L ) = , (3.43) or 2−1 ε2 (δxs xr + δxs xr ) s, r = 1, . . . , n, s 6= r,



91

we write the problem solution as the sum of the functions (sr)

u(x) = u1

(sr)

(x) + u2

(x),

x ∈ D,

s, r = 1, . . . , n,

s 6= r,

which satisfy the estimates ¯ k +k ¯ ¯ ∂ s r ¯ (sr) ¯ ¯ ≤ M [1 + ε1−(ks +kr ) ], u (x) ¯ ∂xks ∂xkr 1 ¯ s r ¯ k +k ¯ ¯ ∂ s r (sr) ¯ £ ¡ ¢¤ ¯ ¯ ≤ M ε−(ks +kr ) min exp −m ε−1 r (x, Γq S Γn+q ) , u (x) ¯ ∂xks ∂xkr 2 ¯ q=s, r s r x ∈ D,

0 ≤ ks + kr ≤ 3,

ks , kr ≤ 2.

(sr)

On the function u1 (x), the operators (∂/∂xq )L(sr) , for q = s, r, are ε(sr) uniformly bounded. On the function u2 (x), the operators (εp ∂ p /∂xpq )L(sr) , for q = s, r and p = 0, 1, are ε-uniformly bounded and when moving away T from the set Γq1 Γq2 , where q1 , q2 =¡s, r, n +s, n¢+r with q1 6= q2 and s 6= r, T they exponentially decrease as ε−1 r x, Γq1 Γq2 → ∞. For the ε-uniform approximation of the operator L(sr) by the operator Λ(sr) on the problem solution, it suffices to ¡use the meshes ω s and ω r that ensure the¢fulfillment of the conditions (3.42) where s is r in the case of the mesh ω r . Lemma 3.4.1 For the solution of the problem and its components in the representations (3.19), (3.11), let the estimates (3.25) hold. Then the operators (sk) Λ(3.41;3.43) , for s, k = 1, . . . , n, defined on the grid Dh(3.31) , where ω s , for s = 1, . . . n, are meshes satisfying the condition (3.42), approximate the operators L(sk) on the problem solution ε-uniformly. Since, in the case of estimates (3.25) the products ε2

∂k u(x), ∂xk11 . . . ∂xknn

x ∈ D,

0≤k≤2

are ε-uniformly bounded, then the products ε2 (∂/∂xs )u(x), x ∈ D, for s = 1, . . . , n, are approximated ε-uniformly by the products of the parameter ε2 and corresponding classical difference derivatives on the grids Dh(3.29) with an arbitrary distribution of nodes. Thus, we approximate the operator L(3.1a) on the grid Dh(3.31) by the operator Λ = Λ(2.46) (L). For the scheme Λz(x) = f (x), x ∈ Dh ,

z(x) = ϕ(x), x ∈ Γh ,

where Λ(3.44) = Λ(2.46) (L), on the piecewise-uniform grid © ª Dh = Dh(3.31) for ω s = ω 1(2.48) (d∗s , d∗s ), s = 1, . . . , n ,


(3.44a)

(3.45)

92


in the case of the estimates (3.25), the conditions similar to those of Theorem 2.3.7 are fulfilled, except for the ε-uniform monotonicity of the operator Λ(3.44) . Note that L∗(2) = L∗(2) (L(3.1) ) = L∗(2)(2.50) (L(2.2) ), Λ∗(2) = Λ∗(2) (Λ(3.44) ) = Λ∗(2)(2.51) (Λ(2.46) ).

(3.44b)

Lemma 3.4.2 Assume that the equation (3.1a) does not contain mixed derivatives, i.e., the following condition ask (x) ≡ 0,

x ∈ D,

s, k = 1, . . . , n,

s 6= k,

(3.46)

is valid. Then the difference scheme (3.44) is ε-uniform monotone on the grid Dh(3.31) and converges ε-uniformly on the grid Dh(3.45) in the case of the estimates (3.25). Lemma 3.4.3 The condition  ε−1 max hsi−1 ,  i ε−1

d∗s <xs ≤d∗s +M ε

max

i ∗ d∗ s −M ε≤xs 0, a sufficiently small neighborhood Q(x∗ ) and n pairs of the functions ρ∗s (x) and ρ∗s (x), for s = 1, . . . , n, for which the relation (2.55) holds. Furthermore, let one have the following condition (similar to (2.65)) h min

Q(x∗ ) r(x,Γ(j...r) )≤m∗

i akk (x) − ν h−1 k −

n X

max

s=1, s6=k s6=j,...,r

Q(x∗ ) r(x,Γ(j...r) )≤m∗

|ask (x)| h−1 s > 0,

(3.50) k, j, . . . , r = 1, . . . , n, k 6= j, . . . , r, j < . . . < r, 1 ≤ |j . . . r| ≤ n − 2. 1

T

1

If on a set D ⊂ D, where Γ(j...r) D 6= ∅, the coefficients of the operator 1 L∗(2) satisfy the relations {(2.55), (3.50)}, where Q(x∗ ) = D , we say that ¡ ¢ 1 the operator L∗(2) has almost canonical form in x1 , . . . , xn on D (by virtue ¡ of (2.55)), and strongly almost canonical form in xs , for s = 1, . . . , n, with ¢ T 1 s 6= j, . . . , r in an m∗ -neighborhood of Γ(j...r) D for 1 ≤ |j . . . r| ≤ n−2 (by virtue of (3.50)), or, briefly, strongly almost canonical form in a neighborhood T 1 of Γ D . The condition (3.48) and the condition the coefficients ask (x) of the operator L∗(2) on the set Q(x∗ ) ⊆ D satisfy the relations {(2.55), (3.50)}

(3.51)

together allow us to construct on the set Q(x∗ ) piecewise-uniform consistent grids and thereby ensure the monotonicity of the operators Λ∗(2)(2.51) and Λ(2.46) . We now give such a mesh. Let the conditions (3.48), (3.51) be fulfilled. We now impose conditions on the parameters of the piecewise-uniform grid Dh(3.45) (piecewise-uniform in all variables) under which the operator Λ∗(2)(2.51) is monotone. Let the step-sizes hs , i.e., maximal step-sizes in the meshes ω s = ω 1(2.48) (d∗s , d∗s ; σs , Ns ),

s = 1, . . . , n

satisfy the conditions {(2.55), (3.50)}. Note that hs = h(2)s , where h(1)s and h(2)s are the step-sizes in the mesh ω s in neighborhoods of the endpoints of the interval [d∗s , d∗s ] and outside it, respectively. Assume that the value σs that defines the transition points in the mesh ω s satisfies the condition σs ≤ m∗(3.50) ,


s = 1, . . . , n,

(3.52a)

Boundary value problems in piecewise-smooth domains and that

ask ∈ C 1 (D),

s, k = 1, . . . , n,

s 6= k.

By virtue of the condition (3.48), one has ¯ ¯ ¡ ¢ ¯ask (x)¯ ≤ M1 σs for r x, Γs S Γs+n ≤ σs , where M1 =

max s,k,r, s6=k, D

95

s 6= k,

¯ ∂ ¯ ¯ ¯ ask (x)¯. ¯ ∂xr

Then ¯ ¯ ¡ ¢ ¯ask (x)¯ h−1 ≤ M1 h−1 σs for r x, Γs S Γs+n ≤ σs , (1)s (1)s

s 6= k.

The inequality X

ν hk−1 −

|ask (x)| h−1 (1)s > 0,

max

Q(x∗ )

s=j,...,r x ∈(d ∗ s/ ∗s +σs , ds −σs )

k, j, . . . , r = 1, . . . , n, k 6= j, . . . , r for r(x, Γ(j...r) ) ≤ m∗ , where Q(x∗ ) is taken from {(2.55), (3.50)}, is satisfied if n X £ ¤ M1 σs h−1 (1)s < ν s=1

min

s=1,...,n

£ −1 ¤ hs ,

i.e., when one has the condition n X

£ ¤ N∗s < M1−1 ν min h−1 , k

s=1

k

k = 1, . . . , n.

(3.52b)

Here N∗s + 1 is the number of nodes in the mesh ω s on each of the intervals −1 [d∗s , d∗s + σs ] and [d∗s − σs , d∗s ]; h(1)s = σs N∗s and hs = h(2)s < (d∗s − −1 d∗s ) (Ns − 2 N∗s ) , with Ns ≤ M N∗s . Let the parameters of the meshes ω s , for s = 1, . . . , n, be chosen to satisfy the conditions (2.55c), (3.52), where hs = h(2)s . Write these meshes ω s in the following form: ω s = {ω s = ω 1(2.48) (d∗s , d∗s ) under the conditions (2.55c), (3.52) }, s = 1, . . . , n.

(3.53)

In the case of (i) the condition (3.48) on the coefficients ask (x) on the boundary Γ, (ii) the condition (3.51) on the coefficients of the operator L∗(2) on the set D, (iii) the conditions (2.55c), (3.52) on the meshes ω s that generate the piecewise-uniform grid Gh(5.45) ,


96


the operators Λ∗(2)(3.44) and Λ(3.44) are ε-uniformly monotone on the grid T Dh(3.45) Q, where Q = Q(x∗ ) is taken from {(2.55), (3.50)}. Thus, in the case of the conditions (3.48), (3.51), the grid ³ ´ Dh = Dh Q(x∗ ), L∗(2) , ω s(3.53) , x1 , . . . , xn = (3.54) n T = Q(x∗ ) Dh(3.45) under the conditions (2.55c), (3.52) o in the case of the relations {(2.55), (3.50)} is consistent on the set Q(x∗ ) in the variables x1 , . . . , xn with the monotonicity condition on the operators Λ∗(2)(3.44) and Λ(3.44) . We consider the difference scheme (3.44) on the grid ³ ´ Dh = Dh(3.54) Q(x∗ ), L∗(2) , ω s(3.53) , x1 , . . . , xn , (3.55) where ω 1(2.48) in the mesh ω s(3.53) (ω 1(2.48) ) is ω 1(2.44) (d∗s , d∗s , m), and m is the constant from the estimates (3.13), (3.20), (3.25) of Theorems 3.2.1 and 3.2.2. In the case of the condition (2.71), one can choose the set Q in (3.55) as Q = D. The following theorems on the convergence are a corollary of Theorem 3.3.2. Theorem 3.4.1 Let the following conditions be satisfied: a) ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n; b) the coefficients ask (x) satisfy the conditions (2.71), (3.48), and {(2.55), (3.50)}, where Q = D; c) for the solution of the problem (3.1) on D(3.2) the estimates of Theorem 3.2.1 (Theorem 3.2.2) hold for K = 3. Then the difference scheme (3.44) on the grid Dh(3.55) , where Q = D, converges ε-uniformly as N → ∞ |u(x) − z(x)| ≤ M (N −1 ln N )ν ,

x ∈ Dh ,

(3.56)

where ν = 1 in the case of the estimates (3.25) and ν = 4−1 in the case of (3.20). Theorem 3.4.2 Let the conditions a), c) of Theorem 3.4.1, and also the condition (3.46) be satisfied. Then the difference scheme (3.44) on the grid Dh(3.31) , where ω s = ω 1(2.44) (d∗s , d∗s ; m) and m = m(3.13) , converges εuniformly as N → ∞ with the estimate (3.56). ¡ Under the hypotheses of Theorems 3.4.1 and 3.4.2, let z(x) = z (2.42) x; ¢ z(·), Dh , x ∈ D, be the interpolant of the function z(x), x ∈ Dh , which is the solution of the difference scheme (3.44) on the grids (3.55) and (3.31), respectively. Then one has the estimate |u(x) − z(x)| ≤ M (N −1 ln N )ν ,


x ∈ D,

ν = ν(3.56) .


3.5

97


When constructing a difference scheme for a boundary value problem in a domain with piecewise-uniform curvilinear boundary in Section 3.6, we shall need piecewise-uniform consistent grids on subdomains, namely, on parallelepipeds; such grids condense in neighborhoods of only some faces of the subdomains-parallelepipeds. Let us now discuss the construction of such consistent grids. Let ³ ´ Dh = Dh Q(x∗ ), L∗(2) , ω s , x1 , . . . , xn be a grid of type Dh(3.54) ; however, this grid differs from the last one such that ¡ ¢ in Dh(3.54) the piecewise-uniform meshes ω s(3.53) d∗s , d∗s along with some u of the xs -axis (or in all directions) are replaced by uniform meshes ¡ ¢ω s = u ∗ ∗ ω s (d∗s , ds ), whose step-size hs equals h(2)s in the mesh ω s(3.53) d∗s , ds ; this grid is consistent in x1 , . . . , xn on Q(x∗ ) with the monotonicity condition for the operators Λ∗(2)(3.44) and Λ(3.44) . The consistency of the grid on Q(x∗ ) is ¡ ¢ preserving also in that case when in Dh(3.54) some meshes ω s(3.53) d∗s¤, d∗s £ that condense in neighborhoods of both ends of the interval d∗s , d∗s are ¡ ¢ ¡ ¢ replaced by the piecewise uniform meshes ω sl d∗s , d∗s or ω sr d∗s , d∗s that condense £ ¤ in a neighborhood, respectively, of the left or right end of the interval d∗s , d∗s ; or by the uniform meshes ω su (d∗s , d∗s ). In this case, it is required ¡ ¢ that the¡ parameters σs , h(1)s , and h(2)s that define the meshes ω sl d∗s , d∗s ¢ ¡ ¢ and ω sr d∗s , d∗s ¡ just as¢the parameter h(2)s = hs be the same as those in the mesh ω s(3.53) d∗s , d∗s . When constructing the consistent grids based on the grid Dh(3.54) , imposed requirements to the coefficients ask (x) and meshes ω s , for s = 1, . . . , n, turn out to be more strong than it is necessary. For example, in the case of a consistent uniform grid ³ ³ ´ ´ u Dh = Dh Q(x∗ ), L∗(2) = Dh Q(x∗ ), L∗(2) , ω su , x1 , . . . , xn , which is the grid (3.54), where ω s is ω su in all directions, it was required that the conditions (3.48) and (3.50) be satisfied. However, for a uniform mesh in order to be consistent, restrictions for ask (x) on the boundary Γ (i.e., the condition (3.48)) and for the parameters of the grid in a neighborhood of the boundary Γ (i.e., the condition (3.50)) are not required. In the case of the domain D(3.2) , we give sufficiently natural conditions imposed on the coefficients ask (x) and on the meshes ω s that guarantee the consistency of the grids Dh that are constructed based on the “elementary b meshes”, i.e., ω u , ω sl , ω sr , and ω sb£; the mesh ¤ ω s condenses in neighborhoods ∗ of both endpoints to the interval d∗s , ds .


98


In the case of the grid Dh , let the piecewise-uniform mesh on the xs axis condense in a neighborhood of the face Γs and/or Γs+n , or this mesh is uniform, for s = 1, . . . , n. We denote by Γe those faces of the boundary Γ in whose the meshes ω s , for s = 1, . . . , n, condense; here ¡ neighborhoods ¢ ω s = ω s d∗s , d∗s . We denote by J ∗ , where ¡ ¢ ¡ ¢ J ∗ = J ∗ D = J ∗ D; Γe

(3.57)

the set of indexes j, related to Γj in whose neighborhoods the grid Dh condenses in the direction orthogonal to the face Γj ; here Γj ⊆ Γe . For example, l ∗ in the case of the uniform © grid Dh ªwe have J = ∅; ∗but if∗ ¡ω s = eω ¢ s , for ∗ s = 1, . . . , n, then J = j = 1, . . . , n . We¡ denote by Js = Js D; Γ , with ¢ s = 1, . . . , 2n, the set of indexes j from J ∗ D , related to the faces Γj ⊆ Γe orthogonal to the xs -axis. We suppose that the coefficients ask (x) of the operator L∗(2) on the boundary Γ satisfy the following condition, which is similar to (3.48): ask (x) = 0, x ∈ Γj , j ∈ J ∗ , s = j or k = j, s 6= k, if J ∗ 6= ∅;

(3.58)

restrictions on ask (x) on the boundary Γ are not imposed if J ∗ = ∅. We also assume that the coefficients ask (x), which define the operator L∗(2) , satisfy the condition (2.55), independent of the structure of the set J ∗ . Furthermore, if J ∗ 6= ∅, we assume that the following additional condition holds, which is similar to (3.50): h min ∗

Q(x ) T r(x, ji Γji )≤m∗ ∗ ji ∈J , i=1,...,r


max ∗

|ask (x)| h−1 s > 0,

Q(x ) s=1 T ∗ s6=k,pji r(x, ji Γji )≤m ∗ ji ∈J , i=1,...,r

k, pj1 , . . . , pjr = 1, . . . , n, k 6= pj1 , . . . , pjr , 1 ≤ r ≤ n − 2, T

ji Γji

(3.59)

∗

6= ∅, J 6= ∅,

where pj = pj (j, n), where j = ji ∈ J ∗ ; pj = j for j ≤ n, and pj = j − n for j > n; m∗ and ν are sufficiently small values. When the relations {(2.55), (3.59)} are fulfilled, we say that the operator L∗(2) has local almost canonical form in x1 , . . . , xn on the set D, and the opera∗[p

,...,p

]

jr tor L(2)j1 has strongly local almost canonical form in xs , for s = 1, . . . , n, T with s 6= pj1 , . . . , pjr , in a neighborhood of the set ji Γji , for i = 1, . . . , r,

∗[p

,...,p

]

jr with ji ∈ J ∗ . Here L(2)j1 is the truncated operator, i.e., the operator L∗(2) that does not contain derivatives in the variables xs , where s = pj1 , . . . , pjr .



99

The condition the coefficients ask (x) of the operator L∗(2) on the set D satisfy the relations (3.58), {(2.55), (3.59)} if J ∗ 6= ∅, ∗

∗

(3.60) ∗

and the relations (2.55) if J = ∅, where Q(x ) ⊆ D, J =

∗ (D), J(3.57)

¡ allows¡us¢¢ on the set Q(x∗ ) to construct piecewise-uniform controlled by the set J ∗ D consistent grids that guarantee the monotonicity of the operators Λ∗(2)(3.44) and Λ(3.44) . Let us give such grids. On the set D we construct consistent grids, which are defined by the structure of the set J ∗ . Let the condition (3.60) hold. ¡ u For J ∗ = ∅, the uniform grid Dh generated by the ¢ meshes ω s , for s = 1, . . . , n, whose step-size satisfies the condition (2.55c) ¡ Dh = Dh Q(x∗ ), L∗(2) , ω s = ω su under the condition (2.55c) ¢ in the case of the relations (2.55) for J ∗ = ∅

(3.61)

is consistent on the set Q(x∗ ). ¡ In the case of the condition J ∗ 6= ∅ i.e., at least one of the meshes ω s , ¢ for s = 1, . . . , n, generating Dh , is piecewise uniform , we suppose that for the step-sizes h(2)s of the piecewise uniform meshes ω s the condition (2.55c) ¡ holds, and¢ furthermore, the meshes ω s satisfy the condition which is similar to (3.52a) : σs = σsl ≤ m∗(3.59) for s = pj if pj = j, j ∈ J ∗ ; σs = σsr ≤ m∗(3.59) for s = pj if pj 6= j, j ∈ J ∗ ,

(3.62a)

¢ and also the following condition (which is similar to (3.52b) : X

∗ N∗s < M1−1 ν min[h−1 k ], k = 1, . . . , n, j ∈ J .

s=pj

k

(3.62b)

Here pj = pj(3.59) , for j ∈ J ∗ , and N∗s + 1 is the number of nodes in the mesh ω s on the interval [d∗s , d∗s +σs ] if the mesh ω s condenses in a neighborhood of the left endpoint of the interval [d∗s , d∗s ] , and on the interval [d∗s − σs , d∗s ] if the mesh ω s condenses in a neighborhood of the right endpoint of the interval, ω s condenses in neighborhoods and M1 = M1(3.52) . In the case when £ the∗mesh ¤ of both endpoints of the interval d∗s , ds , we set σsl = σsr = σs for s = pj1 = pj2 , j1 , j2 ∈ J ∗ , j1 6= j2 .


(3.62c)

100


In the¡ grid Dh , we apply piecewise-uniform or¢ uniform meshes along the xs -axis depending on the structure of the set J ∗  l  ω s if Js∗ = {j = s}     under the conditions   r  ∗  ω if Js = {j = s + n} (2.55c), (3.62)   s   b ∗ if J ∗ 6= ∅, ω s = ω s if Js = {j = s, s + n}     ω su under the condition (2.55c) if Js∗ = ∅, J ∗ 6= ∅,     or if J ∗ = ∅; ¡ ¢ ∗ J ∗ = J(3.57) D; Γe ;

(3.63)

the mesh ω s(3.63) is piecewise uniform on at least one of the xs -axes for J ∗ 6= ∅. The mesh ω s(3.63) is constructed based on the mesh ω 1(2.48) similar to ω 1(2.83) ¡ ¢ ω s = ω 1(2.83) d∗s , d∗s ; ω 1(2.48) . Theorem 3.5.1 Let the condition ask ∈ C 1 (D) and also the condition (3.60) be fulfilled. Then the grid ¡ ¢ ¡ Dh = Dh Q(x∗ ), L∗(2) ; J ∗ = Dh Q(x∗ ), L∗(2) , ω s = ω s(3.63) ,

(3.64)

for s = 1, . . . , n, under the conditions (2.55c) and (3.62) subject to {(2.55), (3.59)} if J ∗ 6= ∅, and under the condition (2.55c) subject to (2.55) if J ∗ = ∅; ∗ J ∗ = J(3.57) (D)

¢

is consistent on the set Q(x∗ ) with the monotonicity condition for the operators Λ∗(2)(3.44) and Λ(3.44) . The grid Dh(3.64) is the grid Dh(3.61) for J ∗ = ∅, and the grid Dh(3.55) for J = {j = 1, . . . , 2n}. ∗

i

i

If in (3.64) one has Q(x∗ ) = D , where D is a subset in D, then the grid i Dh(3.64) is consistent on the set D . In the case when i

D = Di

© Γ , Di = x : di∗s < xs < di∗ s , s = 1, . . . , n}

S i

(3.65)

¡ S is a parallelepiped whose faces belong to the coordinate planes Γi = j Γij , ¢ i for j = 1, . . . , 2n , it is convenient to construct a consistent grid on D based i i i∗ on the meshes ω s , for s = 1, . . . , n, introduced on the intervals [d∗s , ds ]. We show such grids (the construction of the grid is similar to that of (2.89)).


Boundary value problems in piecewise-smooth domains i

i

101

i

For the set D ⊆ D, where D = D(3.65) , by j ∗ we denote the index of the T face Γj ⊆ Γe , for which the condition Γi Γj 6= ∅ holds, with Γj = Γj (D), i i e e Γ = Γ(D ), and Γ = Γ(3.57) . We denote by J i∗ , where J i∗ ¡ ¢ ¡ ¢ J i∗ = J i∗ Di = J i∗ Di ; Γe , (3.66) the set of all indexes j ∗ . ¡ ¢ i∗ i∗ i e We denote by J = J D ; Γ , for s = 1, . . . , 2n, the set of the indexes j s s ¡ i¢ i∗ from J D , related to the face Γj ⊆ Γe orthogonal to the xs -axis. We assume that the following condition (similar to (3.60)) holds: i

the coefficients ask (x) of the operator L∗(2) on the set D ⊆ D satisfy the relations (3.58), {(2.55), (3.59)} if J ∗ 6= ∅,

(3.67)

i

i∗ and the relations (2.55) if J ∗ = ∅, where Q(x∗ ) = D , J ∗ = J(3.66) (Di ). i On the intervals [di∗s , di∗ s ], we construct the meshes ω s , s = 1, . . . , n. The i iu i∗ mesh ω s is uniform, i.e., ω s if s, s + n ∈ / J , or is piecewise uniform, i.e., ω sil , ω sir , or ω sib if, respectively, s ∈ J i∗ , s + n ∈ J i∗ , or s, s + n ∈ J i∗ . Let  il  ω s if Js∗ = {j = s}     under the conditions   ir ∗    ω s if Js = {j = s + n}  (2.55c), (3.62) i ω s = ω sib if Js∗ = {j = s, s + n}  if J ∗ 6= ∅, (3.68a)   iu ∗ ∗  ω s under the condition (2.55c) if Js = ∅, J 6= ∅,     or if J ∗ = ∅; ¡ ¢ i∗ J ∗ = J i∗ = J(3.66) D; Γe . ¡ i ¢ i d∗s , di∗ In the case when the mesh ω s(3.68) is constructed based on the s ¡ i ¢ i i∗ mesh ω 1(2.48) d∗s , ds , the mesh ω s is defined by the relation ¡ ¡ ¢ ¢ ω si ≡ ω si di∗s , di∗ = ω 1(2.88) di∗s , di∗ (3.68b) s s ; ω 1(2.48) . ¡ ¢ Assume that the parameters of the meshes ω si = ω si di∗s , di∗ s , for s = 1, . . . , n, i

i∗ satisfy the conditions (2.55c), (3.62), where Q(x∗ ) = D , J ∗ = J(3.66) .

Theorem 3.5.2 In the case of the condition (3.67), the grid (see (3.64)) ¢ i i¡ i¡ i i Dh = Dh Di , L∗(2) ; J i∗ = Dh Q(x∗ ) = D(3.65) , , L∗(2) , ω si = ω s(3.68) s = 1, . . . , n, under the conditions (2.55c), (3.62) subject to {(2.55), (3.59)} if J ∗ 6= ∅, and under the condition (2.55c) subject to (2.55) if J ∗ = ∅, ¢ i i∗ where Q(x∗ ) = D , J ∗ = J(3.66) (Di ) ,


(3.69)

102

Elliptic reaction-diffusion equations i

is consistent on the set D with the monotonicity condition for the operators Λ∗(2)(3.44) and Λ(3.44) . ¡ ¢ iT ∗ For J i∗ = ∅ i.e., for J(3.57) (D) = ∅, or D Γe = ∅ , the grid Dh(3.69) is uniform. Grids similar to (3.64) and(3.69) are used for the construction of difference schemes for boundary value problems in domains with piecewise-uniform boundaries (see, e.g., Subsection 3.6).

3.6

A difference scheme for the boundary value problem in a domain with piecewise-uniform boundary

We construct a difference scheme for the problem (3.1) in the domain D=D

S

Γ

(3.70)

with a piecewise-uniform boundary Γ; the faces Γj of the set D are, in general, curvilinear. For simplicity, let D be an n-dimensional parallelepiped. Assume that on the set D the coefficients of the elliptic part of the differential operator satisfy the condition of pointwise dominance of the diagonal elements (the condition (2.93)). Suppose also that in parts of the domain adjoining to the boundary, the boundary value problem is reduced to problems (in a local coordinate system) on subdomains which are rectangular parallelepipeds. Furthermore, on the boundaries of these subdomains, which are common with the domain boundary, the canonicity condition (similar to (3.48)) of the elliptic part of the operator in the variables orthogonal to the smooth parts of the domain boundary is fulfilled. Boundary value problems having such properties arise, e.g., in the case of diffusion (heat) processes acting in isotropic mediums when parts of the domain boundary in the local coordinate system are pieces of the coordinate planes [19]. To solve the boundary value problem (3.1), (3.70), we use the grid Schwartz method constructed for the problem (2.2), (2.4). The continual Schwartz method (2.91) in the case of the (3.1), (3.70) converges ε-uniformly as the number of iteration grows. [k]

Theorem 3.6.1 The function u(2.91) (x), i.e., the solution of the continual Schwartz method (2.91) for the boundary value problem (3.1), (3.70), converges ε-uniformly to the solution of the boundary value problem at the rate of a geometric progression as k → ∞: |u(x) − u[k] (x)| ≤ M q k ,


x ∈ D,

q < 1 − m.

(3.71)


103

For the problem {(2.91), (3.1), (3.70)} of the continual Schwartz method, we construct a difference scheme of the decomposition method similar to the one {(2.98), (2.97), (2.114)}. iT Subdomains Di ⊂ D in the case of the condition D Γ = ∅ are chosen as rectangular parallelepipeds whose faces are formed by the planes xs = const, for s = 1, . . . , n. Distance between the sets Di and the boundary Γ is independent of ε. i (j...r) Let x∗ ∈ Γ, and let x∗ ∈ Γ(3.49) , where 1 ≤ |j . . . r| ≤ n. On the set D = Q(x∗ ), i.e., a closure of the neighborhood Q(x∗ ), we pass to new variables i X = X(x). We choose the set DX as a rectangular parallelepiped whose faces are formed by the coordinate planes Xs = const, for s = 1, . . . , n; set iT ΓiL = D Γ, with ΓiL 6= ∅. In the new variables, the boundary value problem (3.1), (3.70) considered S on the set Di Γ iL is transformed into the problem (similar to (2.105))

Here

LX U (X) = F (X),

i X ∈ DX ;

U (X) = Φ(X),

ΓiL X.

LX ≡ ε2 L2X + L0X ; L2X ≡

n X

X∈

(3.72)

L0X = −C 0 (X); n

Ask (X)

s,k=1

X ∂2 ∂ + Bs (X) − C(X); ∂Xs ∂Xk s=1 ∂Xs

set L∗X(2) ≡

n X

Ask (X)

s,k=1

∂2 . ∂Xs ∂Xk

In the case when the operator L∗X(2) is canonical in the variables Xs , for T (j...r)

s = j, . . . , r on the boundary ΓiL ΓX , this operator, by an appropriate X linear transformation X 1 = X 1 (X), Xs1 = Xs , for s = j, . . . , r, can be brought to almost canonical form in the variables Xs1 , for s = 1, . . . , n, with i i s 6= j, . . . , r on the set DXX 1 (choosing (it is possible) the sizes of the set D sufficiently small). The coefficients of the second-order derivatives involving T differentiation in Xs1 , for s = j, . . . , r, are equal to zero on {ΓiL Γ(j...r) }X X 1 , ∗ i.e., the operator LXX 1 (2) remains to be canonical in the variables Xs1 , for s = j, . . . , r on the boundary ΓiL XX 1 i

T (j...r)

ΓXX 1 .

Let the subdomains D , for i = 1, . . . , I, form a covering of the set D, i moreover, the subdomains D , for i = 1, . . . , I1 , adjoin to the boundary Γ, 0 i S and the set D = i D , for i = I1 + 1, . . . , I, has no common points with the 0 boundary Γ. Let the distance between the sets D and Γ be independent of ε. We assume that the condition for the width of overlapping to the subdomains i D , for i = 1, . . . , I, holds, i.e., for all i, where i = 1, . . . , I, the minimum of


104


the minimal distance between the boundary of the set D i and the boundary of the union of all sets which do not contain D i (the minimal overlapping of subdomains that cover D) is independent of the value of the parameter ε. i We suppose that the sets D , for i = I1 + 1, . . . , I, which have no common points with the boundary Γ, are rectangular parallelepipeds formed by i coordinate planes in the variables x1 , . . . , xn . The sets D , for i = 1, . . . , I1 , i adjoining to the boundary Γ, in new variables X i = X i (x) are the sets DX i ; these sets are rectangular parallelepipeds formed by coordinate planes in the variables X1i , . . . , Xni . We assume that the coefficients of the operator L∗(2) on the set D satisfy the condition of pointwise dominance of the diagonal elements (see (2.93)): £ ¤ |ask (x)| < (n − 1)−1 αs (x) αk−1 (x) ass (x), αk (x) αs−1 (x) akk (x) , x ∈ D,

s, k = 1, . . . , n,

s 6= k,

(3.73)

moreover, £ ¤ max αs (x) αk−1 (x) ≤ M,

s, k = 1, . . . , n,

s 6= k.

s,k,D

i

Let the diameters of the sets D , for i = I1 + 1, . . . , I, be sufficiently small i such that the following inequality holds on D (see (2.94)): £ ¤ max |ask (x)| < (n − 1)−1 min αsi (αki )−1 ass (x), αki (αsi )−1 akk (x) , i i D

D

s, k = 1, . . . , n,

s 6= k,

i = I1 + 1, . . . , I,

that ensures a transformation of the operator L∗(2) to almost canonical form i

on D . Next, we consider to be fulfilled the condition (see (2.113)): i

the coefficients ask (x) of the operator L∗(2) on the set D satisfy i

the relations (2.55), where Q(x∗ ) = D , for i = I1 + 1, . . . , I.

(3.74)

We assume that the operators L∗X i (2) satisfy the canonicity condition on the boundaries ΓiL Xi

T (j...r)

ΓX i

, for i = 1, . . . , I1 (see (2.107)):

the coefficients Ask (X i ) of the operator L∗X i (2)

(3.75)

T (j...r)

on the boundaries ΓiL ΓX i satisfy the condition Ask (X i ) = 0, Xi © T (j...r) ª where X i ∈ ΓiL ΓX i , for s, k = j, . . . , r, with s 6= k, Xi and j, . . . , r = 1, . . . , n,

for

ΓiL

T (j...r) Γ

6= ∅,

with

i = 1, . . . , I1 .

By virtue of Lemma 3.4.4, this condition is necessary for the ε-uniform monotonicity of the operator ΛX i = Λ(3.44) (LX i ) on grids, which are ε-uniformly



105

© S ª informative for the solution of the subproblem (3.72) on the set Di Γ iL X i T under the condition ΓiL Γ(j...r) 6= ∅. Set ¡ ¢ ¡ i ¢ i∗ J i∗ = J i∗ Di = J(3.66) D ; Γ , i = 1, . . . , I; (3.76) J i∗ = ∅ for i = I1 + 1, . . . , I, and J i∗ 6= ∅ for i = 1, . . . , I1 . We assume that the following condition holds that is similar to (3.67) (see also (2.108)) i

the coefficients Ask (X i ) of the operator L∗X i (2) on the set DX i satisfy the relations (3.75), {(2.55), (3.59)} if J i∗ 6= ∅;

(3.77)

in {(2.55), (3.59)} ask (x), . . . , Q(x∗ ) is Ask (X i ), . . . , {Q(x∗ )}X i i

i∗ i under the condition {Q(x∗ )}X i = DX i , J i∗ = J(3.76) (DX i ), i = 1, . . . , I1 . i

Conditions (3.74) and (3.77) together allow us to construct on the sets D i and DX i consistent grids and thereby the monotonicity of the operators ¡ ¢ ¡ ensure ¢ Λ = Λ(3.44) L and ΛX i = Λ(3.44) LX i , respectively. i

On the set D , for i = I1 + 1, . . . , I, we construct uniform grids similar to (3.69) for J i∗ = ∅ (see (2.114c)): ¢ i i¡ i¡ Dh = Dh Di , L∗(2) ; J i∗ = Dh Q(x∗ ) = Di , L∗(2) , (3.78) iu ω si = ω s(3.68) , for s = 1, . . . , n under the condition (2.55c) i

in the case of the relations (2.55), where Q(x∗ ) = D , ¢ J i∗ = J i∗ (Di ) = ∅ , i = I1 + 1, . . . , I. i

On the sets DX i we construct piecewise-uniform grids similar to (3.69) for J i∗ 6= ∅ (see (2.110)): ³ ´ i i i ∗ i∗ D X i h = D X i h DX = (3.79) i , LX i (2) ; J ³ i i ∗ = DX i h {Q(x∗ )}X i = DX i , LX i (2) ; i i ωX i s = ωX i s(3.68) , for s = 1, . . . , n, under the conditions

(2.55c), (3.62) subject to {(2.55), (3.59)}, where ask (x), . . . , Q(x∗ ), J ∗ is Ask (X i ), . . . , {Q(x∗ )}iX i , J i∗ i

under the condition {Q(x∗ )}iX i = DX i , ´ ¡ i ¢ i∗ J i∗ = J(3.76) DX i ; ΓX i , J i∗ 6= ∅ , i = 1, . . . , I1 .


106


For the subproblem (3.72) we construct a monotone difference scheme (see the scheme (2.111)) ΛX i Z(X i ) = F (X i ),

X i ∈ DXi i h ,

Z(X i ) = Φ(X i ),

iL X i ∈ ΓX ih,

i

(3.80) i = 1, . . . , I1 .

i

Here ΛX i = Λ(2.46) (LX i ) and DX i h = DX i h(3.79) . Returning to the original variables, we obtain the monotone difference scheme (see (2.112)): Λi z ∗ (x) ≡ Λ∗ z ∗ (x) = f (x),

i x ∈ DX i h {X i }−1 ,

z ∗ (x) = ϕ(x),

x ∈ ΓiL X i h {X i }−1 ,

(3.81) i = 1, . . . , I1 .

Here i i DX i h {X i }−1 = D i

T

i

DX i h {X i }−1 ,

i

DX i h {X i }−1 = {DX i h }{X i }−1 ;

iL ΓiL X i h {X i }−1 = Γ

z ∗ (x) = Z(X i (x)),

T

i

DX i h {X i }−1 , i

x ∈ DX i h {X i }−1 .

We approximate the equation (3.1a) on the set Di , for i = I1 + 1, . . . , I, by the monotone difference scheme Λi z(x) ≡ Λ z(x) = f (x),

x ∈ Dhi ,

i = I1 + 1, . . . , I,

(3.82)

i where Dhi = Dh(3.78) and Λ = Λ(2.46) (L). To solve the problem (3.1), (3.70), we use the iterative scheme of the domain decomposition method {(2.98), (2.97), (2.114)}, where (see (2.114e), (2.114f)):  Di i i = 1, . . . , I1 , i X h {X i }−1 (3.81) for Dh = (3.83) i D for i = I + 1, . . . , I; 1 h(3.78)

( Λi =

Λi(3.81)

for

i = 1, . . . , I1 ,

Λi(3.82)

for

i = I1 + 1, . . . , I.

(3.84)

Set N = min Nsi , for s = 1, . . . , n, and i = 1, . . . , I. i, s

The difference scheme {(2.98), (2.97), (2.114), (3.84), (3.83)} is determined by canonical elements, i.e., the operators Λ(2.46) (L) and Λ(2.46) (LX i ) and the i

i

uniform and piecewise-uniform grids Dh(3.78) and DX i h(3.79) , respectively. The following theorem holds. Theorem 3.6.2 For the boundary value problem (3.1), (3.70), let the condition ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, 2, 3, Γj ∈ C 2 , hold. Let



107

i

the coefficients ask (x), x ∈ D, and also ask (x), x ∈ D , for i = I1 + 1, . . . , I, i and Aisk (X i ), X i ∈ DX i , for i = 1, . . . , I1 , satisfy, respectively, the conditions (3.73), and also (3.74) and (3.75), (3.77); and let the solution of the problem satisfy the estimates of Theorem 3.2.3 for K = 3. Then the solution z [k] (x) of the difference scheme {(2.98), (2.97), (2.114), (3.84), (3.83)} converges to the solution of the boundary value problem u(x) ε-uniformly as N, k → ∞: £ ¤ |u(x) − z [k] (x)| ≤ M (N −1 ln N )ν + q k , x ∈ Dh , q ≤ 1 − m, (3.85) where ν = 1 (ν = 4−1 ) in the case when the solution of the boundary value problem and its components satisfy the estimates (3.28) (the estimates (3.27)). The interpolant zˇ[k] (x), x ∈ D, of the solution of the iterative difference scheme, where [k] zˇ[k] (x) = zˇ(2.114g) (x), x ∈ D, k ≥ 1, converges ε-uniformly |u(x) − zˇ[k] (x)| ≤ M [(N −1 ln N )ν + q k ],

x ∈ D,

ν = ν(3.85) .

The number of iterations k f required for the solution of the iterative difference scheme with the estimate f

|u(x) − z [k ] (x)| ≤ M (N −1 ln N )ν , is ε-uniformly bounded


k f ≤ M ln N.

x ∈ Dh ,

ν = ν(3.85) ,

Chapter 4 Generalizations for elliptic reaction-diffusion equations

4.1

Monotonicity of continual and discrete Schwartz methods

Justification of Theorems 2.5.1–2.5.4 on the convergence of the continual and discrete Schwartz method is deduced using their ε-uniform monotonicity. First, we give the monotonicity principle for the continual Schwartz method. i (0) i i Let on the sets D ⊂ D , for i = 1, . . . , I,with D = D(2.91) , that form (1)

i

(k)

a covering of the set D(2.90) , functions wi (x), x ∈ D , where i = 1, . . . , I and k = 1, 2, 3, . . ., be defined; these functions are 2π-periodic in xs for s = (0) 2, . . . , n. Let w[0] (x), x ∈ D , be a prescribed 2π-periodic function in xs , (0)

[k]

for s = 2, . . . , n; the functions wi (x), w[k] (x), x ∈ D , are defined by the relations ¡ ¢ [k] [k] (k) wi (x) = ui(2.91e) x; {wl (·)}, 1 ≤ l ≤ i, w[k−1] (·) , i = 1, . . . , I; [k]

w[k] (x) = wI (x), x ∈ D [k]

(0)

, k = 1, 2, 3, . . . .

[k]

the functions wi (x), x ∈ D , are 2π-periodic in xs , for s = 2, . . . , n. The following lemma holds. i

(k)

Lemma 4.1.1 Let the functions wi (x), x ∈ D , for i = 1, . . . , I and k = (0)

1, 2, 3, . . ., and also the functions w[0] (x), x ∈ D , satisfy the relations ( T ) 0, x ∈ Γi Γ (k) (k) i Lwi (x) ≤ 0, x ∈ D , wi (x) ≥ , x ∈ Γi , [k] i wi−1 (x), x ∈ Γ \Γ [k]

w[0] (x) ≥ 0, w0 (x) = w[k−1] (x), x ∈ D [k]

(0)

, i = 1, . . . , I, k = 1, 2, 3, . . . .

Then the functions w[k] (x) and wi (x), x ∈ D [k]

w[k] (x), wi (x) ≥ 0,

x∈D

(0)

,

(0)

, satisfy the inequalities

i = 1, . . . , I, k = 1, 2, 3, . . . .

The statement of the lemma is obtained by induction in both k and i for a fixed k.


110

Generalizations for elliptic reaction-diffusion equations

To prove Theorem 2.5.1, we consider the functions (k)

i

(k)

vi (x) = u(x) − ui (x), [k]

x∈D ,

[k]

vi (x) = u(x) − ui (x), v [k] (x) = u(x) − u[k] (x), x ∈ D i = 1, . . . , I,

(0)

,

k = 1, 2, 3, . . . .

(0)

(0)

[k]

Here u[k] (x), x ∈ D , is the solution of the problem (2.91); ui (x), x ∈ D , i (k) and ui (x), x ∈ D , are intermediate functions used for the construction of u[k] (x). i (k) [k] We construct majorant functions for the functions vi (x), x ∈ D , vi (x), (0)

v [k] (x), x ∈ D . i i∗ On each set D , for i = 1, . . . , I, one can choose a subset D such that i∗ (1) i∗ the set of all D , for i = 1, . . . , I, covered D(2.90) , moreover, the sets Dσ ⊂ (0)

i∗

i

D , where Dσ are closed σ-neighborhoods of Di∗ , are contained in D , for i∗ iT i = 1, . . . , I. The sets D adjoin to the boundary Γ if D Γ 6= ∅. The value σ is chosen sufficiently small and independent of ε. i Let the functions wi (x), x ∈ D , be solutions of the problems ( T ) 0, x ∈ Γi Γ i , x ∈ Γi , i = 1, . . . , I. L wi (x) = 0, x ∈ D , wi (x) = 1, x ∈ Γi \ Γ For the functions wi (x) one has the estimate wi (x) ≤ q0 ,

i∗

x∈D .

Here q0 = min w0 (ξ), ξ = (ξ1 , . . . , ξn ), ξ

n X

ξi2 ≤ σ 2 ,

i=1

where w0 (ξ) = 1 − m1 + m1 σ −2

n X

¡ ¢−1 ξi2 , m1 = c0 c0 + 2 n σ −2 a0 + 2 n σ −1 b0 ,

i=1

c0 = c0(2.3) ,

a0 = a0(2.3) ,

b0 = max |bs(2.2) (x)|. s, D

(0)

The value q0 satisfies the estimate q0 = 1 − m1 < 1. As majorants, we use the functions (k)

wi (x) = M q0k wi (x),


i

x∈D ,

Monotonicity of continual and discrete Schwartz methods [k]

w[0] (x) = M, wi (x) = M q0k , x ∈ D

(0)

111

, i = 1, . . . , I, k = 1, 2, 3, . . . .

(k)

i

(k)

Applying Lemma 4.1.1 to the functions wi (x) ± vi (x), x ∈ D , w[0] (x) ± (0)

(u(x) − u[0] (x)), x ∈ D , we verify that Theorem 2.5.1 holds where q(2.92) ≤ q0 . Note that for the functions i

wi∗ (x) = 1 − m2 + m2 wi (x),

x∈D ,

m2 ∈ (0, 1)

one has the inequality L wi∗ (x) ≤ −(1 − m2 ) c0 (x) ≤ −m3 ,

i

x∈D ,

i = 1, . . . , I.

Next, we give the monotonicity principle for the discrete Schwartz method {(2.98), (2.97), (2.96)} in the case of the periodic boundary value problem (2.2), (2.4). i

Let on the sets D ⊂ D

(0)

i

i

, D = D(2.91) , for i = 1, . . . , I, i.e., rectan(1)

gular parallelepipeds, that form a covering of the set D(2.90) , rectangular i

grids Dh(2.96) be introduced, on which the operators Λi = Λi(2.46) ε-uniformly i

i

(k)

monotone. Let on the grid sets Dh the functions wi (x), x ∈ Dh , where i = 1, . . . , I and k = 1, 2, 3, . . ., be defined, which are 2π-periodic in xs , for s = 2, . . . , n. Let w[0] (x), x ∈ Dh , where Dh = Dh(2.97) , be an 2π{i}

[k]

periodic function in xs , for s = 2, . . . , n. The functions wi (x), x ∈ Dh , and w[k] ∈ Dh are defined by the relations ¡ ¢ [k] [k] (k) wi (x) = zi(2.98d) (x) x; {wl (·)}, 1 ≤ l ≤ i, w[k−1] (·) [k]

w[k] (x) = wI (x),

x ∈ Dh ,

i = 1, . . . , I,

[k]

The functions w[k] (x), x ∈ Dh , wi (x), x ∈ D s = 2, . . . , n. The following lemma holds.

{i}

{i}

x ∈ Dh ,

k = 1, 2, 3, . . . .

, are 2π-periodic in xs , for

i

Lemma 4.1.2 In the case of grids Dh , for i = 1, . . . , I, let the operators Λi be i (k) ε-uniformly monotone, and let for the grid functions wi (x), x ∈ Dh , where i = 1, . . . , I and k = 1, 2, . . ., as well as for w[0] (x), x ∈ Dh , the following relations be fulfilled: (k) Λi wi ≤ 0, x ∈ Dhi , ( T ) 0, x ∈ Γih Γ (k) wi (x) ≥ , x ∈ Γih , [k] w ˇi−1 (x), x ∈ Γih \Γ [k]

w[0] (x) ≥ 0, w0 (x) = w[k−1] (x), x ∈ Dh ,


i = 1, . . . , I, k = 1, 2, 3, . . . .

112

Generalizations for elliptic reaction-diffusion equations {i}

[k]

Then the functions w[k] (x), x ∈ Dh , wi (x), x ∈ Dh , satisfy the inequalities w[k] (x) ≥ 0, x ∈ Dh ,

{i}

[k]

wi (x) ≥ 0, x ∈ Dh , i = 1, . . . , I, k = 1, 2, 3, . . . .

The estimate (2.100) is derived similar to (2.92). Let us formulate a monotonicity principle for the noniterative difference scheme of the domain decomposition method {(2.99), (2.97), (2.96)}. i i Let on the grid sets Dh = Dh(2.96) , the operators Λi = Λi(2.46) be defined i

which are ε-uniformly monotone. Let on the grids Dh the functions wi0 (x), i x ∈ Dh , for i = 1, . . . , I, be defined, which are 2π-periodic function in xs , for {i}

s = 2, . . . , n. The functions wi (x), x ∈ Dh , and w(x), x ∈ Dh , are defined by the relations ¢ ¡ {i} wi (x) = zi(2.99d) x; {wl0 (·)}, l = 1, . . . , n, l 6= i , x ∈ Dh , i = 1, . . . , I, w(x) = wI (x),

x ∈ Dh .

i

Lemma 4.1.3 In the case of grids Dh , for i = 1, . . . , I, let the operators Λi i be ε-uniformly monotone, and let for the grid functions wi0 (x), x ∈ Dh , for i = 1, . . . , I, as well as for the function w(x), x ∈ Dh , the following relations be fulfilled: Λi wi0 (x) ≤ 0, x ∈ Dhi ; ( T ) 0, x ∈ Γih Γ 0 , wi (x) ≥ w ˇi−1 (x), x ∈ Γih \Γ w ˇ0 (x) = w(x), ˇ

x ∈ Γih ,

i = 1, . . . , I;

x ∈ Dh . {i}

Then the functions w(x), x ∈ Dh , wi (x), x ∈ Dh , satisfy the inequalities w(x) ≥ 0,

x ∈ Dh ,

wi (x) ≥ 0,

{i}

x ∈ Dh ,

i = 1, . . . , I.

Using Lemma 4.1.3, and under the hypotheses of Theorem 2.5.2, we obtain the estimate (2.101). Statement of Theorem 2.5.2 on the convergence of the difference scheme {(2.98), (2.97), (2.96)} to the solution of the boundary value problem (2.2), (2.4) follows from the estimates (2.100), (2.101).

4.2

Approximation of the solution in a bounded subdomain for the problem on a strip

For the boundary value problem on the slab, i.e., the problem (2.2), (2.4) in Section 2.1, difference schemes are constructed that converge ε-uniformly (see,


Approximation of the solution in a subdomain for the problem on a strip 113 e.g., Theorems 2.4.8-2.4.10, 2.5.2). Under the construction of these schemes, we have used grids with infinite numbers of nodes. Schemes on grids with infinite numbers of nodes, i.e., formal difference schemes, are, in general, inapplicable for computations (for example, schemes for the problem (2.2), (2.4) which is not periodic). In the case of problems in unbounded domains, one needs numerical methods used grids with finite numbers of nodes, i.e., constructive numerical methods that are applicable for computations. In particular, for the nonperiodic problem (2.2), (2.4) we are interested in constructive difference schemes that converge ε-uniformly. In this connection, it is suggested to consider the following computational problem, namely, for a boundary value problem on a slab, to find its solution on a prescribed bounded domain (say on a subdomain of our interest). For such a particular problem, it is required to construct grid approximations that use grids with finite numbers of nodes. In the case of a convectiondiffusion problem, a perturbation of the solution, caused by a perturbation of the data of the problem on some distance from the subdomain of our interest, exponentially decreases as the distance grows. Perturbations of the data far from the subdomain of our interest hardly affect the problem solution on this subdomain. We use this property of the solution of the boundary value problem for the construction of a constructive difference scheme for the solution of the particular problem. It is required that the discrete solution on the prescribed subdomain would converge ε-uniformly with growth of the number of nodes in the meshes used. For the problem (2.2) on the slab (2.4), it is required to find its solution on the prescribed bounded set D0 ⊂ D. (4.1) We assume that the coefficients of the operator L satisfy the conditions (2.57) and (2.71) (the diagonal elements in the matrix of the coefficients of the operator L∗(2)(2.50) dominate on the whole set D). To construct continual approximations of the solution on the set D0 , we consider auxiliary boundary value problems on extending bounded subdomains that contain the set D0 . We denote by Q0 , where ¡ ¢ Q0 = Q0 D0 ; η a rectangular η-neighborhood of the set D0 from Rn ; Q0 is a rectangular parallelepiped whose faces belong to the planes xs =const, for s = 1, . . . , n, moreover, ¡ ¢ min r x, Γ(Q0 ) = η; x∈D

here Γ(Q0 ) is the boundary of the set Q0 . Set ¡ ¢ ¡ ¢T η S D = Dη Γη , Dη = Dη D0 ; η = Q0 D0 ; η D(2.4) ; Dη is a rectangular η-neighborhood (of the set D0 ) from D(2.4) .


(4.2)

114

Generalizations for elliptic reaction-diffusion equations η

On the set D we consider the boundary value problem L u0 (x) = f (x), x ∈ Dη , ( T ) ϕ(x), x ∈ Γη Γ ≡ ϕu0 (x), x ∈ Γ, u0 (x) = ϕ∗ (x), x ∈ Γη \Γ

(4.3)

where L = L(2.2) , Γ = Γ(D), and ϕ∗ (x) is a bounded function; ϕu0 (x) is continuous on Γη . The solution of the problem (4.3), (4.2) considered on D0 satisfies the estimate |u(x) − u0 (x)| ≤ M exp(−m η), x ∈ D0 , (4.4) where m is an arbitrary number in the interval (0, m0 ), with ³ −1 ´1/2 m0 = a0 c0 , a0 = a0(2.3) , c0 = c0(2.3) . Thus, the solution of the problem (4.3), (4.2) converges on the set D0 to the solution of the boundary value problem (2.2), (2.4) ε-uniformly as η → ∞. We consider a grid approximation of the problem (4.3), (4.2) that converges ε-uniformly. η On the set D we construct a rectangular grid ¡ ¢ Dh = Dh Dη = ω 1 × . . . ω n , (4.5a) where ω s is, in general, nonuniform mesh on the interval [d∗s , d∗s ] that is η a projection of the parallelepiped D on the xs -axis, for s = 1, . . . , n; let η Ns + 1 be the number of nodes in the mesh ω s , and N η = mins Nsη , moreover, hs ≤ M (Nsη )−1 (d∗s − d∗s ), where hs is the maximal step-size in the mesh ω s , η for s = 1, . . . , n. The value η that defines the set D satisfies the condition η = m−1 ln N1η ,

m = m(4.4) .

(4.5b)

For the problem (4.3), (4.2) we consider a difference scheme Λ z(x) = f (x), x ∈ Dh , ( T ϕ(x), x ∈ Γh Γ, z(x) = ϕ∗ (x), x ∈ Γh \Γ, T

(4.6)

T

where Λ = Λ(2.46) , Dh = Dη Dh , Γh = Γη Dh , and Γ = Γ(D). As the grid (4.5) we shall use the piecewise-uniform grid Dh = Dh(2.58)

T

η

D .

(4.5c)

The fulfillment of the conditions (2.57) and (2.71) ensures a transformation η of the operator L∗(2) to strongly almost canonical form on the set D . On the η

set D we introduce the grid Dh = Dh(2.70)


T

η

D ,

(4.7a)

Approximation of the solution in a subdomain for the problem on a strip 115 η

where Dh(2.70) = Dh(2.70) (Q = D ), on which the operator Λ(4.6) is monotone. Let the parameters of the grid (4.7) satisfy the relations hs ≈ (N1η )−1 ,

N η = min Nsη = N1η , s

η

Nsη ≈ N1η ln N1η ,

s = 1, . . . , n;

N = min Nsη ≈ N1η ln N1η , s

(4.7b)

s = 2, . . . , n.

Taking into account a priori estimates of Theorem 2.2.1, for the solution of the difference scheme (4.6), (4.7) on the set D0h we obtain the ε-uniform estimate |u(x) − z(x)| ≤ M (N1η )−1 ln N1η , x ∈ D0h , (4.8) T

where D0h = Dh(4.7) D0(4.1) . The following theorem holds. Theorem 4.2.1 Let ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n; the coefficients ask (x) satisfy the conditions (2.57), (2.66), (2.71), and let for the solution of the problem (2.2), (2.4) the estimates of Theorem 2.2.1 hold for K = 3. Then the solution of the difference scheme (4.6) on the grid Dh(4.7) , where ω 1(2.48) = ω 1(2.44) , converges ε-uniformly on the set D0h as N η → ∞ with the estimate (4.8). Remark 4.2.1 The number of nodes in the grid Dh(4.7) , on which the discrete ¢ ¡ problem (4.6), (4.7) is solved, is a value of order O (N1η )n lnn−1 N1η . The ¡ ¢ number of nodes on the set D0h is a value of order O (N1η )n . Thus, the logarithmic multiplier lnn−1 N1η is an additional price for solving of the boundary value problem on the set D0 in the case when the boundary value problem (0) is given on the unbounded domain, i.e., on the n-dimensional slab D(2.4) . For the problem (2.2), (2.4), it is required to find its solution on the set (4.1). We assume that the coefficients of the operator L satisfy the conditions (2.57) and (2.93) (the diagonal elements in the matrix of the coefficients of the operator L∗(2)(2.50) dominate on D pointwisely); periodicity of the problem (2.2), (2.4) is not assumed. When constructing a difference scheme for the problem (4.3), (4.2), we use the discrete Schwartz method {(2.98), (2.97), (2.96)} adapted to the problem (4.3), (4.2) We give the continual Schwartz method for the differential problem (4.3), (4.2). Let the system of overlapping sets, i.e., rectangular parallelepiped, Di , η

S

i = 1, . . . , I,

(4.9a)

cover the set Dη , i.e., D = i D i . We assume that the condition for the width of overlapping to the sets holds, i.e., the minimal distance from the


116


boundary of the set D i , for i¡= 1, . . . , I, to the boundary of an union of the sets which do not contain D i i.e., ¢the minimal overlapping of the subdomains D i , for i = 1, . . . , I, that cover Dη is independent of ε. The distance between iT

the sets Di and the boundary Γ, under the condition that D Γ = ∅, is independent of ε. η On the set D we construct functions u[k] (x), for k = 1, 2, 3, . . ., in the following way. Assume that u[0] (x) = 0,

η

x∈D .

(4.9b)

Let for k ≥ 1 the function u[k−1] (x) be already constructed. Construct u[k] (x). For this, we consider auxiliary functions ( [k] ui (x)

=

(k)

ui (x),

x∈D

[k] ui−1 (x),

η

x∈D \D η

[k]

u0 (x) = u[k−1] (x),

)

i

x∈D ,

,

η

η

x∈D ;

(4.9c)

i = 1, . . . , I.

i

(k)

Here ui (x), x ∈ D , is the solution of the boundary value problem on the i set D (k)

Lui (x) = f (x), x ∈ Di ,   T   ϕ(x), x ∈ Γi Γ     (k) ∗ iT η ϕ (x), x ∈ Γ {Γ \ Γ} ui (x) = ,      u[k] (x), x ∈ Γi \ Γη  i−1 i

(4.9d)

x ∈ Γi ,

η

where Γi = D \ Di . The function u[k] (x), x ∈ D , is defined by the relation [k]

u[k] (x) = uI (x), [k]

η

x∈D .

(4.9e)

η

The function u(4.9e) (x), x ∈ D , is k-th iteration the continual Schwartz method (2.91), k ≥ 1 for the problem (4.3), (4.2). [k]

Theorem 4.2.2 The function u(4.9) (x), i.e., the solution of the continual Schwartz method (4.9) for the boundary value problem (4.3), (4.2), converges η to the solution of the boundary value problem on the set D ε-uniformly as k→∞ η |u0 (x) − u[k] (x)| ≤ M q k , x ∈ D , q < 1. (4.10) For the Schwartz method (4.9), we construct a finite difference scheme and formulate conditions under which the scheme converges ε-uniformly.


Approximation of the solution in a subdomain for the problem on a strip 117 In the case of conditions (2.57) and (2.93), one can choose the diameters of i i the sets D sufficiently small such that the following inequality holds on D : £ ¤ max |ask (x)| < (n − 1)−1 min αsi (αki )−1 ass (x), αki (αsi )−1 akk (x) , i i D

D

s, k = 1, . . . , n,

s 6= k,

i = 1, . . . , I.

This ensures a transformation of the operator

L∗(2)(2.50)

(4.11)

to almost canonical

i

∗[1]

form on D and also a transformation of the operator L(2)(2.62) to strongly iT

almost canonical form if D Γ 6= ∅. We assume that the operator L∗(2) is locally almost canonical on D and i

∗[1]

almost canonical on the subdomains D , and the operator L(2) under the condition D condition

iT

Γ 6= ∅ is strongly almost canonical. We also suppose that the

the coefficients ask (x) of the operator L∗(2) on the set D satisfy the relations {(2.55), (2.65)}, for D the relations (2.55), for D

iT

iT

i

(4.12)

Γ 6= ∅, and i

Γ = ∅, where Q(x∗ ) = D , i = 1, . . . , I,

holds that admits the construction of consistent grids in the variables x1 ,. . ., xn with the monotonicity condition for the difference operator Λ∗(2)(2.51) (L∗(2) ) i

on D . i On the set D we construct the consistent grid ¢ i i¡ Dh = Dh Dh(2.56) , Dh(2.70) ≡ (4.13)   ¡ ¢ iT ∗ i D for D Γ = ∅  h(2.56) Q(x ) = D ≡ , i = 1, . . . , I, ¡ ¢ iT D Q(x∗ ) = D i for D Γ 6= ∅  h(2.70)

that ensures the monotonicity of the operators Λ∗(2) = Λ∗(2)(2.51) (L∗(2) ) and i

i

i

Λ = Λ(2.46) (L) on Dh . The grid Dh(4.13) on the set D is constructed based ¡ ¢ η on the grid Dh(2.56) , or Dh(2.70) introduced on the set D(4.2) and on the i

basis, respectively, the relations (2.55), or {(2.55), (2.65)} for Q(x∗ ) = D . η On the set D we construct the grid {0}

Dh = Dh , where we set 1

D(1)h = Dh , {0}

Dh

iS

D(i)h = Dh

= D(I)h .


n o i D(i−1)h \ D ,

(4.14a)

i = 2, . . . , I,

(4.14b)

118

Generalizations for elliptic reaction-diffusion equations η

{i}

On D we also introduce a set of the grids Dh , for i = 1, . . . , I, {i}

Dh

i S©

= Dh

{i−1}

Dh

\D

iª

,

i = 1, . . . , I.

(4.14c)

On the grid Dh(4.14a) we construct a sequence of grid functions z [k] (x), for k = 1, 2, . . .. Set z [0] (x) ≡ 0, x ∈ Dh . We construct z [k] (x) assuming that the function z [k−1] (x) for k ≥ 1 has already been constructed. Next, we introduce [k] the auxiliary functions zi (x)    z (k) (x), x ∈ D i  {i} h i [k] zi (x) = , x ∈ Dh , (4.15a)  z [k] (x), x ∈ D {i−1} \ D i  h i−1 [k]

z0 (x) = z [k−1] (x),

x ∈ Dh ,

i = 1, . . . , I.

i

(k)

In these relations zi (x), x ∈ Dh , is the solution of the discrete boundary i value problem on the set Dh (k)

Λi zi (x) = f (x), x ∈ Dhi ;   T   ϕ(x), x ∈ Γih Γ     (k) ∗ i T η ϕ (x), x ∈ Γ {Γ \ Γ} zi (x) = , h      zˇ[k] (x), x ∈ Γi \ Γη  i−1

(4.15b)

x ∈ Γih , i = 1, . . . , I.

h

i

Here Λi is Λ = Λ(2.46) (L) on the grid Dh , [k]

zˇ0 (x) = zˇ[k−1] (x),

x ∈ Γ1h \ Γ;

(4.15c)

[k]

the functions zˇi−1 (x), x ∈ D and zˇ[k−1] (x), x ∈ D, are constructed as the {i−1}

[k]

, and z [k−1] (x), x ∈ Dh , for interpolation of the functions zi−1 (x), x ∈ Dh i = 1, . . . , I. The function z [k] (x), x ∈ Dh , is determined by the relation [k]

z [k] (x) = zI (x),

x ∈ Dh .

(4.15d)

[k]

The interpolants zˇi (x) and zˇ [k] (x), x ∈ D, are defined by the relation (2.98f). The function z [k] (x), x ∈ Dh , Dh = Dh(4.14) , for k ≥ 1, is the solution of the difference scheme {(4.15), (4.14), (4.13)}, i.e., the iterative scheme of the domain decomposition method on the overlapping subdomains. The iterative difference scheme {(4.15), (4.14), (4.13)} is defined by canonical elements of the iterative scheme of the domain decomposition method, i.e., the operator Λ(2.46) and the grids Dh(2.56) (ω u1 ) and Dh(2.70) (ω 1(2.48) ), on the basis of which the difference scheme is constructed. Set N i = min Nsi and s


Approximation of the solution in a subdomain for the problem on a strip 119 i

N = min N i , where Nsi + 1 is the number of nodes in the grid Dh along the i

xs -axis, for s = 1, . . . , n. i

The parameters of the grids Dh satisfy the relations i d∗i s − d∗s ≈ 1,

Nsi ≈ N,

s = 1, . . . , n,

i = 1, . . . , I.

(4.16a)

The value η satisfies the condition η = m−1 ln N1 ,

(4.16b)

where N1 = mini N1i , for i = 1, . . . , I, and m = m(4.4) . Thus, the value I, i.e., i

η

the number of the subdomains D covering the set D , depends on N I = I(N )

(4.16c)

and increases unboundedly as N grows. For I one has the estimate I = I(N ) ≤ M lnn−1 N.

(4.17)

For the solution of the difference scheme {(4.15), (4.14), (4.13), (4.16)} under the additional condition ¡ ¢ i In Dh(4.13) Dh(2.56) (ω u1 ), Dh(2.70) (ω 1(2.48) ) (4.16d) the mesh ω 1(2.48) is ω 1(2.44) the following ε-uniform estimate on the set D0h holds: |z(x) − z [k] (x)| ≤ M [N −1 ln N + q k ],

x ∈ D0h ,

(4.18)

T

where D0h = Dh(4.14) D0(4.1) . The number of iterations k f required for the solution of the difference scheme {(4.15), (4.14), (4.13), (4.16)} (where ω 1(2.48) is ω 1(2.44) ) with the estimate f (4.19a) |u(x) − z [k ] (x)| ≤ M N −1 ln N, x ∈ D0h , is ε-uniformly bounded.

k f ≤ M ln N.

(4.19b)

The following theorem on the convergence of the iterative difference scheme holds. Theorem 4.2.3 Let the conditions ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, hold and the coefficients ask (x) satisfy the conditions (2.57), (2.93), (2.95). For the solution of the problem (2.2), (2.4), let the estimates of Theorem 2.2.1 be satisfied for K = 3. Then the iterative difference scheme (4.15), (4.14) on the grids Dh(4.13,4.16) converges ε-uniformly on the set D0h as N, k → ∞ with the estimates (4.18), (4.19).


120


The interpolants zˇ [k] (x) of the solutions of the iterative difference scheme converge ε-uniformly on the set D0 with the estimates £ ¤ |u(x) − zˇ [k] (x)| ≤ M N −1 ln N + q k , x ∈ D0 , f

|u(x) − zˇ [k ] (x)| ≤ M N −1 ln N,

x ∈ D0 ,

k f ≤ M ln N.

Remark 4.2.2 When computing the solutions of the iterative difference scheme {(4.15), (4.14), (4.13), (4.16)} on (one) an iterative step, intermediate solutions ¡ are computed on grids with the common number of nodes of order O N n lnn−1 N ). Taking account of the number of iterations required for convergence of the iterative process, intermediate solutions are computed on ¡ grids with the common number of nodes of order O N n lnn N ). At the same time, the number of nodes in the grid D0h , just as the number of the nodes in ¡ i the grids Dh that belong to D0h , is O N n ). According to the problem formulation, it is required to find approximate solutions of the problem (2.2), (2.4) only on the bounded grid set (the grid) D0h . Thus, the logarithmic ¡ multiplier lnn N is an additional price for computation (with accuracy of O N −1 ln N )) on the set D 0 of the solution of the boundary value problem (2.2), (2.4) when we apply the iterative domain decomposition method.

4.3

Difference schemes of improved accuracy for the problem on a slab

We now construct monotone difference schemes that converge ε-uniformly with an order of convergence close to two. For the differential operator L(2.2) we consider monotone approximations of improved accuracy on grids that condense in the boundary layer. To the operator L(2.2) on the grid Dh we associate the operator: 2

Λ = Λ(L) ≡ ε

½X n

ass (x)δxs xcs + 2−1

s=1

n X £ + ask (x) (δxs xk + δxs xk )+ s,k=1 s6=k

¾ n ¤ X +a− (x)(δ + δ ) + b (x)δ − c(x) − c0 (x), xs xk s x f s sk xs xk

(4.20)

s=1

in which, unlike the operator Λ(2.46) , the first-order derivatives are approximated by central difference derivatives. The operator Λ(4.20) acting on smooth functions, in the case of uniform meshes, approximates the operator L(2.2) with the second order of accuracy.



121

On the grid (2.26) with an arbitrary distribution of nodes, under the condition (2.49) the operator Λ(4.20) is ε-uniformly monotone for £ ¤ N ≥ 2 M max a−1 M = M(2.26) , s = 1, . . . , n. ss (x) |bs (x)| , s,D

We consider an approximation of the operator L by the operator Λ(4.20) on the solution of the problem (2.2), (2.4) in the case of piecewise-uniform grids. For the value ψ11(2.31) (xi ), i.e., the local error of the approximation of 2 the operator ε2 ∂ 2 by the operator ε2 δx1 x1 c on the grid (2.26), the following ∂x1 estimate holds: ¯ 4 ¯ ½ ¯ ∂ ¯ 2 ¯ ¯, ) max ψ11 (xi ) = ψ11 (xi ; u) ≤ M ε2 min (hi1 + hi−1 u(ξ) 1 4 ¯ ¯ ξ ∂x1 ¯ 3 ¯ ¯ 2 ¯¾ ¯ ∂ ¯ ¯ ∂ ¯ i−1 i ¯ ¯ ¯ |h1 −h1 | max ¯ 3 u(ξ)¯ , max ¯ 2 u(ξ)¯¯ , xi ∈ Dh , xi = (xi1 , x2 , . . . , xn ), ξ ξ ∂x1 ∂x1 ξ = (ξ1 , ξ2 , . . . , ξn ),

i+1 ξ1 ∈ [xi−1 1 , x1 ],

ξs = xs , s = 2, . . . , n.

For ψ11 (x; u(·)) on the regular component U (x) in (2.7) in the case of the piecewise-uniform grid Dh(2.58) , we have ( M N1−2 , x1 6= d∗1 + σ, d∗1 − σ, (4.21a) |ψ11 (x; U (·))| ≤ M ε N1−1 , x1 = d∗1 + σ, d∗1 − σ, x ∈ Dh . On the singular component V (x) in (2.7), one has ( |ψ11 (x; V (·))| ≤

M ε−2 h2(1) ,

x1 ∈ / [d∗1 + σ, d∗1 − σ],

(4.21b)

M exp(−m ε−1 σ), x1 ∈ [d∗1 + σ, d∗1 − σ], x ∈ Dh .

For the operator Λ(4.20) on the components U (x) and V (x) in the case of the piecewise-uniform grid Dh = Dh(2.58) , (4.22a) where

(1)

ω 1 = ω (2.43) (d∗1 , d∗1 ; 2−1 m), m < m0(2.12) ,

(4.22b)

by virtue of (4.21), one obtains the estimates ( M N −2 , x1 6= d∗1 + σ, d∗1 − σ, ¡ ¢ | L − Λ U (x)| ≤ M [ε N1−1 + N −2 ], x1 = d∗1 + σ, d∗1 − σ, x ∈ Dh ; ¡ ¢ | L − Λ V (x)| ≤ M N −2 ln2 N, x ∈ Dh . In the case when the difference scheme Λ z(x) = f (x), x ∈ Dh ,


z(x) = ϕ(x), x ∈ Γh ,

(4.23)

122


where Λ = Λ(4.20) , is ε-uniformly monotone on the grids (4.22), for its solution we obtain the estimate |u(x) − z(x)| ≤ M N −2 ln2 N,

x ∈ Dh .

Here for the construction of the majorant we use the function h ¡ ¢ ¡ ¢i w(x) = 1 − max exp − m ε−1 r(x, Γ) , exp − m ε−1 σ) ,

(4.24)

x ∈ Dh ,

where m < m0(2.14) . Theorem 4.3.1 Let for the solution of the boundary value problem (2.2), (2.4) the estimates of Theorem 2.2.1 hold for K = 4. Then the solution of the monotone difference scheme (4.23), (4.20), (4.22) satisfies the estimate (4.24). We now study conditions that ensure the monotonicity of the operator Λ(4.20) ; the fulfillment of the condition (2.49) is not assumed. We write the operator Λ(4.20) as the sum of the operators L = L1 + L2 .

(4.25a)

Here ½ 1

1

2

L = L (λ) ≡ ε

¾ n X ∂2 ∂2 (1 − λ) ass (x) 2 + ask (x) ≡ (4.25b) ∂xs ∂xs ∂xk s=1 n X

s,k=1 s6=k

≡ ε2

n X s,k=1

aλsk (x)

∂2 ; ∂xs ∂xk

½ X ¾ n n X ∂2 ∂ L2 = L2 (λ) ≡ ε2 λ ass (x) 2 + bs (x) − c(x) − c0 (x), (4.25c) ∂xs s=1 ∂xs s=1 the parameter λ, which takes values in the interval (0, 1] and depends on the coefficients of the operator L2(2.2) , is chosen below. The operator L1 = L1 (λ) = L∗(2)(2.50) (L1 ) is strongly elliptic if the parameter λ satisfies the condition λ ∈ [0, λ0 ),

(4.26a)

where λ0 is a constant in the interval (0, 1]. For the value λ0 , the following upper estimate holds: λ0 ≤ a0 (a0 )−1 , (4.26b)



123

where a0 and a0 are the constants from the ellipticity condition (2.3). The value λ0 can be chosen to satisfy the condition λ0 = λ0 (L) → 1 as max |ask (x)| → 0, s, k = 1, . . . , n, s 6= k.

(4.26c)

s,k,D

In the case when the equation does not contain mixed derivatives we set λ = 1. The operator n ¡ ¢ X Λ1 = Λ1 L1(4.25, 4.26) ≡ aλss (x) δxsc xs +

(4.27)

s=1

+2−1

n X £ λ+ ¤ ask (x) (δxs xk + δxs xk ) + aλ− sk (x) (δxs xk + δxs xk ) , s,k=1 s6=k

aλsk (x) = aλsk(4.25b) (x), s, k = 1, . . . , n, is defined only by the coefficients aλsk (x) of the operator L1(4.25, 4.26) . The operator Λ1 is ε-uniformly monotone in the following cases: (i) on the uniform grid ¡ ¢ Dh = Dh Q(x∗ ), L1(4.25, 4.26) (λ) =

(4.28)

¡

= Dh(2.56) Q(x∗ ), L∗(2)(2.50) (L1 ) = L1(4.25, 4.26) (λ), x1 , . . . , xn ; ¢ under the condition (2.55c) subject to (2.55) (the fulfillment of the conditions (2.49), (2.59), or (2.57) is not assumed); (ii) and also on piecewise-uniform grids in the case of the conditions: (iia )

(2.57), on the grid ¡ Dh = Dh(2.70) Q(x∗ ), L∗(2)(2.50) (L1 ) = L1(4.25, 4.26) (λ), x1 , . . . , xn ; ω 1 = ω 1(4.22) under the conditions (2.67) and (2.55c) ¢ subject to {(2.55), (2.65)} ;

(2.59), on the grid ¡ Dh = Dh(2.63) Q(x∗ ), L∗(2)(2.50) (L1 ) = L1(4.25, 4.26) (λ), x2 , . . . , xn ;

(4.29)

(iib )

ω 1 = ω 1(4.22) (iic )

(4.30)

¢ under the condition (2.61c) subject to (2.61)

and (2.49), on the grid Dh = Dh(2.58) (ω 1(4.22) ).


(4.31)

124


Consistent grids (4.28)–(4.31) are constructed depending on the coefficients aλsk (x), for s, k = 1, . . . , n, of the operator L1(4.25) , at that time as consistent grids in Section 2.1 have been constructed depending on the coefficients ask (x) of the operator L∗(2)(2.50) . The operator ¡ ¢ ¡ ¢ Λ2 = Λ2 L2(4.25, 4.26) (λ) ≡ Λ L2(4.25, 4.26) (λ) = (4.32) ½ X ¾ n n X = ε2 λ ass (x) δxsc bs (x) δxfs − c(x) − c0 (x), λ = λ(4.26) xs + s=1

s=1

on the grid Dh(2.26) with an arbitrary distribution of nodes is ε-uniformly monotone under the condition £ ¤ N ≥ 2 M λ−1 max a−1 (4.33) ss (x) |bs (x)| , M = M(2.26) , s = 1, . . . , n, s, D

where λ > 0. Under the condition (4.33) the operator Λ(4.20) = Λ1 + Λ2 , ¡ ¢ ¡ ¢ where Λ1 = Λ1(4.27) L1(4.25, 4.26) and Λ2 = Λ2(4.32) L2(4.25, 4.26) , preserves the monotonicity property of the operator Λ1(4.27) on the grids (4.28)–(4.31). For the operator Λ(4.20) statements similar to that of Theorems 2.4.2, 2.4.4, 2.4.5 are valid. Theorem 4.3.2 The operator Λ(4.20) under the condition (4.33) is ε-uniformly monotone on the uniform grid (4.28) and also on the piecewise-uniform grids (4.29), (4.30), and (4.31) in the case of the conditions (2.57), (2.59), and (2.49), respectively. (0)

We associate the scheme (4.23) with the problem (2.2) on D(2.4) . Assume that the condition (4.33) holds. Depending on the fulfillment of the conditions (2.57), (2.59), and (2.49), we chose the grid, respectively, Dh(4.29) , Dh(4.30) , and Dh(4.31) ; otherwise, we chose the grid Dh(4.28) . On each of these grids the operator Λ(4.23) is ε-uniformly monotone. The following theorems similar to those of 2.4.8, 2.4.9, 2.4.10 are valid. Theorem 4.3.3 Let the conditions ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, be fulfilled and the coefficients ask (x) and aλsk(4.25) (x) satisfy, respectively, the conditions (2.57) and (2.66), (2.71) with ask(2.66,2.71) (x) = aλsk(4.25) (x) and Q(x∗ ) = D. For the solution of the problem (2.2), (2.4), let the estimates of Theorem 2.2.1 be satisfied for K = 4. Then the difference scheme (4.23) on the grid Dh(4.29) , where Q = D, converges ε-uniformly at the rate O(N −2 ln2 N ) as N → ∞: |u(x) − z(x)| ≤ M N −2 ln2 N,


x ∈ Dh .

Domain-decomposition method for improved iterative schemes

125

Theorem 4.3.4 Let ask , bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, hold and the coefficients ask (x) and aλsk(4.25) (x) satisfy, respectively, the conditions (2.59) and (2.61), (2.73) with ask(2.61,2.73) (x) = aλsk(4.25) (x), Q(x∗ ) = D. For the solution of the problem (2.2), (2.4), let the estimates of Theorem 2.2.1 be satisfied for K = 4. Then the difference scheme (4.23) on the grid Dh(4.30) , where Q = D converges ε-uniformly at the rate O(N −2 ln2 N ) as N → ∞. Theorem 4.3.5 Let the coefficients ask (x) satisfy the condition (2.49), and let ass , bs , c, c0 , f ∈ C(D), for s = 1, . . . , n. For the solution of the problem (2.2), (2.4), let the estimates of Theorem 2.2.1 hold for K = 4. Then the difference scheme (4.23) on the grid Dh(4.31) converges ε-uniformly at the rate O(N −2 ln2 N ) as N → ∞. Under the hypotheses of Theorems 4.3.3, 4.3.4, and 4.3.5, for the interpolant z(x) = z (2.42) (x; z(·), Dh ), x ∈ D, where z(x), x ∈ Dh , is the solution of the difference scheme (4.23), respectively, on the grids (4.29), (4.30), and (4.31), one has the estimate |u(x) − z(x)| ≤ M N −2 ln2 N,

x ∈ D.

4.4 Domain-decomposition method for improved iterative schemes We now consider difference schemes of the domain-decomposition method that converge ε-uniformly with the convergence order close to two. For the boundary value problem (2.2), (2.4), let the coefficients a1s (x) satisfy only the condition (2.57). The fulfillment of the condition (2.71) is not assumed, but the condition (2.93) holds. To solve the boundary value problem, we apply an iterative difference scheme similar to {(2.98), (2.97), (2.96)}. i In the case of condition (2.93), one can choose the diameter of the set D i and the values λ sufficiently small such that the following inequality holds: £ ¤ i |aλsk (x)| < (n − 1)−1 min αsi (αki )−1 ass (x), αki (αsi )−1 akk (x) , max i i D

D

i

i

aλsk (x) = aλsk(4.25) (x),

s, k = 1, . . . , n,

s 6= k.

This ensures a transformation of the regular operator L∗R (2) , i.e., ¡ −2 1 ¢ ∗R L∗R L(4.25) (λ i ) ≡ ε−2 L1(4.25) (λ i ), (2) = L(2)(2.50) ε


(4.34a)

126

Generalizations for elliptic reaction-diffusion equations i

to almost canonical form on D and also a transformation of the operator ∗R[1] L(2) , i.e., ¢ ¡ ∗R[1] ∗[1] L(2) = L(2)(2.62) L∗R (4.34b) (2)(4.34a) , iT

to strongly almost canonical form if D Γ 6= ∅ and the coefficients a1s (x) on the boundary Γ satisfy the condition (2.57). Assume that the operator L∗R (2)(4.34a) is locally almost canonical on D, morei

over, on the subdomains D it is almost canonical, and under the condition iT ∗R[1] D Γ 6= ∅, the operator L(2)(4.34b) is strongly almost canonical. Suppose also that the condition i

λ the coefficients ask (x) of the operator L∗R (2)(4.34a) on the set D

satisfy the relations {(2.55), (2.65)} if D and the relations (2.55) if D i

iT

iT

i

Γ 6= ∅,

(4.35)

Γ = ∅, i

where Q(x∗ ) = D , ask (x) = aλsk(4.25) (x), for i = 1, . . . , I, holds. Then the above conditions together with the condition (2.57) admit us i to construct on D consistent grids in the variables x1 , . . . , xn with respect to the monotonicity condition for the difference regular operator Λ∗R (2) ¡ ¢ ∗ ∗R −2 1 Λ∗R Λ(4.27) L1(4.25) (λ i ) . (2) = Λ(2)(2.51) (L(2)(4.34a) ) ≡ ε i

On the set D we construct the consistent grid  ¡ ¢ iT ∗ i D if D Γ = ∅, h(4.28) Q(x ) = D i Dh = ¡ ¢ iT D ∗ i if D Γ 6= ∅, h(4.29) Q(x ) = D

(4.36) i = 1, . . . , I,

∗ ∗R that ensures the monotonicity of the operators Λ∗R (2) = Λ(2) (L(2) ) and Λ = i

Λ(4.20) (L) on Dh . (0)

i

i

On the slab D = D(2.4) on the basis of the grids Dh = Dh(4.36) , for i = 1, . . . , I, we construct the grid ¡ i ¢ {0} Dh = Dh(2.97) Dh(4.36) , i = 1, . . . , I . With regard to the relations (2.98), where ¡ ¢ i Λi = Λi(2.98) ≡ Λ(4.20) L(4.25) ; λ(4.26) ,

i = 1, . . . , I,

(4.37)

(4.38)

solving the discrete problems (2.98b), where Λi = Λi(4.38) , we find the functions [k] z [k] (x) = z(2.98) (x), x ∈ Dh , k = 1, 2, 3, . . .



127

and the interpolant [k]

zˇ[k] (x) = zˇ(2.98) (x),

x ∈ D, k = 1, 2, 3, . . . .

(4.39)

The function z [k] (x), x ∈ Dh , is the solution of the iterative difference scheme {(2.98), (2.97), (4.38), (4.36)}, i.e., the iterative scheme of the domain decomposition method on the overlapping subdomains. The iterative difference scheme {(2.98), (2.97), (4.38), (4.36)} is defined by canonical elements, i.e., the operator Λ(4.20) and the grids Dh(4.28) (ω u1 ) and Dh(4.29) (ω 1(4.22) ), on the basis of which the difference scheme is constructed. Set N i = min Nsi and N = min N i , where Nsi + 1 is the number of nodes in s i Dh

i

the grid along the xs -axis, for s = 1, . . . , n. Assume that the following condition holds, which is similar to (4.33): £ ¤ N ≥ 2 M max(λi )−1 max a−1 M = M(2.26) , ss (x) |bs (x)| , i

s, D

s = 1, . . . , n,

i = 1, . . . , I.

Theorem 4.4.1 Let ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n; i the coefficients ask (x) and aλsk(4.25) (x) satisfy, respectively, the conditions (2.57), (2.93), and (4.35). For the solution of the problem (2.2), (2.4), let the estimates of Theorem 2.2.1 hold for K = 4. Then the iterative difference scheme {(2.98), (2.97), (4.38), (4.36)} on the grid Dh(4.37) converges ¡ ¢ ε-uniformly at the rate O N −2 ln2 N + q k as N, k → ∞: |u(x) − z [k] (x)| ≤ M [N −2 ln2 N + q k ],

x ∈ Dh .

The interpolant zˇ(4.39) (x) of the solution of the iterative difference scheme converges ε-uniformly as N, k → ∞ with the estimate |u(x) − zˇ [k] (x)| ≤ M [N −2 ln2 N + q k ],

x ∈ D.

The number of iterations k f required for the solution of the iterative difference scheme with the estimate f

|u(x) − zˇ[k ] (x)| ≤ M N −2 ln2 N,

x ∈ D,

is ε-uniformly bounded. The value k f satisfies the estimate k f ≤ M ln N. We now construct a difference scheme for the boundary value problem (2.2) in the domain D(2.103) with a curvilinear boundary. The fulfillment of the dominance condition of the diagonal elements of the elliptic operator, i.e., the condition (2.71), as well as restrictions for the coefficients of the mixed


128


derivatives on the domain boundary, is not assumed. To solve the problem, we apply an iterative difference scheme similar to {(2.98), (2.97), (2.114)}. S Using an appropriate transformation of the coordinates on the set D i ΓiL in a neighborhood of the domain boundary, the boundary value problem (2.2), (2.103) can be brought to the form (2.105):

Here DXi

S iL Γ

LX U (X) = F (X),

i X ∈ DX ,

(4.40a)

U (X) = Φ(X),

X ∈ ΓiL X.

(4.40b)

is the rectangular parallelepiped adjoining to the boundary

X

LX ≡ ε2 L2X + L0X , L2X ≡

n X

L0X = −C 0 (X), n

Ask (X)

s,k=1

X ∂2 ∂ + Bs (X) − C(X). ∂Xs ∂Xk s=1 ∂Xs

The coefficients Ask of the operator LX satisfy the strong ellipticity condition A0

n X

n X

ξs2 ≤

s=1

Ask (X) ξs ξk ≤ A0

n X

i

ξs2 ,

X ∈ DX ,

A0 > 0.

(4.40c)

s=1

s,k=1

On the domain boundary one has the condition A1s (X) = 0,

X ∈ ΓiL X,

s = 2, . . . , n.

(4.40d)

We write the operator LX as the decomposition similar to (4.25): LX = L1X + L2X .

(4.41a)

Here L1X = L1X (λi ) ≡ ε2

n X

i

Aλsk (X)

s,k=1

∂2 , ∂Xs ∂Xk

L2X = L2X (λi ) ≡ n ≡ ε 2 λi

n X

Ass (X)

s=1

(4.41b) (4.41c)

o ∂2 ∂ + B (X) − C(X) − C 0 (X), s ∂Xs2 s=1 ∂Xs n X

where i

Aλss (X) = (1 − λi ) Ass (X),

i

Aλsk (X) = Ask (X), s 6= k.

By virtue of the conditions (4.40c), (4.40d), the regular operator L∗R X(2) , i.e., ¡ −2 1 ¢ ∗ L∗R LX(4.41b) (λi ) ≡ ε−2 L1X(4.41b) (λi ) X(2) = LX(2)(2.50) ε


(4.42a)


129

i

can be brought to almost canonical form on DX (for sufficiently small diamei ∗R[1] ters of the sets DX and the values λi ), and the operator LX(2) , i.e., ¡ ¢ ∗R[1] ∗[1] LX(2) = LX(2)(2.62) L∗R X(2)(4.42a) ,

(4.42b)

can be brought to strongly almost canonical form in X2 , . . . , Xn in a neighborhood of the boundary ΓiL X. Assume that the operator L∗R X(2)(4.42a) has almost canonical form in the i

∗R[1]

variables X1 , . . . , Xn on DX , and the operator LX(2)(4.42b) has strongly almost canonical form in X2 , . . . , Xn in a neighborhood of ΓiL X . Let the coefficients © ª i ∗R i of the operator LX(2) defined on DX = X : d∗s ≤ Xs ≤ di∗ s , s = 1, . . . , n satisfy the condition (similar to (2.108): i

i

the coefficients Aλsk (X) of the operator L∗R X(2)(4.42a) on the set D X satisfy the relations (2.78), {(2.55), (2.79)} if J i∗ 6= ∅, and the relations (2.55) if J i∗ = ∅, i where ask (x), . . . , Q(x ) is Aλsk (X), . . . , {Q(x∗ )}X , i i under the condition {Q(x∗ )}X = DX , J i∗ = J i∗ (DX )

(4.43)

∗

= {k = 1}.

i The condition J i∗ (DX ) = {k = 1} means that the piecewise-uniform grid ω X1 in the X1 -axis £condenses ¤ (for small ε) in a neighborhood of the left endpoint of the interval di∗1 , di∗ 1 . By virtue of (4.40d), the coefficients of the operator L∗X(2)(4.41a) on the face iL ΓX satisfy the condition i

Aλ1s (X) = 0,

X ∈ ΓiL X,

s = 2, . . . , n,

i.e., the condition (2.78), where the coefficients a1k (x) on the set Γk , for i iL k ∈ J ∗ , are the coefficients Aλ1k (X) on the set ΓX . In the case of the operator ¡ i ∗ LX(2) , the following condition similar to (2.55c) for Q(x∗ ) = D , where ¢ i D ⊆ D imposed on the parameters hs takes the form: £ ¤ ∗ ∗ ∗ ρ (X) h , min hs ∈ max ρ (X) h , s = 1, . . . , n. (4.44) ∗s s i i DX

DX

i

The fulfillment of the condition (4.43) allows us to construct on the set DX a piecewise-uniform grid that ensures the monotonicity of the operators ¡ ∗R ¢ ¡ ¢ ∗ −2 1 Λ∗R ΛX(4.27) L1X(4.41b) (λi ) , X(2) = ΛX(2)(2.51) LX(2)(4.42a) ≡ ε and also

¡ ¢ ¡ ¢ ΛX = ΛX LX(4.41) (λi ) ≡ ΛX(4.20) LX(4.41a) = ¡ ¢ ¡ ¢ = Λ1X(4.27) L1X(4.41b) (λi ) + Λ2X(4.32) L2X(4.41c) (λi ) .


(4.45)

130


∗R The operators Λ∗R X(2) and ΛX are constructed using the operators LX(2)(4.42a) and LX(4.41a) , respectively. i In the case of the conditions (4.43) and (2.82), where J ∗ = J i∗ (DX ) = l {j = 1}, imposed on the parameters of the meshes ω X1 = ω X1 (ω (4.22) ) and ¡ ¢ ω Xs = ω uXs , for s = 2, . . . ,¡n, where ω Xs = ω iXs di∗s , di∗ s = 1, . . . , n, s , for ¢ the piecewise-uniform grid the grid (2.89), where J ∗ = {j = 1} :

¢ i i ¡ i i∗ DXh = DXh {Q(x∗ )}X = DX , L∗R = X(2)(4.42a) ; J ³ i i = DXh(2.89) {Q(x∗ )}X = DX , L∗R X(2)(4.42a) ; i l ω X1 = ω X1 (ω (4.22) ),

(4.46)

i u ω Xs = ω Xs , for s = 2, . . . , n,

under the conditions (4.44), (2.82), J i∗ = {j = 1} in the case of the condition (4.43); in the relations (2.78), {(2.55), (2.79)} from (2.89) ´ i ask (x) and Q(x∗ ) are replaced by Aλsk (X) and {Q(x∗ )}X , ¡ i which is constructed directly on the set DX , is consistent in the variables ¢ i i X1 , . . . , Xn on the set DX . The grid DXh(4.46) is generated by the meshes l l u ω X1 = ω X1 (ω (4.22) ) and ω Xs , for s = 2, . . . , n, i.e., i

l u u DXh(4.46) = ω X1 × ω X2 × . . . , ω Xn .

For the subproblem (4.40) we construct the monotone difference approximation i ΛX Z(X) = F (X), X ∈ DXh ;

iL . Z(X) = Φ(X), X ∈ ΓXh

i

(4.47)

i

Here ΛX = ΛX(4.20) (LX ) and DXh = DXh(4.46) . Returning to the variables x = (x1 , . . . , xn ), we obtain the monotone difference scheme i Λ∗ z ∗ (x) = f (x), x ∈ DXh X −1 ;

z ∗ (x) = ϕ(x), x ∈ ΓiL XhX −1 .

(4.48a)

Here i i DXhX −1 = D i

T i

i

DXh X −1 ,

DXh X −1 = {DXh }X −1 ;

T

i

ΓiL Xh X −1 = Γ D Xh X −1 , z ∗ (x) = Z(X(x)), i

(4.48b)

i

x ∈ DXh X −1 .

We assume that on subdomains D , which have no common points with the boundary Γ, the operator L is almost canonical; let on these subdomains



131

the following condition hold (similar to (2.87), where J ∗ = ∅): i

the coefficients aλsk (x) of the operator L∗R (2)(4.34a) on the set D satisfy the relations (2.55), i

i

(4.49) i

where Q(x∗ ) = D , ask (x) = aλsk(4.25) (x). i

Such condition admits the construction of the uniform grids Dh , i.e., ¡ ¢ i Dh = Dh(4.28) Q(x∗ ) = D i , L1(4.25, 4.26) (λi ) , (4.50) that are consistent with the monotonicity condition for the operators ¡ ¢ Λ i = Λ L(4.25, 4.26) (λi ) ≡ Λ(4.20) . (4.51) Next, for the construction of iterative scheme we use the algorithm {(2.98), (2.97), (2.114)}. i Let the subdomains D , for i = 1, . . . , I , form a covering of the set D, i moreover, the subdomains D , for i = 1, . . . , I1 , adjoin to the boundary Γ, i and the subdomains D , for i = I1 + 1, . . . , I, have no common points with the boundary Γ; and let the distance between these sets and Γ be independent of ε. We assume that the condition for the width of overlapping to subdomains holds, i.e., the minimal overlapping of subdomains that cover D is independent of the value of the parameter ε. i The sets D for i = I1 + 1, . . . , I, are rectangular parallelepipeds formed i by coordinate planes xs =const, for s = 1, . . . , n. The sets D , for i = i 1, . . . , I1 , in new variables X i = X i (x) are the sets DX i which are rectangular parallelepipeds formed by coordinate planes Xsi = const, s = 1, . . . , n. i Note that the coefficients of the operator LX i (4.41) on the set DX i , for i = 1, . . . , I1 , satisfy the condition (4.43), and the coefficients of the operator i L(4.25, 4.26) (λi ) on the set D , for i = I1 +1, . . . , I, satisfy the condition (4.49). Next, we construct the grid sets  Di i i h(2.114a) for i = 1, . . . , I1 , Dh ≡ Dh(2.114e) = (4.52a) Di h(2.114c) for i = I1 + 1, . . . , I and the difference operators ( Λi ≡ Λi(2.114f) =

Λi(2.114b) for i = 1, . . . , I1 , Λi(2.114d) for i = I1 + 1, . . . , I.

Here i

i

i

i

Dh(2.114a) = DX i h{X i }−1 (4.48b) for Dh(2.114c) = Dh(4.50)


for

i = 1, . . . , I1 , i = I1 + 1, . . . , I;

(4.52b)

132

Generalizations for elliptic reaction-diffusion equations i

Λi(2.114b) = Λ∗(4.48a) with x ∈ DX i h{X i }−1 , i

Λi(2.114d) = Λi(4.51) with x ∈ Dh ,

for

i = 1, . . . , I1 ,

for

i = I1 + 1, . . . , I. [k]

In the iterative process (2.98), (2.97), (2.114), the interpolants zˇi (x), x ∈ D, for k ≥ 1, are defined by the relations (2.114g). ¡ i [k] The function z [k] (x) = z(2.98e) (x), x ∈ Dh , where Dh = Dh(2.97a) Dh(4.52a) , ¢ i = 1, . . . , I , is the solution of the iterative difference scheme {(2.98), (2.97), (2.114), (4.52)}. i We denote by Nsi +1 the number of nodes in the grid DX i h , for i = 1, . . . , I1 i (the grid Dh , for i = I1 + 1, . . . , I), along the Xsi -axis (respectively, along the xs -axis), for s = 1, . . . , n. Set N i = mins Nsi and N = mini N i . Assume that the following condition holds: n £ ¤ i i N ≥ 2 M max(λi )−1 maxi A−1 ss (X ) |Bs (X )| i=1,...,I1 , i

s,i,D

o £ ¤ max(λi )−1 max a−1 (x) |b (x)| , M = M(2.26) . s ss i=I1 +1,...,I i

s,D

The iterative difference scheme is defined by canonical elements, i.e., the operators Λ(4.51) (L(λi )) and ΛX i (4.45) (LX i (4.41) ) and the consistent uniform i

i

and piecewise-uniform grids Dh(4.50) and DX i h(4.46) . The following theorem on convergence holds. Theorem 4.4.2 Let ask ∈ C 1 (D), bs , c, c0 , f ∈ C(D), for s, k = 1, . . . , n, i i i i Γ ∈ C 2 , the coefficients aλsk (x), x ∈ D , and Aλsk (X i ), X i ∈ DX i , satisfy, respectively, the conditions (4.49) for i = I1 + 1, . . . , I, and (4.43), where X = X i , for i = 1, . . . , I1 . For the solution of the problem (2.2), (2.103), let the estimates of Theorem 2.2.2 hold for K = 4. Then the difference scheme {(2.98), (2.97), (2.114), (4.52)} on the grid Dh(2.97a,2.114e) , in which the mesh i

l i ωX i 1 in D X i h , for i = 1, . . . , I1 , is defined by the mesh ω 1(4.22) , converges ¡ −2 2 ¢ ε-uniformly at the rate O N ln N + q k as N, k → ∞: ¡ ¢ |u(x) − z [k] (x)| ≤ M N −2 ln2 N + q k , x ∈ Dh . [k]

For the interpolant zˇ[k] (x), zˇ[k] (x) = zˇ(2.114g) (x), x ∈ D, one has the estimate ¡ ¢ |u(x) − zˇ[k] (x)| ≤ M N −2 ln2 N + q k , x ∈ D. The number of iterations required for the solution of the iterative difference scheme with the estimate f

|u(x) − z [k ] (x)| ≤ M N −2 ln2 N, satisfies the estimate


k f ≤ M ln N.

x ∈ Dh ,

Chapter 5 Parabolic reaction-diffusion equations

In this chapter, a method for the construction of ε-uniformly convergent difference schemes for parabolic reaction-diffusion equations is developed using classical finite difference approximations on special condensing grids. Conditions are given that are sufficient for the ε-uniform convergence of the difference schemes. For problems on a slab and a parallelepiped, monotone ε-uniformly convergent difference schemes are constructed.

5.1

Problem formulation

In an n-dimensional domain D, we consider the boundary value problem for the parabolic reaction-diffusion equation L u(x, t) = f (x, t),

(x, t) ∈ G,

(5.1a)

u(x, t) = ϕ(x, t),

(x, t) ∈ S.

(5.1b)

Here G = D × (0, T ], L ≡ ε2 L2 + L1 , L2 ≡

n X

S = S(G) = G \ G;

L1 ≡ −c1 (x, t) − p(x, t)

∂ , ∂t

n

ask (x, t)

s,k=1

X ∂2 ∂ + bs (x, t) − c(x, t). ∂xs ∂xk s=1 ∂xs

The coefficients of the equation satisfy the ellipticity condition a0

n X s=1

ξs2 ≤

n X

ask (x, t) ξs ξk ≤ a0

n X

ξs2 ,

(x, t) ∈ G,

a0 > 0, (5.1c)

s=1

s,k=1

and also the conditions p(x, t) ≥ p0 > 0,

c(x, t),

c1 (x, t) ≥ 0,

(x, t) ∈ G.

The coefficients and the right-hand side f on the set G are assumed to be sufficiently smooth, just as the boundary function ϕ on the closures of the smooth


134

Parabolic reaction-diffusion equations

parts of the boundary S, i.e., the data of the problem (5.1) are assumed to be sufficiently smooth. We suppose that the problem data are bounded in the case of an unbounded domain D. Consider the boundary value problem on the slab D = {x : d∗1 < x1 < d∗1 ,

|xs | < ∞,

s = 2, . . . , n}

(5.2)

and on the parallelepiped D = {x : d∗s < xs < d∗s ,

s = 1, . . . , n}.

(5.3)

Let Γj = Γj(3.6) , where j = 1, . . . , J, be the faces of the set D, and let S = SL

S

S0 ,

S L = Γ × (0, T ],

(5.4a)

where S L and S0 are the lateral and the lower parts of the boundary, here S S L = j Sj , Sj = Γj × (0, T ], j = 1, . . . , J and S0 = S 0 . Denote by S c a set that is formed by mutual intersection of smooth parts of the boundary S, and S c = S Lc Lc

c

S

S0c ,

(5.4b)

S0c L

= Γ × {t = 0} are sets of “edges” on the where S = Γ × (0, T ] and lateral part of the boundary S and “edges” (boundaries) on the lower part of the boundary S0 . In some cases, as well as the continuity condition for the function ϕ(x, t) on the set S c , additional compatibility conditions shall be imposed to guarantee the sufficient smoothness of the solution to the boundary value problem. For the boundary value problem (5.1), it is required to construct a difference scheme that converges ε-uniformly.

5.2


Let us estimate the solution of the problem (5.1) on the slab (5.2) and on the parallelepiped (5.3). When deriving estimates, we use a priori estimates for regular problems [67, 37]. We denote by C l0 ,l0 /2 (G), where l0 = l + α, l ≥ 0 is an integer number, and α ∈ (0, 1), the Banach space whose elements are continuous functions u(x, t) in G that have in G continuous derivatives (∂ k+k0 /∂xk11 . . . ∂xknn ∂tk0 ) u(x, t),

k + 2k0 ≤ l,

which are in the sense of Hölder continuous with exponent α. For these elements a finite value is taken by the norm ¯ ¯ l ¯ ¯ X X ∂ k+k0 ¯ ¯ |u|l+α = |u|α + u ¯ k1 ¯ . ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ k+2 k0 =1 k1 +...+kn =k α



135

Here |u|α = |u|0 + H α (u), H α (u) =

|u|0 = sup |u(x, t)|, G

sup (x,t),(x0 ,t0 )∈G

¯ ¯ ¯u(x, t) − u(x0 , t0 )¯ ¡ ¢ , dα (x, t), (x0 , t0 )

¡ ¢ and d (x, t), (x0 , t0 ) is the distance between the points (x, t) and (x0 , t0 ) ¡ ¢ ¡ ¢1/2 d (x, t), (x0 , t0 ) = |x − x0 |2 + ε2 |t − t0 | . Let the function u(x, t), (x, t) ∈ G on the boundary S take the values of ϕ(x, t); moreover, on the set S g ⊂ S the function ϕ(x, t) is “good”, i.e., g ϕ ∈ C l0 ,l0 /2 (S g ), and also, S g ∈ C l0 ,l0 /2 , S g = S , where l0 = l + α, l ≥ 0 and α ∈ (0, 1). The set S g can coincide with either S0 or S j , or can be the empty set. We denote by C l0 ,l0 /2 (G) the Banach space whose elements S are continuous functions in G S g that take “good” boundary values on the set S g , and have continuous derivatives (∂ k+k0 /∂xk11 . . . ∂xknn ∂tk0 ) u(x, t) for k + 2k0 ≤ l, which are in the sense of Hölder continuous with exponent α. For these elements a finite value is taken by the norm ¯ ¯g l ¯ ¯ X X ∂ k+k0 ¯ k+2 k0 ¯ g g |u|l+α = |u|α + u ¯d ¯ . ¯ ∂xk11 . . . ∂xknn ∂tk0 ¯ k+2 k0 =1 k1 +...+kn =k

α

Here |u|gα = |u|0 + Hαg (u), Hαg (u)

=

sup (x,t),(x0 ,t0 )∈G

dα (x,t),(x0 ,t0 )

|dp v|gα = |dp v|g0 + Hαg (dp v), Hαg (dp v) =

sup (x,t),(x0 ,t0 )∈G

¯ ¯ ¯u(x, t) − u(x0 , t0 )¯ ¡ ¢ , dα (x, t), (x0 , t0 )

|dp v|g0 = sup dp(x,t) |v(x, t)|, (x,t)∈G

¯ ¯ ¯v(x, t) − v(x0 , t0 )¯ p+α ¢, d(x,t),(x0 ,t0 ) α ¡ d (x, t), (x0 , t0 )

¡ ¢ ¡ ¢1/2 d (x, t), (x0 , t0 ) = |x − x0 |2 + ε2 |t − t0 | , ¡ ¢ g d(x,t) = d(x,t) = 0 0inf g d (x, t), (x0 , t0 ) , (x ,t )∈S\S

¡ ¢ d(x,t),(x0 ,t0 ) = dg(x,t),(x0 ,t0 ) = min d(x,t) , d(x0 ,t0 ) . For the function ϕ(x, t), (x, t) ∈ S satisfying the condition ϕ ∈ C l0 ,l0 /2 (S g ) for S g = S0 , we also shall use equivalent notations ϕ(·, 0) ∈ C l0 (D) and ϕ ∈ C l0 (S0 ).


136


We give a definition. Set ϕ0 (x) = ϕ(x, t), (x, t) ∈ S0 . Let the function ϕ(x, t), (x, t) ∈ S satisfy the condition ϕ(·, 0) ∈ C l0 (D) (i.e., ϕ0 ∈ C l0 (D)), L and for the function ϕ(x, t) considered on S , the derivatives (∂ k0 /∂tk0 )ϕ(x, t), where k0 ≤ l/2, are defined for (x, t) ∈ S0c . Using the function ϕ0 (x) prescribed on the set S0 and the equation (5.1a), we find the derivative in t of the function u(x, t) on S0 . We denote it by (∂/∂t) ϕ0,t=0 (x). Furthermore, differentiating the equation (5.1a) in x and t, we find the derivatives in t up to order k0 ≤ [l/2]i , where [a]i is the integer part of the number a ≥ 0; we denote these derivatives by (∂ k0 /∂tk0 ) ϕ0,t=0 (x), x ∈ D. We say that the data of the boundary value problem satisfy a compatibility condition on the set S0c guaranteeing the continuity of the derivatives in t up to order K0 of the function u(x, t) on S0c , or, briefly, the problem data satisfy a compatibility condition on S0c for the derivatives in t up to order K0 [67, 37], if one has the condition ∂ k0 ∂ k0 ϕ(x, t) = ϕ0,t=0 (x), k ∂t 0 ∂tk0

(x, t) ∈ S0c , 0 ≤ k0 ≤ K0 .

(5.5)

Under the conditions given above, we have K0 ≤ [l/2]i . Using a maximum principle, one can verify that the problem solution is stable, i.e., that £ ¤ |u(x, t)| ≤ M max max |f (x, t)|, max |ϕ(x, t)| , (x, t) ∈ G, G

S

and that it continuously approaches the boundary data on S L (the lateral boundary of the set G), i.e., ¡ ¢ |u(x, t) − ϕ(x∗ , t)| ≤ M ε−1 r x, Γ , (x, t) ∈ G, (5.6) L

∗ ∗ where (x ¡ ¢ , t) is a point on S nearest to the point (x, t) ∈ G, thus x = ∗ x x, Γ . Under the condition ask , bs , c, c1 , p, f ∈ C l0 ,l0 /2 (G), ϕ ∈ C(S), we have u ∈ C l1 ,l1 /2 (G), where l1 = l0 + 2, l0 = l + α, l ≥ 0 is an integer number, and α ∈ (0, 1). Under the additional condition ϕ ∈ C α1 (S0 ), ϕ ∈ C α1 ,α1 /2 (S j ), j = 1, . . . , J, where J = 2 in the case of the slab while J = 2 n in the case T of the parallelepiped, we have u ∈ C α2 ,α2 /2 (G) C l1 ,l1 /2 (G), α2 ∈ (0, α1 ), l1 l1 ,l1 /2 α1 ∈ (0, 1) [67, 37]. Let ϕ ∈ C (S0 ), ϕ ∈ C (S j ), j = 1, . . . , J, i.e., g g S0 = S0 , S j = S j in the case of the function u(x, t), (x, t) ∈ G. Taking into account interior a priori estimates and the estimates up to the smooth parts of the boundary (derived using Schauder estimates in the variables ξ, t, where ξs = ε−1 xs , s = 1, . . . , n), we find ¯ ¯ ¯ ¯ £ ¡ ¢¤ £ ¡ ¢¤ ∂ k+k0 ¯ ¯ u(x, t) ¯ k1 ¯ ≤ M ε−k + r−k x, Γ 1 + ε2k0 r−2k0 x, Γ , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯

(x, t) ∈ G,

x∈ / Γ for k + k0 > 0,


0 ≤ k + 2k0 ≤ K,

(5.7)


137

where K = l + 2. But if in the case of the problem on the slab (5.2), the boundary function ϕ(x, t) is sufficiently smooth on each smooth part of the boundary S, namely, ϕ ∈ C l0 +2 (S0 ) and ϕ ∈ C l0 +2,l0 /2+1 (S j ), where j = 1, 2, and, moreover, for the data of the problem on the set S0c compatibility conditions up to order K0 are fulfilled, where K0 = [l/2]i + 1, then u ∈ C l0 +2,l0 /2+1 (G) [67, 37]. Using the a priori estimates up to the boundary (derived in the variables ξ, t), we obtain the estimate ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ u(x, t) (5.8) ¯ k1 ¯ ≤ M ε−k , (x, t) ∈ G, 0 ≤ k + 2k0 ≤ K, k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ where K = l + 2. Let us find estimates in a neighborhood of the boundary layers. The derivation of the estimates of the solutions for problems on the slab and on the parallelepiped is analogous to those in Theorems 2.2.1 and 3.2.1. Consider the problem on the slab (5.2). Assume that the problem data are sufficiently smooth and the boundary function ϕ(x, t) is continuous, but the fulfillment of compatibility conditions on the set S0c is, in general, not assumed. Write the problem solution as the sum of the functions u(x, t) = U (x, t) + V (x, t),

(x, t) ∈ G,

(5.9)

where U (x, t) and V (x, t) are the regular and singular parts of the solution of the problem. The function U (x, t) is the restriction to G of the function e e S U e (x, t), (x, t) ∈ G , where G = Ge S e , Ge = De × (0, T ] and De = IRn . e e The function U (x, t), (x, t) ∈ G is the bounded solution of the problem Le U e (x, t) = f e (x, t), (x, t) ∈ Ge ,

U e (x, t) = ϕ e (x), (x, t) ∈ S e . (5.10)

The domain Ge is an extension of the domain G beyond the boundary S L . The coefficients and the right-hand side in (5.10) are smooth extensions of those in (5.1a), preserving their properties. Let Le ≡ ε2 Le2 + Le1 , and assume that Le2 = 4 and Le1 = −c−∂/∂t outside an m-neighborhood of the set G and that there the function f e (x, t) vanishes. The function ϕ e (x) is sufficiently smooth on the boundary S e = S0e and coincides with the function ϕ(x, t) on the set S0 ; the function ϕ e (x) is assumed to be equal to zero outside an mneighborhood of the set D. The function V (x, t), (x, t) ∈ G, is the solution of the problem LV (x, t) = 0, (x, t) ∈ G,

V (x, t) = ϕV (x, t), (x, t) ∈ S,

where ϕV (x, t) = ϕ(x, t)−U (x, t), (x, t) ∈ S. The function V (x, t) vanishes on the set S0 . Write the function U e (x, t) as the sum of the functions U e (x, t) = U0e (x, t) + v1e (x, t),


e

(x, t) ∈ G .

138


The functions U0e (x, t) and v1e (x, t) are solutions of the following problems: Le1 U0e (x, t) = f e (x, t),

(x, t) ∈ Ge ,

U0e (x, t) = ϕe (x),

(x, t) ∈ S e ;

Le v1e (x, t) = −ε2 Le2 U0e (x, t), v1e (x, t) = 0,

(x, t) ∈ G e ,

(x, t) ∈ S e .

Let the data of the problem (5.1), (5.2) satisfy the condition ask , bs , c, c1 , p, f ∈ C l0 ,l0 /2 (G), ϕ ∈ C l0 (S0 ), l0 = l + α, l ≥ 2 where α ∈ (0, 1), and let the e e extensions of these data to the set G belong to the class C l0 ,l0 /2 (G ). Then e for the functions U0e (x, t) and v1e (x, t), one has U0e , v1e ∈ C l0 ,l0 /2 (G ). The functions U0e (x, t) and v1e (x, t) satisfy the estimates ¯ ¯ ¯ ¯ ∂ k+k0 e ¯ ¯ e U (x, t) ¯ k1 ¯ ≤ M, |v1e (x, t)| ≤ M ε2 , (x, t) ∈ G , 0 ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ 0 ≤ k + 2k0 ≤ K. Taking into account interior a priori estimates, for derivatives of v1e (x, t) we find the estimate ¯ ¯ ¯ ¯ ∂ k+k0 e ¯ ¯ e v (x, t) ¯ k1 ¯ ≤ M ε2−k , (x, t) ∈ G , 0 ≤ k + 2k0 ≤ K. 1 k n k ¯ ∂x1 . . . ∂xn ∂t 0 ¯ Thus U ∈ C l0 ,l0 /2 (G), and for the function U (x, t) we have the estimate ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ U (x, t) (5.11) ¯ k1 ¯ ≤ M 1 + ε2−k , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ (x, t) ∈ G,

0 ≤ k + 2k0 ≤ K,

where K = l. For the function V (x, t), one gets the estimate ¡ ¢ |V (x, t)| ≤ M exp − m ε−1 r(x, Γ) , (x, t) ∈ G, where m is an arbitrary constant. For the data of the boundary value problem (5.1), (5.2), besides the conditions given above, let the condition ϕ ∈ C l0 ,l0 /2 (S j ), j = 1, 2 also be fulfilled. Moreover, for the problem data on the set S0c , compatibility conditions up to order K0 , where K0 = [l/2]i , are assumed to be fulfilled. In that case V ∈ C l0 ,l0 /2 (G). With regard to the a priori estimates up to the boundary (derived in the variables ξ, t, where ξs = ε−1 xs and s = 1, . . . , n), we obtain ¯ ¯ ¯ ¯ ¡ ¢ ∂ k+k0 ¯ ¯ V (x, t) ¯ k1 ¯ ≤ M ε−k exp − m ε−1 r(x, Γ) , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ (x, t) ∈ G,


0 ≤ k + 2k0 ≤ K.


139

The derivation of ϕV and our previous estimates yield a bound on the derivaL tives of ϕV (x, t), (x, t) ∈ S (not including differentiation in x1 ) ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ ϕ (x, t) (5.12) ¯ k2 ¯ ≤ M 1 + ε2−k , V ¯ ∂x2 . . . ∂xknn ∂tk0 ¯ L

(x, t) ∈ S ,

0 ≤ k + 2k0 ≤ K,

k = k2 + . . . + kn .

Taking (5.12) into account, we find the refined estimate ¯ ¯ ¯ ¯ £ ¤ ¡ ¢ ∂ k+k0 ¯ ¯ V (x, t) ¯ k1 ¯ ≤ M ε−k1 + ε2−k exp − m ε−1 r(x, Γ) , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ (x, t) ∈ G,

0 ≤ k + 2k0 ≤ K,

(5.13)

where K = l and m is an arbitrary constant. In the case when the fulfillment of compatibility conditions on S0c is not assuL Lg T med, we have V ∈ C α1 ,α1 /2 (G) C l0 ,l0 /2 (G), moreover, S0 = S0g , S = S , l0 = l + α, l ≥ 0, and α, α1 ∈ (0, 1). For the function V (x, t), one obtains the estimate ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ V (x, t) ¯ k1 ¯ ≤ M ε−k + r−k (x, Γ) × k n k ¯ ∂x1 . . . ∂xn ∂t 0 ¯ £ ¤ ¡ ¢ × 1 + ε2k0 r−2k0 (x, Γ) exp − m ε−1 r(x, Γ) , x∈ / Γ for k + k0 > 0,

(x, t) ∈ G,

0 ≤ k + 2k0 ≤ K.

Note that the function V (x, t) is equal to zero on S0 , and for the function ϕV (x, t) the estimate (5.12) holds. Taking into account the a priori estimate up to the smooth parts of the boundary, we find the estimate ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ V (x, t) (5.14) ¯ k1 ¯ ≤ M ε−k1 + ε1−k + r−k (x, Γ) × k n k ¯ ∂x1 . . . ∂xn ∂t 0 ¯ £ ¤ ¡ ¢ × 1 + ε2k0 r−2k0 (x, Γ) exp − m ε−1 r(x, Γ) , (x, t) ∈ G, x∈ / Γ for k + k0 > 0,

0 ≤ k + 2k0 ≤ K,

where K = l, and m is an arbitrary constant. The following theorem holds. Theorem 5.2.1 Let ask , bs , c, c1 , p, f ∈ C l0 ,l0 /2 (G), s, k = 1, . . . , n, ϕ ∈ L T T C l0 ,l0 /2 (S ) C l0 (S0 ) C(S), l0 = l + α, l ≥ K, K ≥ 2, α ∈ (0, 1). Then for the function u(x, t), i.e., the solution of the problem (5.1), (5.2) on the slab, and for the components in the representation (5.9), the estimates (5.6), (5.7),


140


(5.11), and (5.14) hold. In the case when on the set S0c the problem data satisfy the compatibility conditions (5.5) up to order [K/2]i , the estimates (5.8), (5.11), (5.13) hold. In the case of the problem on the parallelepiped (5.3), we write its solution as the decomposition u(x, t) = U (x, t) + V (x, t),

(x, t) ∈ G.

(5.15)

The function U (x, t), i.e., the regular component, is constructed similarly to U (x, t) in the representation (5.9). The function V (x, t), i.e., the singular component of the solution, is the solution of the problem LV (x, t) = 0, (x, t) ∈ G,

V (x, t) = ϕV (x, t), (x, t) ∈ S,

where ϕV (x, t) = ϕ(x, t) − U (x, t), (x, t) ∈ S. T Under the conditions ask , bs , c, c1 , p, f ∈ C l0 ,l0 /2 (G), ϕ ∈ C l0 (S0 ) C(S), l0 = l + α, l ≥ 2, α ∈ (0, 1), we have U ∈ C l0 ,l0 /2 (G) and V ∈ C l0 ,l0 /2 (G). For the functions U (x, t) and V (x, t), one obtains the estimates ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ U (x, t) ¯ k1 ¯ ≤ M 1 + ε2−k , (x, t) ∈ G; k n k ¯ ∂x1 . . . ∂xn ∂t 0 ¯ ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ V (x, t) ¯ k1 ¯ ≤ M ε−k + r−k (x, Γ) × ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ £ ¤ ¡ ¢ × 1 + ε2k0 r−2k0 (x, Γ) exp − m ε−1 r(x, Γ) , x∈ / Γ for k + k0 > 0,

(x, t) ∈ G,

0 ≤ k + 2k0 ≤ K.

where K = l, and m is an arbitrary constant. Further, we need more refined estimates for the singular component of the solution. Let for the problem data as well as the condition given above, the additional condition ϕ ∈ C l0 ,l0 /2 (S j ) holds, where j = 1, . . . , 2n. In that case, V ∈ C α1 ,α1 /2 (G), where α1 ∈ (0, 1). Taking into account the condition ϕV ∈ g C l0 ,l0 /2 (S j ), we have S j = S j , j = 1, . . . , 2n in the case of the function V (x, t), (x, t) ∈ G. Set G(j...r) = D(j...r) × (0, T ], j, . . . , r = 1, . . . , 2n, S

where D(j...r) = D(j...r)(3.6) and G(j...r) = G(j...r) S(j...r) . Write the function V (x, t), (x, t) ∈ G, as the sum of the functions: V (x, t) = V0 (x, t) + v2 (x, t),

(x, t) ∈ G,

(5.16)

where V0 (x, t) and v2 (x, t) are the main term of the singular part of the solution and the remainder term. The functions V0 (x, t) and v2 (x, t), (x, t) ∈ G are solutions of homogeneous equations with homogeneous conditions.



141

Let us present the function V0 (x, t), (x, t) ∈ G, as the sum of the boundary layer functions of dimension from 1 to n: X X V0 (x, t) = V(j) (x, t) + V(j...r) (x, t), (x, t) ∈ G, (5.17) j=1,...,2n

j,..., r=1,...,2n j 0,

0 ≤ k + 2k0 ≤ K.

Thus the solution of the boundary value problem can be presented as the sum of the functions u(x, t) = U(0) (x, t) + V0 (x, t),

(x, t) ∈ G,

U(0) (x, t) = U (x, t) + v2 (x, t),

(x, t) ∈ G.

(5.18)

where For the function u(x, t) and its components U(0) (x, t) and V(j...r) (x, t) in the representations (5.18) and (5.17), the following estimates ¯ ¯ ¯ ¯ h ¡ ¢i ∂ k+k0 ¯ ¯ −k −k u(x, t) ≤ M ε + r x, Γ × (5.19a) ¯ k1 ¯ k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ h ¡ ¢i × 1 + ε2k0 r−2k0 x, Γ , ¯ ¯ ¯ ¯ h ¡ ¢i ∂ k+k0 ¯ ¯ 2−k 2 −k U (x, t) ≤ M 1 + ε + ε r x, Γ × (5.19b) ¯ k1 ¯ (0) k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ h ¡ ¢i × 1 + ε2k0 r−2k0 x, Γ , ¯ ¯ ¯ ¯ h ∂ k+k0 ¯ ¯ V (x, t) ≤ M ε−(kj +...+kr ) + ε1−k + (5.19c) ¯ k1 ¯ (j...r) k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ ih ¡ ¢i + r−(kj +...+kr ) (x, Γ) + ε r−k (x, Γ) 1 + ε2k0 r−2k0 x, Γ × ¡ ¢ T × exp − m ε−1 r(x, q=j,...,r Γq ) , 0 ≤ k + 2k0 ≤ K,


(x, t) ∈ G, x ∈ / Γ for k + k0 > 0,

j, . . . , r = 1, . . . , 2n, 1 ≤ |j . . . r| ≤ n;

144

Parabolic reaction-diffusion equations |u(x, t) − ϕ(x∗ , t)| ≤ M ε−1 r (x, Γ) ,

(x, t) ∈ G,

(5.19d)

are valid, where x∗ = x∗(5.6) (x, Γ) is a point on Γ(D) nearest to the point P x ∈ D, (x, t) ∈ G. In (5.19) k p = k p(3.16) (ks ), p = j, . . . , r, k = ns=1 ks , K = l, and the constant m can be chosen arbitrarily. The following theorem holds. Theorem 5.2.2 Let ask , bs , c, c1 , p, f ∈ C l0 ,l0 /2 (G), s, k = 1, . . . , n, ϕ ∈ T T C l0 ,l0 /2 (S j ) C l0 (S0 ) C(S), j = 1, . . . , 2n, l0 = l + α, l ≥ K, K ≥ 2, α ∈ (0, 1). Then for the function u(x, t), i.e., the solution of the problem (5.1) on the parallelepiped (5.3), and for the components U(0) (x, t) and V(j...r) (x, t) in the representations (5.18) and (5.17) of the function u(x, t), the estimates (5.19) hold. The smoothness of the data of the problem (5.1), (5.3) does not, in general, imply the smoothness of the solution on the set G (see, e.g., [67, 37]). In some problems, for the function u(x, t), one has the inclusion u ∈ C l0 ,l0 /2 (G),

l0 = l + α,

l = K,

α ∈ (0, 1),

(5.20)

if on the set S c (G), the data of the problem (5.1), (5.3) satisfy special compatibility conditions. In some cases, the inclusion (5.20) allows us to obtain estimates of derivatives of the components in (5.18) and (5.17) on the set G. In the case of the problem (5.1), (5.3), assume that compatibility conditions in t up to order K0 = [K/2]i are satisfied on the set S0c . In that case, for the solution of the boundary value problem and for the components in (5.18) and (5.17), the estimates (5.19) are satisfied, where r(x, Γ) is r(x, Γc ). This estimate is used outside an (mε)-neighborhood of the set S Lc . In the cases n = 2, 3, compatibility conditions in x on the set S Lc , which are necessary and sufficient for (5.20) for K = 2, can be written out based on results of [212, 213]. For example, if L ≡ ε2 4 − c1 (x, t) − p(x, t) r (x, Γc ) ≤ m ε,

∂ , ∂t

(x, t) ∈ G,

(5.21)

t ∈ [0, T ],

then in a neighborhood of the set S Lc one rewrites the parabolic problem corresponding to (5.21) as an elliptic problem where the term p(x, t) ∂/∂t has been moved to the right-hand side and t is regarded as a parameter. Applying compatibility conditions from [212, 213] to the elliptic problem and also to the problem obtained by differentiation in t of the problem (5.1), (5.21), it is not difficult to write down compatibility conditions on S c for the problem (5.1), (5.21), which guarantee that (5.20) holds. Using the a priori estimates of the ¡ ¢ solution up to the smooth parts of the boundary (for ¡r x, Γ¢c ≥ m1 ε) and the estimates in a neighborhood of “edges” S Lc (for r x, Γc ≤ m2 ε, where



145

e m2 > m1 ), then for the functions V(j...r) (x, t) and v2 (x, t) for n = 2, 3, one can justify the estimates ¯ ¯ ¯ ¯ h i ∂ k+k0 ¯ ¯ e V (x, t) ≤ M ε−(kj +...+kr ) + ε2−k × ¯ k1 ¯ (j...r) k n ¯ ∂x1 . . . ∂xn ∂tk0 ¯ ³ ¡ T ¢´ × exp − m ε−1 r x, q=j,...,r Γq (D(j...r) ) , (x, t) ∈ G(j...r) ,

¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ v (x, t) ¯ k1 ¯ ≤ M ε2−k , 2 k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯

(x, t) ∈ G,

0 ≤ k + 2k0 ≤ K,

where K = l. In the case of the conditions (5.21), if the membership (5.20) holds, the functions u(x, t), U(0) (x, t) and V(j...r) (x, t) then satisfy the estimates ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ u(x, t) (5.22a) ¯ k1 ¯ ≤ M ε−k , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ U (x, t) (5.22b) ¯ k1 ¯ ≤ M 1 + ε2−k , (0) ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ V (x, t) ≤ M ε−(kj +...+kr ) + ε2−k × (5.22c) ¯ k1 ¯ (j...r) k n ¯ ∂x1 . . . ∂xn ∂tk0 ¯ ³ ¢´ ¡ T × exp − m ε−1 r x, q=j,...,r Γq , (x, t) ∈ G, 0 ≤ k + 2k0 ≤ K, j, . . . , r = 1, . . . , 2n, 1 ≤ |j . . . r| ≤ n. In (5.22) we have k p = k p(3.16) (ks ) for p = j, . . . , r, while K = l, and the constant m can be chosen arbitrarily. The following theorem holds. Theorem 5.2.3 Let n ≤ 3. Let the assumptions of Theorem 5.2.2, the condition (5.21), and the membership (5.20) be fulfilled. Then the function u(x, t), i.e., the solution of the problem (5.1) on the parallelepiped (5.3), and the components U(0) (x, t) and V(j...r) (x, t) in the representations (5.18) and (5.17) satisfy the estimates (5.22).

5.3


When constructing ε-uniformly convergent difference schemes for singularly perturbed parabolic reaction-diffusion equations, we use a technique that


146


is similar to the one used for singularly perturbed elliptic reaction-diffusion equations. For the ε-uniform convergence of difference schemes whose construction is based on classical difference approximations of the boundary value problem (5.1), it is necessary to use meshes that condense in a neighborhood of the boundary layers. In order that the difference schemes on such meshes be ε-uniformly monotone, it is required that the coefficients of the mixed derivatives in the equation at the domain boundary are subject to special conditions as are the relations between the step-sizes in the different space coordinates.

5.3.1

Grid approximations of the boundary value problem

In the case of the parallelepiped (5.3) on the set G, we introduce the rectangular grid Gh = Dh × ω 0 , Dh = ω 1 × . . . × ω n , (5.23a) where ω 0 is a mesh on the interval [0, T ] on the t-axis with an arbitrary distribution of nodes, while ω s , for s = 1, . . . , n, is a mesh on the interval [d∗s , d∗s ] on the xs -axis with an arbitrary distribution of nodes. In the case of the slab (5.2), we use the grid Gh = Dh × ω 0 ,

Dh = ω 1 × ω2 × . . . × ωn ,

(5.23b)

where ω 0 = ω 0(5.23a) , ω 1 = ω 1(5.23a) and ωs , for s = 2, . . . , n, is a mesh on the xs -axis. Set ht = max htj , where hjt = tj+1 − tj with tj , tj+1 ∈ ω 0 . Let j

N0 + 1 be the number of nodes in the mesh ω 0 . Set h = max hs , hs = max his , s

i

his = xi+1 − xis . Let Ns + 1 be the number of nodes in the mesh ω s(5.23a) , s s = 1, . . . , n, in the case of the problem on the parallelepiped; in the case of the problem on the slab N1 + 1 is the number of nodes in the mesh ω 1(5.23b) and Ns + 1 is the minimal number of nodes in the mesh ω s(5.23b) , s = 2, . . . , n, per unit length; N = min Ns , s = 1, . . . , n. Assume that the conditions s

T

T

h ≤ M N −1 and ht ≤ M N0−1 are fulfilled. Set Gh = G Gh and Sh = S Gh . On the grid Gh(5.23) for the problem (5.1), we consider the difference scheme Λ z(x, t) = f (x, t), (x, t) ∈ Gh , Here Λ = Λ(5.25) (L) ≡ ε2

z(x, t) = ϕ(x, t), (x, t) ∈ Sh .

½X n

ass (x, t)δxs xcs +

(5.24)

(5.25)

s=1

+2

−1

n h X

i − (a+ sk (x, t) (δxs xk + δxs xk ) + ask (x, t)(δxs xk + δxs xk ) +

s,k=1 s6=k

+

n h X

¾ i − − c(x, b+ (x, t)δ + b (x, t)δ t) − c1 (x, t) − p(x, t)δt , xs xs s s

s=1


ε-uniformly convergent difference schemes ¡ ¢−1 £ ¤ where δt z(x, t) = hj−1 z(x, tj ) − z(x, tj−1 ) , t tj , tj−1 ∈ ω 0 . The operator L∗(2) ≡

n X

147 hj−1 = tj − tj−1 , t

ask (x, t)∂ 2 /∂xs ∂xk

(5.26)

s,k=1

is the elliptic part of the operator L(5.1) . We approximate it by the discrete operator Λ∗(2)

=

Λ∗(2) (L∗(2) )

≡

n X

ass (x, t)δxs xcs +

(5.27)

s=1

+2−1

n h X

i − a+ (x, t)(δ + δ ) + a (x, t)(δ + δ ) xs xk sk sk xs xk xs xk xs xk .

s,k=1 s6=k

The difference scheme (5.24), (5.25), (5.23)} is, in general, not ε-uniformly monotone. The monotonicity of the operator Λ(5.25) on the grid Gh is violated, e.g., when one has ask (x, t) ≥ 0,

(x, t) ∈ G,

s, k = 1, . . . , s 6= k,

but the following condition is violated: n h ¡ ¢−1 X ¡ s ) ¢−1 i min akk (x, t) 2 hikk + hikk −1 − ask (x, t) h(i ≥ 0, s (i )

hs s ,k,Gh

s=1 s6=k

x = (xi11 , . . . , xinn ),

(x, t) ∈ G, (i )

k = 1, . . . , n,

(i )

where hs s = hiss , or hs s = hiss −1 , s = 1, . . . , n. Imposing some conditions on the differential operator L∗(2) , we construct grids Gh with a special distribution of nodes in the meshes xs , for s = 1, . . . , n, on which the difference operators Λ∗(2)(5.27) and Λ(5.25) and, hence, the difference scheme (5.24) are ε-uniformly monotone. These conditions are given below.

5.3.2

Consistent grids on a slab

Let us consider the problem (5.2) on the slab D(5.2) . On the set G we introduce the uniform grid Gh = Dh × ω 0 ,

Dh = ω 1 × ω2 × . . . × ωn ,

(5.28)

where ωs , for s = 1, . . . , n, are uniform meshes with step-size hs and ω 0 is a uniform mesh with step-size ht . Let Q be a convex subdomain of G. On the set Q we introduce the uniform grid T

Gh (Q) = {Q }h = Q Gh(5.28) .


(5.29)

148


The discrete operator Λ∗(2)(5.27) corresponds to the operator L∗(2)(5.26) on the grid Gh(5.29) (Q). The elliptic operator L∗(2)(5.26) , considered at the point (x0 , t0 ), can be brought to canonical form by an appropriate linear transformation X = X(x, t0 ). In the transformed operator L∗(2)X , the maxima of the coefficients of the mixed derivatives at the point (X, t) depend on the distance to the point (X0 , t0 ), where X0 = X(x0 , t0 ), and these maxima, in general, increase as this distance increases. Let L∗(2)(5.26) be already the canonical operator at the point (x0 , t0 ). Then in a small neighborhood of (X0 , t0 ), the operator L∗(2) has “almost canonical” form, i.e., the coefficients of the mixed derivatives are small compared with the coefficients of the second-order derivatives in each variable. We shall assume that for an arbitrary point (x∗ , t∗ ) ∈ G, one can find (i) a sufficiently small neighborhood Q(x∗ , t∗ ), where Q(x∗ , t∗ ) ⊆ G, (ii) n pairs of functions ρ∗s (x, t) and ρ∗s (x, t), (x, t) ∈ Q(x∗ , t∗ ), for s = 1, . . . , n, such that (iii) for these functions one has max ρ∗s (x, t) ≤

Q(x∗ , t∗ )

min ρ∗s (x, t),

Q(x∗ , t∗ )

s = 1, . . . , n,

(5.30a)

(iv) the coefficients ask (x, t), for s, k = 1, . . . , n, satisfy the relation min akk (x,

Q(x∗ , t∗ )

t)h−1 k

−

n X s=1 s6=k

max |ask (x, t)| h−1 s > 0,

Q(x∗ , t∗ )

(5.30b)

k = 1, . . . , n,

where (v) the values hs , for s = 1, . . . , n, in (5.30b) satisfy the condition " # hs ∈

max

Q(x∗ , t∗ )

ρ∗s (x, t) h∗ ,

min

Q(x∗ , t∗ )

ρ∗s (x, t) h∗ ,

s = 1, . . . , n,

(5.30c)

and h∗ > 0 is an arbitrary number. The functions ρ∗s (x, t) and ρ∗s (x, t) are defined by the coefficients of the operator L∗(2)(5.26) : ³ ´ ³ ´ ρ∗s (x, t) = ρ∗s x, t; L∗(2) , ρ∗s (x, t) = ρ∗s x, t; L∗(2) , s = 1, . . . , n. The values ρ∗s (x, t) and ρ∗s (x, t) can be chosen to satisfy the relations max ρ∗s (x, t) → 0,

Q(x∗ ,t∗ )

for


max

s,k,s6=k, Q(x∗ , t∗ )

min ρ∗s (x, t) → ∞

Q(x∗ ,t∗ )

|ask (x, t)| → 0.


149

In the general case, one can ensure the fulfillment of the relations (5.30) if one brings the operator L∗(2)(5.26) to the canonical form at the point (x∗ , t∗ ) (or close to the canonical form) and chooses the neighborhood Q(x∗ , t∗ ) to be sufficiently small. We say that the operator L∗(2)(5.26) on the set G has local almost canonical form (in the variables x1 , . . . , xn ), if for an arbitrary point (x∗ , t∗ ) ∈ G one can find a neighborhood Q(x∗ , t∗ ) and n pairs of the functions ρ∗s (x, t) and ρ∗s (x, t), x ∈ Q(x∗ , t∗ ), such that the relations (5.30) hold for the coefficients of the operator L∗(2) . Note that when defining the local almost canonical operator, in the relations (5.30) we use the values hs , for s = 1, . . . , n, which, in general, have no relations to grids on the set Q(x∗ , t∗ ). Grids on the set Q(x∗ , t∗ ) are considered below. T For the set Q Gh and the operator Λ∗(2)(5.27) (the operator Λ(5.25) ), we consider the node (x0 , t0 ) as interior, if all nodes of the stencil, for which the point (x0 , t0 ) is centre, of the operator Λ∗(2)(5.27) (the operator Λ(5.25) ) belong T to the set Q Gh . Theorem 5.3.1 Assume that the grid Gh is uniform on the set Q(x∗ , t∗ ) in each of the variables, i.e., ¡ ¢ ¡ ¢ Gh = Gh Q(x∗ , t∗ ), L∗(2) = Gh Q(x∗ , t∗ ), L∗(2) , x1 , . . . , xn = (5.31) n© o ª = Q(x∗ , t∗ ) h(5.29) under the condition (5.30c), and when (5.30) holds . Then the operator Λ(5.25) (L) (the operator Λ∗(2)(5.27) (L∗(2) )) is ε-uniformly monotone on the set of the interior nodes. Thus, the uniform grid Gh(5.31) is consistent on Q(x∗ , t∗ ) in x1 , . . . , xn with the monotonicity condition for the operators Λ∗(2)(5.27) and Λ(5.25) . We are interested in monotone difference approximations of the boundary value problem (5.1) on the slab (5.2). The following statement is valid. Lemma 5.3.1 For the mesh ω 1 that defines the grid Gh(5.23b) , the condition ε−1 ε−1

max i

 hi−1  1

maxi

hi1 ,

d∗1 <x1 ≤d∗1 +M ε ∗ d∗ 1 −M ε≤x1 0, (ii) a neighborhood Q(x∗ , t∗ ), and (iii) n pairs of functions ρ∗s (x, t) and ρ∗s (x, t), for s = 1, . . . , n, such that



151

(iv) the coefficients ask (x, t), for s, k = 1, . . . , n, of the operator L∗(2) sat¡ isfy the relations (5.30) i.e., the operator L∗(2) is locally almost canonical in ¢ x1 , . . . , xn on the set G ; (v) the values m∗ , ν > 0 do not appear in (5.30). L We modify (iv) in an m∗ -neighborhood of S , where we assume that the coefficients ask (x, t) satisfy (5.30) for s, k = 2, . . . , n, and that one has the additional relation h min

Q(x∗ , t∗ )

n i X akk (x, t) − ν h−1 − k

max

Q(x∗ , t∗ )

|ask (x, t)| h−1 s > 0,

(5.36)

s=2 r(x,Γ)≤m∗ s6=k

r(x,Γ)≤m∗

k = 2, . . . , n. Thus, in the above mentioned relations (5.30) and (5.36), considered in a L neighborhood of S , the coefficients ask (x, t) are involved only for s, k ≥ 2, i.e., only the coefficients of the truncated operator ∗[1]

∗[1]

L(2) = L(2) (L∗(2) ) ∗[1]

appear. In the case when the coefficients of the operator L(2) satisfy the ∗[1]

conditions {(5.30), (5.36)} for s, k ≥ 2, the operator L(2) is strongly local L

almost canonical in x2 , . . . , xn in a neighborhood of the set S . The fulfillment of {(5.30), (5.36)} in the case of the condition (5.34) can be ensured by a linear transformation of the variables x2 , . . . , xn that brings L the operator L∗(2) to the canonical form at the point (x∗ , t∗ ) ∈ S (or to a form close to canonical), and by choosing the neighborhood Q(x∗ , t∗ ) and the values m∗ , ν to be sufficiently small. The condition (5.34) and the condition the coefficients ask (x, t) of the operator L∗(2) on the set Q(x∗ , t∗ ) ⊆ G satisfy the relations {(5.30), (5.36)}

(5.37)

together allow us to construct grids on the set Q(x∗ , t∗ ) that are consistent piecewise-uniform on the x1 -axis and thereby ensure the monotonicity of the operators Λ∗(2)(5.27) and Λ(5.25) . Let the conditions (5.34) and (5.37) be satisfied. Consider the grid T

Gh (Q) = {Q }h = Q Gh(5.33) ,

(5.38)

which is piecewise-uniform in x1 . We shall provide conditions on the parameters of this grid under which the operator Λ∗(2)(5.27) is monotone. First, consider the operator Λ∗(2)(5.27) on the grid Gh(5.33) . Let the stepsizes hs in the meshes ωs , for s = 2, . . . , n, and the step-size h1 , i.e., the


152


maximal step-size in the mesh ω 1(2.48) , satisfy the condition (5.30c). Note that h1 = h(2) in the mesh (2.48). Let the value σ = σ(2.48) satisfy the condition σ ≤ m∗(5.36) ,

(5.39a)

and let the condition (5.35) hold. By virtue of conditions (5.34), (5.35) one has ¯ ¯ ¯ ∂ ¯ ¡ L¢ |a1s (x, t)| ≤ M1 σ for r (x, t), S ≤ σ, where M1 = max ¯¯ a1s (x, t)¯¯, s, s6=1, G ∂x1 and it follows that ¯ ¯ ¯a1k (x, t)¯ h−1 ≤ M1 h−1 σ, (1) (1)

k 6= 1,

£ ¤ where step-size in the mesh ω 1(2.48) on the intervals d∗1 , d∗1 + σ ¤ £ ∗h(1) is the and d1 − σ, d∗1 . The inequality ν h−1 k −

max

Q(x∗ , t∗ ) x1 ∈(d / ∗1 +σ, d∗ 1 −σ)

is fulfilled if M1 h−1 (1) σ < ν

|a1k (x, t)| h−1 (1) > 0,

k = 2, . . . , n,

min [h−1 k ], i.e., when one has

k=2,...,n

−1 N∗1 ≡ σ h−1 (1) < M1 ν

min [h−1 k ],

k=2,...,n

(5.39b)

where N∗1 + 1 is the number of nodes in the mesh ω 1(2.48) on the interval [d∗1 , d∗1 + σ] or [d∗1 − σ, d∗1 ]. Under the above conditions on the values σ and N∗1 , the operators Λ∗(2)(5.27) and Λ(5.25) are ε-uniformly monotone on the grid Gh(5.38) (Q), where Q is Q(x∗ , t∗ ) from {(5.30), (5.36)}. Thus, in the case of {(5.30), (5.36)}, the condition (5.39) that is imposed on the parameters of the mesh ω 1(2.48) is a sufficient condition for the local ε-uniform monotonicity of the operators Λ∗(2) and Λ on the piecewise-uniform mesh Gh(5.38) (Q). In the case of the conditions (5.34), (5.35), (5.37), when constructing the consistent mesh (5.38) we use the meshes © ª ω 1 = ω 1(2.48) under the conditions (5.30c), (5.39) . (5.40) Then the grid ¡ ¢ Gh = Gh Q(x∗ , t∗ ), L∗(2) = (5.41) ¡ ¢ = Gh Q(x∗ , t∗ ), L∗(2) , ω 1(5.40) , x1 , . . . , xn = © T = Gh(5.33) Q(x∗ , t∗ ), ω 1 = ω 1(2.48) , ωs = ωsu , s = 2, . . . , n,



153 ª

under the conditions (5.30c), (5.39) in the case of {(5.30), (5.36)}

is consistent on the set Q(x∗ , t∗ ) in the variables x1 , . . . , xn with the monotonicity condition for the operators Λ∗(2)(5.27) and Λ(5.25) . Theorem 5.3.3 Let the conditions (5.34), (5.35), (5.37) hold.¢ Then the ope¡ rator Λ(5.25) = Λ(L) on the grid Gh = Gh(5.41) Q(x∗ , t∗ ), L∗(2) is ε-uniformly monotone. Remark 5.3.1 Let the coefficients of the mixed derivatives of the operator L∗(2) satisfy the condition £ ¤ max |ask (x, t)| < (n − 1)−1 min αs αk−1 ass (x, t), αk αs−1 akk (x, t) , (5.42) G

G

s, k = 1, . . . , n,

s 6= k,

where α1 , . . . , αn are some positive numbers, i.e., we have dominance of the diagonal terms in the matrix of the coefficients of the operator L∗(2)(5.26) on the whole set G. Then, under the conditions (5.34), (5.35), one can choose the value ν and the functions ρ∗s (x, t) and ρ∗s (x, t), for s = 1, . . . , n, such that the condition (5.37) is valid for Q(x∗ , t∗ ) = G.

(5.43)

The following theorem holds. Theorem 5.3.4 Let ask ∈ C 1 (G), bs , c, c1 , p, f ∈ C(G), for s, k = 1, . . . , n. Let the coefficients ask (x, t) satisfy the conditions (5.42), (5.34), and (5.37), where Q = G, and let the solution of the problem (5.1), (5.2) satisfy the estimates (5.6), (5.11), (5.14) (or the estimates (5.11), (5.13)) of Theorem 5.2.1 for K = 4. Then the difference scheme (5.24), (5.25) on the grid Gh(5.41) , where Q = G, ω 1(2.48) = ω 1(2.44) , and m1(2.44) = m1(5.14) , converges ε-uniformly. For the discrete solutions, the following estimate is valid: £ ¤ν (5.44) |u(x, t) − z(x, t)| ≤ M N −1 ln N + N0−1 , (x, t) ∈ Gh , where ν = 5−1 in the case of the estimates (5.6), (5.11), (5.14), while ν = 1 in the case of the estimates (5.11), (5.13). Under the hypotheses of Theorem 5.3.4, let ¡ ¢ z(x, t) = z (2.42) x, t; z(·), Gh ,

(x, t) ∈ G,

be the interpolant of the function z(x, t), (x, t) ∈ Gh , which is the solution of the difference scheme (5.24), (5.25) on the grid (5.41), (5.43). Then one has the estimate £ ¤ν |u(x, t) − z(x, t)| ≤ M N −1 ln N + N0−1 , (x, t) ∈ G, ν = ν(5.44) .


154

5.3.3

Parabolic reaction-diffusion equations Consistent grids on a parallelepiped

In the case of the boundary value problem (5.1) on the parallelepiped (5.3), we are interested in monotone difference approximations on piecewise-uniform meshes. We shall consider the difference scheme (5.24), (5.25) on the grids ¡ ¢ Gh = Gh ω s = ω 1(2.48) (d∗s , d∗s ), s = 1, . . . , n , (5.45) that condense in a neighborhood of the faces Sj , for j = 1, . . . , 2n. The following statements are valid. Lemma 5.3.2 The condition  i−1 ε−1 max h  s i ε−1

d∗s <xs ≤d∗s +M ε

maxi

∗ d∗ s −M ε≤xs 2) can be brought to canonical form in the variables X1 , . . . , Xp−1 , Xp+1 , . . . , Xn , where the coefficients of the mixed derivatives not involving differentiation in Xp vanish at the point (X0 , t0 ) with X0 = X(x0 , t0 ). Note that the coefficients of the mixed derivatives involving differentiation in Xp are equal (p)

to zero on S X , i.e., on the faces orthogonal to the Xp -axis. Now that faces have been dealt with, we move on to edges of the parallelepiped. (j...r) At the point (x0 , t0 ) ∈ G in a neighborhood of the set S , where (j...r) (j...r) (j...r) (j...r) S = Γ × (0 T ] and Γ = Γ(3.49) (when |j . . . r| ≤ n − 2), using a transformation of the variables xs , for s = 1, . . . , n, and s 6= j, . . . , r (written as X = X(x, t0 ), where Xs = xs , with s = j, . . . , r), one can bring the operator L∗(2) to canonical form in the variables Xs , for s = 1, . . . , n, with s 6= j, . . . , r, where the coefficients of the mixed derivatives not involving differentiation in Xs , for s = j, . . . , r vanish at the point (X0 , t0 ). The coefficients of the mixed derivatives involving differentiation in Xs , for s = j, . . . , r, (j...r) are equal to zero on S X . Assume that the operator L∗(2) has the above canonical form at the point (x0 , t0 ) ∈ G, which depends on the relative position of the point (x0 , t0 ) and the faces Sj ⊂ S L . If a neighborhood of the point (x0 , t0 ) is sufficiently¡small then in this neighborhood the operator L∗(2) has almost canonical form in all ¢ spatial variables , i.e., the coefficients of the mixed derivatives ∂ 2 /∂xs ∂xk , for s, k = 1, . . . , n, with s 6= k, are small compared with the coefficients of the second-order derivatives ∂ 2 /∂x2s , for s = 1, . . . , n. Under the condition (5.47), let the operator L∗(2)(5.26) have local almost canonical form in x1 , . . . , xn in the set G and local almost canonical form in (p) x1 , . . . , xp−1 , xp+1 , . . . , xn in a neighborhood of the faces S , and in xs , for (j...r)

s = 1, . . . , n, with s 6= j, . . . , r, in a neighborhood of the edges S . (j...r) We shall assume that, in a neighborhood of S , the condition of strongly local almost canonicity of the operator L∗(2) is fulfilled. That is, in the case of the condition (5.47), for an arbitrary point (x∗ , t∗ ) ∈ G, one can find sufficiently small values m∗ and ν > 0, a sufficiently small neighborhood Q(x∗ , t∗ ) and n pairs of the functions ρ∗s (x, t) and ρ∗s (x, t), for s = 1, . . . , n, for which the relation (5.30) holds, and one also has the following condition (similar to (5.36)): h min

i akk (x, t) − ν h−1 k −

Q(x∗ , t∗ ) r(x,Γ(j...r) )≤m∗

n X s=1, s6=k s6=j,...,r

max

|ask (x, t)| h−1 s > 0,

Q(x∗ , t∗ ) r(x,Γ(j...r) )≤m∗

k, j, . . . , r = 1, . . . , n, k 6= j, . . . , r, j < . . . < r, 1 ≤ |j . . . r| ≤ n − 2.


(5.48)

156


The condition (5.47) and the condition the coefficients ask (x, t) of the operator L∗(2) on the set Q(x∗ , t∗ ) ⊆ G satisfy the relations {(5.30), (5.48)}

(5.49)

together allow us to construct on the set Q(x∗ , t∗ ) piecewise-uniform (in all variables) consistent grids and thereby ensure the monotonicity of the operators Λ∗(2)(5.27) and Λ(5.25) . In the case of the conditions (5.47), (5.49), we now impose conditions on the parameters of the piecewise-uniform grid Gh(5.45) under which the operator Λ∗(2)(5.27) is monotone. Let the step-sizes hs , i.e., maximal step-sizes in the meshes ω s = ω 1(2.48) (d∗s , d∗s ; σs , Ns ),

s = 1, . . . , n,

satisfy the conditions (5.30), (5.48). Note that hs = h(2)s in the mesh ω s . Assume that the value σs that defines the transition points in the mesh ω s satisfies the condition σs ≤ m∗(5.48) , and that

ask ∈ C 1 (G),

s = 1, . . . , n,

s, k = 1, . . . , n,

s 6= k.

By virtue of the condition (5.47), one has ¯ ¯ ¡ ¢ ¯ask (x, t)¯ ≤ M1 σs for r (x, t), S s S S s+n ≤ σs , where M1 =

max s,k,r, s6=k, G

(5.50a)

s 6= k,

¯ ∂ ¯ ¯ ¯ ask (x, t)¯. ¯ ∂xr

Then ¯ ¯ ¯ask (x, t)¯ h−1 ≤ M1 h−1 σs (1)s (1)s

¡ ¢ S for r (x, t), S s S s+n ≤ σs .

The inequality ν h−1 k −

X s=j,...,r

max

Q(x∗ , t∗ ) xs ∈(d / ∗s +σs , d∗ s −σs )

|ask (x, t)| h−1 (1)s > 0,

k, j, . . . , r = 1, . . . , n, k 6= j, . . . , r for

¡ (j...r) ¢ r (x, t) S ≤ m∗ ,

where Q(x∗ , t∗ ) is taken from (5.30), (5.48), is satisfied if M1

n X £ ¤ σs h−1 (1)s < ν s=1


min

s=1,...,n

£ −1 ¤ hs ,


157

i.e., when one has the condition n X

£ ¤ N∗s < M1−1 ν min h−1 , k

s=1

k

k = 1, . . . , n.

(5.50b)

Here N∗s + 1 is the number of nodes in the mesh ω s on each of the intervals −1 [d∗s , d∗s + σs ] and [d∗s − σs , d∗s ]; h(1)s = σs N∗s and hs = h(2)s ≤ (d∗s − −1 d∗s ) (Ns − 2 N∗s ) , with Ns ≤ M N∗s . Let the parameters of the meshes ω s , for s = 1, . . . , n, that generate the piecewise-uniform mesh Gh(5.45) be chosen to satisfy the conditions (5.30c), (5.50), where hs = h(2)s . Write these meshes ω s in the following form: © ω s = ω s = ω 1(2.48) (d∗s , d∗s ; σs , Ns ) under the conditions (5.51) ª (5.30c), (5.50) , for s = 1, . . . , n. In the case of L (i) the condition (5.47) on the coefficients ask (x, t) on the boundary S , (ii) the condition (5.49) on the coefficients of the operator L∗(2) and the parameters hs , m∗ , ν on the set G, (iii) the conditions (5.30c), (5.50) on the meshes ω s that generate the piecewise-uniform grid Gh(5.45) , the operators Λ∗(2)(5.27) and Λ(5.25) are ε-uniformly monotone on the grid T Gh(5.45) Q, where Q = Q(x∗ , t∗ ) is taken from {(5.30), (5.48)}. Thus, in the case of the conditions (5.47), (5.49), the grid ³ ´ Gh = Gh Q(x∗ , t∗ ), L∗(2) , ω s(5.51) , x1 , . . . , xn = (5.52) n T = Q(x∗ , t∗ ) Gh(5.45) under the conditions (5.30c), (5.50) o in the case of the relations {(5.30), (5.48)} is consistent on the set Q(x∗ , t∗ ) in the variables x1 , . . . , xn with the monotonicity condition on the operators Λ∗(2)(5.27) and Λ(5.25) . We consider the difference scheme (5.24), (5.25) on the grid ³ ´ Gh = Gh(5.52) Q(x∗ , t∗ ), L∗(2) , ω s(5.51) , x1 , . . . , xn , (5.53) where ω s = ω 1(2.48) is ω 1(2.44) (d∗s , d∗s ; m), and m is an arbitrary constant. In the case of the condition (5.42), one can choose the set Q in (5.53) as Q = G. The following theorem holds. Theorem 5.3.6 Let the following conditions be fulfilled: a) ask ∈ C 1 (G), bs , c, c1 , p, f ∈ C(G), for s, k = 1, . . . , n; b) the coefficients ask (x, t) satisfy


158


the conditions (5.42), (5.47) and {(5.30), (5.48)}, where Q = G; c) for the solution of the problem (5.1), (5.3), the estimates of Theorem 5.2.2 (Theorem 5.2.3) are valid for K = 4. Then the difference scheme (5.24), (5.25) converges ε-uniformly on the grid G(5.53) , where Q = G. The discrete solutions satisfy the estimate £ ¤ν |u(x, t) − z(x, t)| ≤ M N −1 ln N + N0−1 , (x, t) ∈ Gh , (5.54) where ν = 5−1 in the case of the estimates (5.19) of Theorem 5.2.2, while ν = 1 in the case of the estimates (5.22) of Theorem 5.2.3. Under the hypotheses of Theorem 5.3.6, let ¡ ¢ z(x, t) = z (2.42) (x, t); z(·), Gh ,

(x, t) ∈ G,

be the interpolant of the function z(x, t), (x, t) ∈ Gh , which is the solution of the difference scheme (5.24), (5.25) on the grid G(5.53) , where Q = G. Then one has the estimate £ ¤ν |u(x, t) − z(x, t)| ≤ M N −1 ln N + N0−1 , (x, t) ∈ G, ν = ν(5.54) .

5.4


In Section 5.3, we have constructed special meshes that condense in a neighL borhood of all faces of the lateral boundary S . These meshes guarantee the monotonicity of the operator Λ∗(2)(5.27) . For later use (see, e.g., Chapter 7), we shall need special meshes that condense in a neighborhood of only some L faces of the lateral boundary S and also guarantee the monotonicity of the ∗ operator Λ(2)(5.27) . Let us now discuss the construction of such consistent meshes.

5.4.1

The problem on a slab

For the boundary value problem (5.1) considered on subdomains of the slab (5.2), we impose conditions on both the coefficients ask (x, t) and the meshes ω 1 , ω2 , . . ., ωn that generate consistent meshes guaranteeing the monotonicity of the operator Λ∗(2)(5.27) . The mesh ω 1 on the subdomains is constructed based on the elementary meshes on the x1 -axis, i.e., ω 1u , ω 1l , ω 1r , or ω 1b . Set S S L = j Sj , Sj = Γj × (0, T ], j = 1, 2, (5.55) where Γ1 and Γ2 are the left and right parts of the boundary Γ. Consider a grid Gh that is piecewise-uniform in x1 and condenses in a neighborhood of



159

one or both of the sides S 1 and S 2 , or is uniform. We denote by Γe those parts of the boundary Γ of the set D(5.2) in whose neighborhoods it is required to condense the mesh ω 1 . Let J ∗ = J ∗ (D) = J ∗ (D; Γe )

(5.56)

be the set of indexes j related to Γj in whose neighborhood the mesh ω 1 condenses; here Γj ⊆ Γe . L

Suppose that on the boundary S the coefficients ask (x, t) of the operator L∗(2) , where L∗(2) = L∗(2)(5.26) (L2(5.1) ), satisfy the condition ask (x, t) = 0, (x, t) ∈ S j , j ∈ J ∗ , s = 1 or k = 1, s 6= k, if J ∗ 6= ∅; restrictions on ask (x, t) on S

L

are not imposed if J ∗ = ∅,

(5.57)

i.e., the operator L∗(2) is canonical on the side S j for j ∈ J ∗ , where J ∗ 6= ∅. Assume that on the set Q(x∗ , t∗ ) ⊆ G the coefficients ask (x, t) satisfy the relations (5.30) (the condition of local almost canonicity of the operator L∗(2) ). In the case when J ∗ 6= ∅, we assume that the following additional condition holds, which is similar to (5.36): h min Q(x∗ , t∗ ) r(x,Γj )≤m∗ j∈J ∗

n i X akk (x, t) − ν h−1 − k

max

∗ ∗ s=2 Q(x , t ) ∗ s6=k r(x,Γj )≤m j∈J ∗

k = 2, . . . , n,

|ask (x, t)| h−1 s > 0,

J ∗ 6= ∅.

(5.58)

The condition (5.30) together with (5.58) is the condition of strongly local ∗[1] almost canonicity of the truncated operator L(2) in a neighborhood of the set S j , where j ∈ J ∗ . The condition the coefficients ask (x, t) of the operator L∗(2) on the set G

(5.59)

satisfy the relations (5.57), {(5.30), (5.58)} if J ∗ 6= ∅, or ∗ the relations (5.30) if J ∗ = ∅, where Q(x∗ , t∗ ) ⊆ G, J ∗ = J(5.56) (D),

allows us on the set Q(x∗ , t∗ ) to construct piecewise-uniform consistent grids that ensure the monotonicity of the operators Λ∗(2)(5.27) and Λ(5.25) . We now construct such meshes. Let the condition (5.59) hold. In the case of the condition J ∗ 6= ∅ (when the mesh ω 1 is piecewise uniform), we assume that the step-sizes h1 = h(2)1 and hs in the meshes ω 1 and ωs , for s = 2, . . . , n, satisfy the condition (5.30c) on the set Q(x∗ , t∗ ) and, moreover, for the mesh ω 1 , one has the condition


160


(similar to (5.39a)) σ = σ l ≤ m∗(5.58) , for j = 1, j ∈ J ∗ ,

(5.60a)

σ = σ r ≤ m∗(5.58) , for j = 2, j ∈ J ∗ , if J ∗ 6= ∅; and also the following condition (similar to (5.39b)): −1 N∗1 ≡ σ h−1 (1) < M1 ν

∗ min [h−1 k ], j ∈ J ,

k=2,...,n

(5.60b)

where N∗1 + 1 is the number of nodes on the interval [d∗1 , d∗1 + σ] in the case {j = 1} ⊆ J ∗ , and on the interval [d∗1 − σ, d∗1 ] for {j = 2} ⊆ J ∗ ; here M1 = M1(5.39) . In the case when J ∗ = {j = 1, 2}, set σ l = σ r . Furthermore, we assume that the condition (5.59) is fulfilled. In the mesh Gh along the x1 -axis, we use the piecewise-uniform or uniform meshes  l  ω 1 if J ∗ = {j = 1}     under the conditions   ω r if J ∗ = {j = 2} (5.30c), (5.60) 1  ω1 = (5.61a)  b ∗  if J ∗ 6= ∅, ω 1 if J = {j = 1, 2}     u ω 1 under the condition (5.30c) if J ∗ = ∅; the mesh ω 1(5.61) condenses only in a neighborhood of the sets Γj for which j ∈ J ∗ , where J ∗ 6= ∅. On the slab, when constructing consistent meshes that guarantee the monotonicity of the operator Λ∗(2) = Λ∗(2)(5.27) (Λ(5.25) ), we apply the piecewiseuniform mesh ¡ ¢ ¡ ¢ ω 1(5.61) d∗1 , d∗1 = ω 1(2.83) d∗1 , d∗1 ; ω 1(2.48) . (5.61b) ¡ ¢ (∗) The parameters σ, h(1)1 , and h(2)1 of the mesh ω 1(2.83) d∗1 , d1 are the same as those in the meshes ω 1l and ω 1r . Theorem 5.4.1 In the case of the conditions (5.35), (5.59), the grid ¡ ¢ Gh = Gh Q(x∗ , t∗ ), L∗(2) ; J ∗ = (5.62) ³ = Gh Q(x∗ , t∗ ), L∗(2) , ω 1 = ω 1(5.61) (ω 1(2.48) ), ωs = ωsu , s = 2, . . . , n,

under the conditions (5.30c), (5.60)

in the case of the relations (5.57), {(5.30), (5.58)} if J ∗ 6= ∅, and under the conditions (5.30c) in the case of the relations (5.30) if J ∗ = ∅;

´ ∗ J ∗ = J(5.56) (D; Γe ) ,

is consistent on the set Q(x∗ , t∗ ) with the monotonicity condition for the operator Λ∗(2)(5.27) .



5.4.2

161

The problem on a parallelepiped

For the boundary value problem (5.1) considered on subdomains in G(5.3) , we impose conditions on both the coefficients ask (x, t) and the meshes ω s that ensure, on the subdomains, the consistency of the meshes that are constructed based on the elementary meshes ω su , ω sl , ω sr , and ω sb . In the case of the grid Gh , let the piecewise-uniform mesh on the xs -axis condense in a neighborhood of the face S s and/or S s+n , or this mesh is uniform, for s = 1, . . . , n. We denote by Γe those faces of the boundary Γ of the set Dh(5.3) in whose neighborhoods the meshes ω s , for s = 1, . . . , n, ¡ ¢ condense; here ω s = ω s d∗s , d∗s . We denote by J ∗ , where ¡ ¢ ¡ ¢ J ∗ = J ∗ D = J ∗ D; Γe ,

(5.63)

the set of indexes j, related to Γj in whose neighborhoods the grid Dh condenses in the direction orthogonal to the face Γj ; here Γj ⊆ Γe . We assume that the coefficients ask (x, t) of the operator L∗(2) on the bound¡ L ary S satisfy the following condition, which is similar to (5.47) the condition ¢ of the canonicity of the operator L∗(2) on the faces S j , for j ∈ J ∗ ask (x, t) = 0,

(x, t) ∈ S j ,

j ∈ J ∗;

(5.64)

s = pj , or k = pj , s 6= k, j ∈ J ∗ , if J ∗ 6= ∅; restrictions on ask (x, t) on S

L

are not imposed if J ∗ = ∅,

where pj = pj(3.59) (j, n). Suppose also that the coefficients ask (x, t), which define the operator L∗(2) , satisfy the condition (5.30) on the set Q(x∗ , t∗ ) ⊆ G, independent of the structure of the set J ∗ . Furthermore, if J ∗ 6= ∅, we assume that the following additional condition holds, which is similar to (5.48): h min ∗

∗

Q(x , t ) T r(x, ji Γji )≤m∗ ji ∈J ∗ , i=1,...,r

n i X akk (x, t) − ν h−1 − k

max ∗

∗

|ask (x, t)| h−1 s > 0,

Q(x , t ) s=1 T ∗ s6=k,pji r(x, ji Γji )≤m ji ∈J ∗ , i=1,...,r

k, pj1 , . . . , pjr = 1, . . . , n, k 6= pj1 , . . . , pjr , 1 ≤ r ≤ n − 2, T

ji Γji

(5.65)

6= ∅, J ∗ 6= ∅,

where pj = pj(3.59) (j, n), and m∗ , ν are sufficiently small values. The condition (5.30) together with (5.65) is the condition of strongly local almost canonicity T T ∗[p ,...,pjr ] of the operator L(2)j1 in a neighborhood of the set S j1 . . . S jr , where ∗ j1 , . . . , jr ∈ J .


162


The condition the coefficients ask (x, t) of the operator L∗(2) on the set G satisfy the relations (5.64), {(5.30), (5.65)} if J ∗ 6= ∅,

(5.66)

∗

and the relations (5.30) if J = ∅, ∗ where Q(x∗ , t∗ ) ⊆ G, J ∗ = J(5.63) (D)

allows us on the set Q(x∗ , t∗ ) to construct piecewise-uniform consistent grids that guarantee the monotonicity of the operators Λ∗(2)(5.27) and Λ(5.25) . Next, on the set Q(x∗ , t∗ ) ⊆ G, we construct consistent grids that are defined by the structure of the set J ∗ . Let ¡ the condition (5.66) hold. In the case of the condition J ∗ 6= ∅ i.e., when generating the grid¢ Gh , at least one of the meshes ω s , for s = 1, . . . , n, is piecewise uniform , we assume that the step-sizes h(2)s in the piecewise-uniform meshes ω s satisfy the ¡ condition (5.30c); ¢ furthermore, for the meshes ω s the following condition similar to (5.50a) holds: σs = σsl ≤ m∗(5.65) for s = pj , if pj = j, j ∈ J ∗ ; σs = σsr ≤ m∗(5.65) for s = pj , if pj 6= j, j ∈ J ∗ , ¡ ¢ and also the following condition similar to (5.50b) is satisfied: X ∗ N∗s < M1−1 ν min[h−1 k ], k = 1, . . . , n, j ∈ J . s=pj

k

(5.67a)

(5.67b)

Here M1 = M1(5.50) , pj = pj(3.59) , for j ∈ J ∗ , and N∗s + 1 is the number of nodes in the mesh ω s on the interval [d∗s , d∗s +σs ] if the mesh ω s condenses in a neighborhood of the left endpoint of [d∗s , d∗s ] , or on the interval [d∗s −σs , d∗s ] if the mesh ω s condenses in a neighborhood of the right endpoint of the ω s condenses in neighborhoods of both interval. In the case when £ the mesh ¤ endpoints of the interval d∗s , d∗s , we set σsl = σsr = σs for s = pj1 = pj2 , where j1 , j2 ∈ J ∗ , and j1 6= j2 .

(5.67c)

When constructing consistent grids, we apply piecewise-uniform¢ or uniform ¡ grids along the xs -axis depending on the structure of the set J ∗ :  l  ω s if Js∗ = {j = s}     under the conditions   r  ∗  ω if Js = {j = s + n} (5.30c), (5.67)   s   b ∗ if J ∗ 6= ∅, ω s = ω s if Js = {j = s, s + n} (5.68)    u ∗ ∗  ω s under the condition (5.30c) if Js = ∅, J 6= ∅,     or if J ∗ = ∅; ¡ ¢ ∗ J ∗ = J(5.63) D; Γe .



163

The mesh ω s(5.68) is piecewise uniform on at least one of the xs -axes for J ∗ 6= ∅. The mesh ω s(5.68) is constructed based on the mesh ω 1(2.48) similar to ω 1(2.83) ¡ ¢ ¡ ¢ ω s = ω s d∗s , d∗s ≡ ω 1(2.83) d∗s , d∗s ; ω 1(2.48) . Theorem 5.4.2 In the case of the conditions (5.35), (5.66), the grid ³ ´ ³ Gh = Gh Q(x∗ , t∗ ), L∗(2) ; J ∗ = Gh Q(x∗ , t∗ ), L∗(2) , ω s = ω s(5.68) , for s = 1, . . . , n under the conditions (5.30c), (5.67) in the case of the relations (5.64), {(5.30), (5.65)} if J ∗ 6= ∅, (5.69) and under the condition (5.30c) in the case of the relations (5.30) if J ∗ = ∅;

¡ ¢´ ∗ J ∗ = J(5.63) D; Γe

is consistent on the set Q(x∗ , t∗ ) with the monotonicity condition for the operator Λ∗(2)(5.27) .


Chapter 6 Elliptic convection-diffusion equations

In this chapter, we develop a method for the construction of ε-uniformly convergent schemes for singularly perturbed convection-diffusion equations. In the present section, we consider boundary value problems for elliptic equations in domains that do not contain characteristic boundary parts.

6.1

Problem formulation

In an n-dimensional domain D with a boundary Γ we consider the Dirichlet problem for an elliptic convection-diffusion equation L u(x) = f (x),

x ∈ D,

(6.1a)

u(x) = ϕ(x),

x ∈ Γ.

(6.1b)

Here L ≡ εL2 + L1 ,

L1 ≡

n X

b1s (x)

s=1 n X

L2 ≡

∂ − c1 (x), ∂xs

(6.1c)

n

ask (x)

s,k=1

X ∂2 ∂ + bs (x) − c(x). ∂xs ∂xk s=1 ∂xs

The coefficients of the operator L satisfy the ellipticity condition a0

n X

n X

ξs2 ≤

s=1

ask (x) ξs ξk ≤ a0

n X

ξs2 ,

x ∈ D,

a0 > 0,

(6.2a)

s=1

s,k=1

as well as the condition n ³ X

b1s (x)

´2

≥ b20 ,

b0 > 0,

c(x), c1 (x) ≥ 0,

x ∈ D.

(6.2b)

s=1

The coefficients and the right-hand side in the equation are assumed to be sufficiently smooth; the parameter ε ∈ (0, 1].


166

Elliptic convection-diffusion equations

Assume that the solution of the problem is bounded ε-uniformly and the characteristics of the operator L1 are of a limited length if the domain D is unbounded. Let Γj , with j = 1, . . . , J, be smooth faces that generate the boundary Γ, S where Γ = j Γj and Γj = Γj . We denote by Γc the set of vertexes and edges of the domain D. The function ϕ(x) is sufficiently smooth on each from the faces Γj , j = 1, . . . , J, and is continuous on Γ. The fulfillment of other compatibility conditions on the set Γc is, in general, not assumed (under these assumptions, in general, u 6∈ C 2 (D)). Depending on the behaviour of the characteristics of the operator L1 in a neighborhood of the boundary Γ, the boundary is divided into the subsets Γ+ , Γ− , and Γ0 . We define positive direction of the characteristics of the operator L1 by the vector b, where b(x) = −(b11 (x), . . . , b1n (x)). The set Γ− (or Γ+ ) is the part of the boundary Γ through which the characteristics leave (enter) the S domain D. The set Γ0 is defined by the relation Γ0 = Γ\{Γ+ Γ− }. This set is generated by the characteristics of the reduced equation. We assume that each face Γj , with j = 1, . . . , J, of the domain D belongs entirely to one of the sets Γ+ , Γ− , or Γ0 . The characteristics of the reduced equation intersect S the faces Γj in Γ+ Γ− with nonzero angles. Assume that Γ0 = ∅. For small values of the parameter ε, a boundary layer appears in a neighborhood of the set Γ− . We shall consider the boundary value problem (6.1) on a slab n o D = x : d∗1 ≤ x1 ≤ d∗1 , |xs | < ∞, s = 2, . . . , n (6.3) and on an n-dimensional rectangular parallelepiped D = {x : d∗s ≤ xs ≤ d∗s , s = 1, . . . , n}.

(6.4)

For the boundary value problem (6.1) on the sets D(6.3) and D(6.4) , it is required to construct a difference scheme that converges ε-uniformly.

6.2


In this chapter, we use spaces that were introduced in Chapter 2.

6.2.1

The problem solution on a slab

First, we consider the problem (6.1) on the slab D(6.3) . The coefficients and the right-hand side f in (6.1a), as well as the boundary function ϕ are assumed to be sufficiently smooth and bounded on D and Γ, respectively. We assume that b11 (x) ≥ b0 > 0, x ∈ Γ. (6.5)


Estimates of solutions and derivatives Thus, Γ = Γ−

167

S +

Γ ,

where Γ− = Γ1 and Γ+ = Γ2 are the left and right parts of the boundary Γ. Write the problem solution as the sum of the functions u(x) = U (x) + V (x),

x ∈ D,

(6.6)

where U (x) and V (x) are the regular and singular parts of the solution. The e function U (x), x ∈ D, is the restriction of the function U e (x), x ∈ D , to the set D. Here U e (x) is the solution of the problem Le U e (x) = f e (x),

x ∈ De ,

(6.7a)

U e (x) = ϕe (x),

x ∈ Γe .

(6.7b)

The domain De is an extension of the domain D beyond the boundary Γ− ; the distance between the boundaries Γ− and Γe− is strictly greater than zero, and is independent of ε, here Γ+ = Γe+ , Γe+ = Γ+ (De ), and Γe− = Γe− (De ). The coefficients and the right-hand side in (6.7a) are smooth continuations of those in the equation (6.1a), preserving their properties. The boundary function ϕe (x) is sufficiently smooth and coincides with the function ϕ(x) on the set Γ+ . The function V (x), x ∈ D, is the solution of the boundary value problem L V (x) = 0,

x ∈ D,

(6.8a)

V (x) = ϕV (x) ≡ ϕ(x) − U (x),

x ∈ Γ.

(6.8b)

One can represent the function U (x) by the following expansion U (x) = U0 (x) + ε U1 (x) + v1 (x) = U (1) (x) + v1 (x),

x ∈ D.

The functions U0 (x), U1 (x), and v1 (x) are restrictions to the set D of the functions U0e (x), U1e (x), and v1e (x) that are solutions of the problems Le1 U0e (x) = f e (x),

e

U0e (x) = ϕe (x), x ∈ Γe+ ;

e

U1e (x) = 0,

x ∈ D \ Γe+ ,

Le1 U1e (x) = −Le2 U0e (x), x ∈ D \ Γe+ , Le v1e (x) = −ε2 Le2 U1e (x),

x ∈ De ,

x ∈ Γe+ ;

v1e (x) = ϕe (x) − U (1)e (x), x ∈ Γe .

Here Lei = Li , x ∈ D, i = 1, 2, L1 = L1(6.1) , L2 = L2(6.1) . When the problem data are sufficiently smooth then the function U (1) (x) is sufficiently smooth, and its derivatives satisfy the estimates ¯ ¯ ¯ ¯ ∂k ¯ ¯ (1) U (x) ¯ k1 ¯ ≤ M, x ∈ D. ¯ ∂x1 . . . ∂xknn ¯


168


For the function v1 (x) one has the estimate |v1 (x)| ≤ M ε2 ,

x ∈ D.

Taking into account the a priori estimates (in the variables ξ = ξ(x) and ξs = ε−1 xs , for s = 1, . . . , n), we obtain ¯ ¯ ¯ ¯ ∂k ¯ ¯ v (x) (6.9) ¯ k1 ¯ ≤ M ε2−k , x ∈ D. 1 ¯ ∂x1 . . . ∂xknn ¯ Hence, for the function U (x) the following estimate holds: ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ U (x) ¯ k1 ¯ ≤ M 1 + ε2−k , x ∈ D, 0 ≤ k ≤ K. k n ¯ ∂x1 . . . ∂xn ¯ For the function V (x) the estimate ³ ¯ ¯ ¡ ¢´ ¯V (x)¯ ≤ M exp − m1 ε−1 r x, Γ− ,

x ∈ D,

(6.10)

(6.11a)

is valid, where m1 is an arbitrary number in the interval (0, m0 ), and the constant m0 is given by the relation m0 = min[b11 (x)a−1 11 (x)].

(6.11b)

D

With regard the a priori estimates (in the variables ξ = ξ(x)), we have ¯ ¯ ¯ ¯ ³ ¡ ¢´ ∂k ¯ ¯ −k −1 − V (x) ≤ M ε exp − m ε r x, Γ , x ∈ D. (6.12) ¯ k1 ¯ 1 k ¯ ∂x1 . . . ∂xnn ¯ Taking into account the estimate ¯ ¯ ¯ ¯ £ ¤ ∂k ¯ ¯ ϕ (x) ¯ k2 ¯ ≤ M 1 + ε2−k , V k n ¯ ∂x2 . . . ∂xn ¯

x ∈ Γ− ,

k = k2 + . . . + kn ,

we have ¯ ¯ ¯ ¯ ³ £ −k1 ¤ ¡ ¢´ ∂k ¯ ¯ 2−k −1 − V (x) ≤ M ε + ε exp − m ε r x, Γ , ¯ k1 ¯ 1 ¯ ∂x1 . . . ∂xknn ¯ x∈D

(6.13)

0 ≤ k ≤ K,

where m1 = m1(6.11) . The following theorem holds. Theorem 6.2.1 Let the condition (6.5) hold, and let ask , bs , b1s , c, c1 , f ∈ C l0 (D), for s, k = 1, . . . , n, and ϕ ∈ C l0 (Γ), where l0 = l + α with l ≥ K + 2, K ≥ 2 and α ∈ (0, 1). Then for the components of the solution to the problem (6.1), (6.3) in the representation (6.6), the estimates (6.10) and (6.13) are fulfilled.



6.2.2

169


Let us estimate the solution of the problem (6.1) on the parallelepiped (6.4). For simplicity, we assume that the following condition is fulfilled: b1s (x) ≥ b0 > 0,

x∈D

s = 1, . . . , n.

(6.14)

In this case Γ = Γ−

S +

Γ ,

Γ− =

S

Γ+ =

q=1,...,n Γq ,

S

q=n+1,...,2n Γq ,

where Γq = Γq(3.6) , for q = 1, . . . , 2n. The derivation of a priori estimates is analogous to one in the case of the boundary value problem (3.1), (3.2). Let the data of the problem (6.1), (6.4) satisfy the condition ask , bs , b1s , c, c1 , f ∈ C l+α (D), s, k = 1, . . . , n, ϕ ∈ C 2 (Γj ), j = 1, . . . , 2n. Then

T

u ∈ C α1 (D) C l+2+α (D),

l > 0,

α, α1 ∈ (0, 1).

The solution of the problem (6.1), (6.4) satisfies the estimates similar to (3.3)–(3.5): h i |u(x)| ≤ M max max |f (x)|, max |ϕ(x)| , x ∈ D; (6.15) Γ

D

¯ ¯ ¯ ¯ h ¡ ¢i ∂k ¯ ¯ −k −k u(x) ≤ M ε + r x, Γ , x ∈ D, ¯ k1 ¯ k ¯ ∂x1 . . . ∂xnn ¯ x∈ / Γ for k > 0, 0 ≤ k ≤ K; ¯ ¯ ¡ ¢ ¯u(x) − ϕ(x∗ (x))¯ ≤ M ε−1 r x, Γ , x ∈ D, where x∗ (x) = x∗(3.5) (x) is a point on the set Γ nearest to the point x ∈ D, and K = l + 2. Let us investigate the behaviour of the solution in a neighborhood of the boundary layer and outside of it. Assume that D(j...r) =

T

q=j,...,r D(q) ,

D[j...r] = D(j...r) where

T

j, . . . , r ≤ 2n;

D(n+1,...,2n) , j, . . . , r ≤ n,

D(j) = {x : d∗j < xj < ∞} D(j) = {x : −∞ < xj−n
n.

On the set D, we write the problem solution as the sum u(x) = U (x) + V (x),


x ∈ D,

(6.16)

170


where U (x) and V (x) are the regular and singular parts of the solution. The e function U (x), x ∈ D, is the restriction of the function U e (x), x ∈ D , to the e set D, where D = Dn+1,...,2n . Here U e (x) is the solution of the problem Le U e (x) = f e (x),

x ∈ De ,

(6.17a)

U e (x) = ϕe (x),

x ∈ Γe .

(6.17b)

The domain De is an extension of the domain D beyond the boundary Γ− ; the faces Γej0 contain Γj0 , for j0 = n + 1, . . . , 2n. The coefficients and the right-hand side of the equation (6.17a) are smooth continuations of those in the coefficients and the right-hand side of the equation (6.1a), preserving their properties. We assume that the functions f e (x) and ϕe (x) equal to zero in a m-neighborhood of the set D. The faces from the set Γe (x) belong to the set Γe+ (x). The boundary function ϕe (x) is sufficiently smooth on each of the faces Γej (x) and coincides with the function ϕ(x) on the set Γ+ . The function V (x), x ∈ D, is the solution of the problem L V (x) = 0, x ∈ D,

V (x) = ϕ(x) − U (x), x ∈ Γ.

We now estimate the functions U (x), V (x) and their derivatives. Let the data of the problem (6.1), (6.4) satisfy the condition ask , bs , b1s , c, c1 , f ∈ C l+α (D), j = 1, . . . , 2n,

l ≥ 4,

ϕ ∈ C l+α (Γj ),

α ∈ (0, 1).

The function U (x) is bounded ε-uniformly: |U (x)| ≤ M,

x ∈ D.

We represent the function U (x) by the formal expansion U (x) = U (1) (x) + v1 (x),

x ∈ D,

where U (1) (x) = U0 (x) + εU1 (x) is the main term of the regular component of the solution to the boundary value problem. The functions U0 , U1 , and v1 are obtained as solutions of the following problems: L1 U0 (x) = f (x),

x ∈ D \ Γ+ ,

L1 U1 (x) = −L2 U0 (x), x ∈ D \ Γ+ , L v1 (x) = −ε2 L2 U1 (x),

x ∈ D,

U0 (x) = ϕ(x), x ∈ Γ+ ;

(6.18)

x ∈ Γ+ ;

(6.19)

U1 (x) = 0,

v1 (x) = U (x) − U (1) (x), x ∈ Γ.

The equations from (6.18), (6.19) are hyperbolic. Solutions of the problems (6.18), (6.19) and, hence, the function U (1) (x) are sufficiently smooth in the case when on the intersection of the faces Γj from Γ+ , the data of the problems



171

(6.18), (6.19) satisfy special conditions, namely, compatibility conditions for a hyperbolic system. Let us give the definition of compatibility conditions for the functions f (x) and ϕ(x) in the case of the hyperbolic system (6.18), (6.19). Let the functions u1 (x), u2 (x) be solutions of the following problems L1 u1 (x) = f (x),

x ∈ D \ Γ+ ,

L1 u2 (x) = f 2 (x), x ∈ D \ Γ+ ,

u1 (x) = ϕ(x),

x ∈ Γ+ ;

(6.20)

u2 (x) = ϕ2 (x), x ∈ Γ+ ,

(6.21)

where f 2 (x) = −L2 u1 (x), x ∈ D, and ϕ2 (x) = 0, x ∈ Γ+ . On each of the faces Γj+n ⊂ Γ+ , the derivatives are defined: ∂k ∂xk11

kj−1 kj+1 . . . ∂xj−1 ∂xj+1

. . . ∂xknn

ui (x) =

∂k ∂xk11

kj−1 kj+1 . . . ∂xj−1 ∂xj+1

. . . ∂xknn

ϕi (x),

x ∈ Γj+n , i = 1, 2. By virtue of the equations (6.20), (6.21), we obtain the derivatives ∂xk11

∂k ui (x), . . . ∂xknn

x ∈ Γj+n .

We say that the data of the problems (6.20), (6.21), i.e., coefficients of the operators L1 and L2 and the functions f (x), x ∈ D, and ϕ(x), x ∈ Γ+ , satisfy on the set Γc compatibility conditions guaranteeing the continuity of the derivatives up to order K of the function U (1) (x) on Γ+c if the derivatives ∂x1k1

∂k ui (x), . . . ∂xknn

0 ≤ k ≤ K + 2 (1 − i),

i = 1, 2,

are continuous on Γ+c , where Γ+c is a set that is formed by mutual intersecT tions of all faces Γj from Γ+ , i.e., Γ+c = Γc (D(n+1,...,2n) ) Γ+ . Assume that the following condition is fulfilled: The data of the problems (6.18), (6.19) on the set Γ+c satisfy compatibility conditions of the derivatives up to order l − 2 of the function U

(6.22) (1)

(x).

Under the above assumptions, the functions U0 (x) and U1 (x) satisfy the condition U0 , U1 ∈ C K+α (D), α ∈ (0, 1), K ≥ 0; furthermore, for the function U (1) (x) the following estimate is fulfilled: ¯ ¯ ¯ ¯ ∂k ¯ ¯ (1) U (x) (6.23a) ¯ k1 ¯ ≤ M, x ∈ D, 0 ≤ k ≤ K, ¯ ∂x1 . . . ∂xknn ¯


172


where K = l − 2. For l ≥ 4, for the function v1 (x) one has the estimate |v1 (x)| ≤ M ε2 ,

x ∈ D.

(6.23b)

By virtue of interior a priori estimates, for derivatives of v1e (x) we find the estimates ¯ ¯ ¯ ¯ ∂k ¯ ¯ v (x) ¯ k1 ¯ ≤ M ε2 [ε−k + r−k (x, Γ)], x ∈ D, (6.23c) 1 ¯ ∂x1 . . . ∂xknn ¯ x∈ / Γ for k > 0,

0 ≤ k ≤ K.

The estimate for derivatives of the function U (x) follows from the estimates (6.23) for the derivatives of the functions U (1) (x) and v1 (x). Write the function V (x), x ∈ D, as a sum of the functions V (x) = V0 (x) + v2 (x),

x ∈ D.

Here the functions V0 (x) and v2 (x) are solutions of the problems L V0 (x) = 0, x ∈ D,

V0 (x) = ϕ(x) − U (1) (x),

L v2 (x) = 0, x ∈ D,

v2 (x) = −U (x) + U (1) (x), x ∈ Γ.

For the function v2 (x) one has the estimate ¯ ¯ ¯ ¯ h ¡ ¢i ∂k ¯ ¯ v (x) ¯ k1 ¯ ≤ M ε2 ε−k + r−k x, Γ , 2 k n ¯ ∂x1 . . . ∂xn ¯ x∈ / Γ for k > 0,

x ∈ Γ;

x ∈ D,

0 ≤ k ≤ K.

We represent the function V0 (x) as the sum of regular and corner boundary layers X X V0 (x) = V(j) (x) + V(j...r) (x), x ∈ D, (6.24) j=1,...,n

j,...,r=1,...,n j 0, x ∈ D), it is necessary that the following condition was fulfilled: ε−1 h∗1 − →0 ε

as N1 → ∞,

where h∗1 is step-size in the mesh ω 1 on the interval [d∗1 , d∗1 ]. In the case when a difference scheme constructed by a classical approximation of a boundary value problem resolves the boundary value problem ε-uniformly, terms of the grid equation are not bounded ε-uniformly. The following Theorem holds. Theorem 6.3.1 In the case of the problem (6.1), (6.3), (6.5), for a class of difference schemes on the grids Dh(6.34) satisfying the condition (6.37), whose difference equations are constructed by a term-wise classical approximation of differential equations, there exist no grids on which the difference scheme, on the interior mesh nodes, approximates termwise the differential equation ε-uniformly. To prove Theorem 6.3.1 it is sufficient to consider the expression (∂/∂x1 − δx1 )u(x) for x ∈ (xi1 , x2 , . . . , xn ) ∈ Dh , where d∗1 < xi1 < d∗1 + M ε for the value i corresponding to the maximal hi1 . Classical difference schemes do not have the property of the ε-uniform approximation of the boundary value problem (6.1) that does not allow to use Statement A from Section 2.1 in order to justify the ε-uniform convergence of the schemes. Thus, for the boundary value problem (6.1) there arises a necessity to develop a technique to investigate convergence of constructing schemes (and deriving a priori estimates of errors of approximate solutions), that allows to establish the ε-uniform convergence of the schemes. The same difficulties appear in the case of boundary value problems for parabolic convection-diffusion equations. We now discuss difference schemes on piecewise uniform meshes. Let ω 1 = ω 1 (d1 , d2 ) = ω 1 (d1 , d2 ; σ, N1 ) (6.38a) be a piecewise-uniform mesh on the interval [d1 , d2 ] on the x1 -axis; the stepsizes in the mesh ω 1(6.38a) on the intervals [d1 , d1 + σ] and [d1 + σ, d2 ], where 0 ≤ σ ≤ d2 − d1 , are h(1) and h(2) , respectively; here N1 + 1 is the number of nodes in the mesh ω 1 . Using the meshes ω 1(6.38a) , one can construct the simplest meshes that condense in the boundary layer.


On construction of ε-uniformly convergent difference schemes

179

Lemma 6.3.2 Let the scheme (6.35) be constructed by a classical approximation of the boundary value problem (6.1), (6.3), (6.5) on the grid Dh(6.34) , where ω 1 = ω 1(6.38a) (d∗1 , d∗1 ; σ, N1 ). For σ = σ(ε) (σ is independent of N1 ), there exist no grids on which the scheme converges ε-uniformly. The proof of this lemma is similar to that of Lemma 2.3.4. Theorem 6.3.2 Let the scheme (6.35) be constructed by a classical approximation of the boundary value problem (6.1), (6.3), (6.5) on the grid Dh(6.34) , where ω 1 = ω 1(6.38a) (d∗1 , d∗1 ; σ, N1 ). In order that the scheme be convergent ε-uniformly it is necessary for the value σ to be dependent on both ε and N1 , i.e., σ = σ0 (ε, N1 ). In the class of the functions σ = σ0 (ε, N1 ) = min[m1 , σ ∗ (ε, N1 )],

σ ∗ (ε, N1 ) = ε σ1 (N1 ),

(6.38b)

where m1 is an arbitrary constant from the interval (0, d∗1 −d∗1 ), the fulfillment of the condition σ1 (N1 ) → ∞,

N1−1 σ1 (N1 ) → 0 as N1 → ∞

(6.38c)

is necessary for the scheme to be convergent ε-uniformly. On the interval [d∗1 , d∗1 ], we shall consider the meshes ω 1 = ω 1(6.38a−c) (d∗1 , d∗1 ; σ, N1 ),

(6.38d)

whose step-sizes h(1) and h(2) satisfy the condition h(1) ≤ h(2) . The mesh ω 1(6.38) is similar to ω 1(2.40) ; however it condenses only in a neighborhood of the left endpoint of [d∗1 , d∗1 ]. The grid Dh = Dh(6.34) , where ω 1 = ω 1(6.38) (d∗1 , d∗1 ), (6.39) is ε-uniformly informative for the solution of the boundary value problem (6.1), (6.3). In the case of the grid operator Λ(6.36) , for the solution of the boundary value problem on the grid Dh(6.39) under the condition |V (x∗ )| ≥ m1 , x∗ ∈ Γ− , one obtains the estimate |(L − Λ) u(x)| ≥ m2 ε−2 h(1) − M N −1 ≥ £ ¤ ≥ m min ε−2 N1−1 , ε−1 N1−1 σ1 (N1 ) − M N −1 , x ∈ Dh ,

x1 ≤ d1∗ + M1 ε,

xs = x∗s ,

s = 2, . . . , n.

Thus, we have max |(L − Λ) u(x)| ≥ m ε−1 N1−1 for ε = O(N1−1 ), Gh = Gh(6.39) . Gh


180


Theorem 6.3.3 In the case of the boundary value problem (6.1), (6.3), for a class of difference schemes, whose discrete equations are constructed by a classical approximation of differential equations on the grids Dh(6.39) piecewise-uniform in x1 , there are no grids on which the difference scheme approximates the boundary value problem ε-uniformly. Difficulties in the construction of ε-uniformly convergent difference schemes that are discussed in this section are also observed in the case of the problem on a parallelepiped.

6.3.2

The problem on a slab

Let us estimate the solution error for the difference scheme (6.35), (6.36) on the grid (6.39). In this subsection, we assume that the operator Λ(6.36) on the grid Dh(6.39) is ε-uniformly monotone. Write the solution of the difference scheme (6.35), (6.36), (6.39) as the decomposition z(x) = zU (x) + zV (x), x ∈ Dh , (6.40a) that corresponds to the decomposition (6.6) of the solution to problem (6.1), (6.3). Here zU (x) and zV (x) are the regular and singular components of the discrete solution, and they are solutions of the problems Λ zU (x) = f (x), x ∈ Dh ,

zU (x) = U (x), x ∈ Γh ;

(6.40b)

Λ zV (x) = f (x), x ∈ Dh ,

zV (x) = V (x), x ∈ Γh .

(6.40c)

For the components in the representation (6.6), let the estimates of Theorem 2.2.1 for K = 3 be satisfied. Taking into account the estimates of derivatives of the function U (x), we find |U (x) − zU (x)| ≤ M N −1 ,

x ∈ Dh .

Let us consider the discrete boundary layer zV (x), x ∈ Dh . Let the following condition be fulfilled: σ = ε σ1 (N1 ),

σ = σ(6.38) ,

σ1 (N1 ) = σ1(6.38) (N1 ),

moreover, σ < m1(6.38) . When we estimate V (x) − zV (x) outside a neighborhood of the set Γ− , it is suitable to use that the functions V (x) and zV (x) decay towards zero as x moves away from the set Γ− . In the case of the grid Dh(6.39) , as a majorant of the discrete boundary layer zV (x), we can take the solution of the one-dimensional problem Λ v(x1 ) ≡ {ε δx1x1 c + m1 δx1 } v(x1 ) = 0,

x1 ∈ ω 1e ,

v(x1 ) = 1,

x1 = d∗1 ,


(6.41)


181

where the function v(x1 ) tends to zero as x1 → ∞, and m1 is an arbitrary number in the interval (0, m0 ) with m0 = m0(6.11) . The mesh ω 1e on the set [d∗1 , ∞) is the extension of the mesh ω 1(6.38) (d∗1 , d∗1 ) for x > d∗1 . The step-size in the mesh ω 1e for x1 ≥ d∗1 + σ equals to h(2)(6.38) . The solution of this problem considered on the set ω 1 is the function ( 1 − α (1 − ψ1−n1 ), x1 ≤ d∗1 + σ, v(x1 ) = [1 − α (1 − ψ1−n1 )] ψ2−n2 , x1 > d∗1 + σ, x1 ∈ ω 1 . Here ω 1 = ω 1(6.38) (d∗1 , d∗1 ),

ψi = ψi (ε, h(i) , m1 ) = 1 + m1 ε−1 h(i) ,

n1 = n1 (x1 ) = (h(1) )−1 (x1 − d∗1 ),

i = 1, 2,

n1 = n1 (σ),

n2 = n2 (x1 ) = (h(2) )−1 (x1 − d∗1 − σ),

m1 = m1(6.41) ,

σ = σ1(6.38) ,

the value α = α(ε, h(1) , h(2) , σ) is found from the relation ¡ ¢ (1 − α) ε + 2−1 m1 (h(1) + h(2) ) − 2−1 m1 α (h(2) − h(1) ) ψ1−n1 = 0. The function zV (x) satisfies the estimate ¯ ¯ ¡ ¢ ¯zV (x)¯ ≤ M1 v(σ) ≤ M1 ψ −n1 ≤ M exp − m1 σ1 (N1 ) , 1 x ∈ Dh ,

(6.42)

x1 ≥ d∗1 + σ,

where m1 is an arbitrary number in the interval (0, m0(6.11) ) satisfying the condition m1 ≤ m1(6.41) , and σ = σ0(6.38) (ε, N1 ). Taking into account the estimate (6.13), we find ¡ ¢ |V (x) − zV (x)| ≤ M exp − m1 σ1 (N1 ) , (6.43) x ∈ Dh ,

x1 ≥ d∗1 + σ,

where m1 ≤ m1(6.42) . In an σ-neighborhood of the set Γ− , one has the estimate ¯ ¡ ¢¯ £ ¤ ¯Λ V (x) − zV (x) ¯ ≤ M ε−1 N −1 σ1 (N1 ) + N −1 × 1 ³ ¡ ¢´ × exp − m1 ε−1 r x, Γ− , x ∈ Dh , x1 < d∗1 + σ. Using as a majorant (up to a constant multiplier) the function ¡ ¢ ¡ ¢ w(x) = exp − m1 ε−1 r(x, Γ− ) , x ∈ Dh , r x, Γ− ≤ σ,

(6.44)

and taking account of the estimate (6.43), in an σ-neighborhood of the set Γ− , we obtain the estimate h ¡ ¢i (6.45) |V (x) − zV (x)| ≤ M N −1 + N1−1 σ1 (N1 ) + exp − m1 σ1 (N1 ) , x ∈ Dh ,


x1 ≤ d∗1 + σ.

182


Note that the operator Λ on the function w(x) is not ε-uniformly bounded. By virtue of (6.43), (6.45), for the component zV (x) we have the estimate h ¡ ¢i |V (x) − zV (x)| ≤ M N −1 + N1−1 σ1 (N1 ) + exp − m1 σ1 (N1 ) , (6.46) x ∈ Dh . Using majorant functions that are already defined on the set Dh , we obtain similar estimates in the case when the parameters of the mesh ω 1(6.38) satisfy the condition σ = m1(6.38) . Thus, the component zV (x) converges ε-uniformly with the estimate (6.46). For the solution of the difference scheme (6.35), (6.36), (6.39), we have the estimate h ¡ ¢i |u(x) − z(x)| ≤ M N −1 + N1−1 σ1 (N1 ) + exp − m1 σ1 (N1 ) , (6.47) x ∈ Dh . It is convenient to consider the difference scheme on the grids Dh = ω 1 × ω2 × . . . × ωn , where ω s , for s = 2, . . . , n, are uniform meshes, and the mesh ¡ ¢ ¡ ¢ ω 1 = ω 1 d∗1 , d∗1 = ω 1 d∗1 , d∗1 ; m1 , m2 ¡ ¢ is the mesh ω 1(6.38) d∗1 , d∗1 ; σ, N1 , where £ ¤ σ = σ(6.38b) = min m1 , ε σ1 (N1 ) , σ1 (N1 ) = m−1 2 ln N1 ;

(6.48)

(6.49a)

(6.49b)

here m2 is an arbitrary constant from the interval (0, m0 ), with m0 = m0(6.11) , and m1 = m1(6.38) . We denote by N1σ + 1 the number of nodes in the mesh ω 1(6.49) on the interval [d∗1 , d∗1 + σ], where σ = min[m1 , m−1 2 ε ln N1 ]. For the values m1 and N1σ , one has the condition ¡ ¢−1 N1σ = m1 d∗1 − d∗1 N1 . (6.49c) For the solution of the difference scheme (6.35), (6.36), (6.39) on the grid (6.48) the following estimate holds: |u(x) − z(x)| ≤ M N −1 ln N,

x ∈ Dh .

(6.50)

Theorem 6.3.4 For the components of the solution of the boundary value problem (6.1), (6.3) in the representation (6.6), let the estimates of Theorem 6.2.1 be satisfied for K = 3. Then the difference scheme (6.35), (6.36), (6.39), provided that the operator Λ(6.36) is ε-uniformly monotone, converges ε-uniformly. For the discrete solution in the case of the grid (or the grid (6.48)) the estimate (6.47) (the estimate (6.50)) holds.



183

When the difference scheme (6.35), (6.36), (6.39) is ε-uniformly monotone then the ε-uniform convergence of the grid singular component zV (x) is obtained using a special form of majorant functions for the components V (x), zV (x) and for their difference V (x) − zV (x).

6.3.3


We investigate difference schemes on the meshes that condense in a neighborhood of boundary layers in the case of the boundary value problem on a parallelepiped. Assume that for the solution of the problem (6.1), (6.4) the estimates of Theorem 6.2.2 are fulfilled. On the parallelepiped D(6.4) we introduce the rectangular mesh Dh = ω 1 × . . . × ω n ,

(6.51)

where ω s is a mesh on the interval [d∗s , d∗s ], for s = 1, . . . , n. On the grid Dh(6.51) for the problem (6.1), (6.4) we consider the difference scheme Λ z(x) = f (x), x ∈ Dh ,

z(x) = ϕ(x), x ∈ Γh ,

(6.52)

where Λ = Λ(6.36) , and find sufficient conditions for its ε-uniform convergence. We assume that the operator Λ is ε-uniformly monotone. We shall consider the problem (6.1), (6.4) in the case when the problem solution satisfies the estimates (6.33) for K = 3. T Let the function v(x), x ∈ D, where v ∈ C 3 (D) C(D), be one of the components in the decompositions (6.27), (6.24). We denote by zv (x) the solution of the difference problem Λ z(x) = L v(x), x ∈ Dh ,

z(x) = v(x), x ∈ Γh .

(6.53)

Write the solution of the difference scheme (6.52), (6.51) as the decomposition z(x) = zU(0) (x) + zV0 (x), X X zV0 (x) = zV(j) (x) + zV(j...r) (x), j

x ∈ Dh ,

j,...,r

that corresponds to the decomposition (6.27), (6.24) of the solution of the boundary value problem. Let us consider the difference scheme on the grid Dh(6.51) , (6.54) ¡ ¡ ¢ ¢ where ω s = ω s d∗s , d∗s , σs , Ns = ω 1(6.49) d∗s , d∗s , σs , Ns ; m1s , m2 , for s = 1, . . . , n, where m2 and m1s are arbitrary numbers from the intervals (0, m0 ) and (0, d∗s − d∗s ), respectively, and m0 = m0(6.26) .


184


We find the estimates U(0) (x) − zU(0) (x), V(j) (x) − zV(j) (x), and V(j...r) (x) − zV(j...r) (x) similar to the derivation of the estimates U (x) − zU (x) and V (x) − zV (x) for the problem on the slab. In the case of the estimates (6.33) for the solution of the difference scheme (6.52), (6.54), one obtains the estimate |u(x) − z(x)| ≤ M N −1 ln N,

x ∈ Dh .

(6.55)

For the solution of the problem and its components let the estimates (6.26) be fulfilled for K = 3. For the solution of the difference scheme (6.52), (6.54), we find the estimate ¯ ¡ ¢¯ ¡ ¢ ¯z(x) − ϕ x∗ (x) ¯ ≤ M ε−1 r x, Γ , x ∈ Dh , x∗ (x) = x∗ (x). (3.5) With regard to the above estimate, we find ¡ ¢ |u(x) − z(x)| ≤ M ε−1 r x, Γ ,

x ∈ Dh .

¡ ¢ Estimating u(x) − z(x) for r x, Γ ≥ r0 , where r0 > 0 is an arbitrary value, we obtain h i ¡ ¢ |u(x) − z(x)| ≤ M ε3 r0−3 N −1 ln N + ε−1 r0 , x ∈ Dh , r x, Γ ≥ r0 . Thus, for the solution of the difference scheme in the case of the a priori estimates (6.15), (6.26), the estimate holds ¡ ¢ν |u(x) − z(x)| ≤ M N −1 ln N , x ∈ Dh , (6.56) where ν = 4−1 . Let the function uλ (x) = uλ(6.28) (x) approximate the function u(x), i.e., the solution of the boundary value problem (6.1), (6.4). Consider the difference scheme (6.52), (6.54) in the case when for uλ (x) and for its components in the representations (6.27), (6.24), the estimates (6.29) and (6.26), (6.30) are fulfilled; in (6.27) and (6.26), u(x) is replaced by uλ (x). Let z λ (x), x ∈ Dh , be the solution of the difference scheme Λ z λ (x) = f λ (x), λ

λ

z (x) = ϕ (x),

x ∈ Dh , x ∈ Γh ,

where Λ = Λ(6.52) ; f λ (x) and ϕλ (x) are approximations of the functions f (x) and ϕ(x), respectively. For the function z λ (x) the estimate ¯ ¯ ¯z(x) − z λ (x)¯ ≤ M λ, x ∈ Dh , (6.57) holds that is similar to (6.29) for the function uλ (x); here and below the constant M is independent of λ.


Monotone ε-uniformly convergent difference schemes

185

Taking into account the a priori estimates for the function uλ (x), we find the following estimate similar to the derivation of the estimate (6.56): ¯ λ ¯ ¡ ¢ ¯u (x) − z λ (x)¯ ≤ M λ−1 N −1 ln N ν , x ∈ Dh , where ν = ν(6.56) . With regard to the estimates (6.57) and (6.29), we obtain h i ¡ ¢ν |u(x) − z(x)| ≤ M λ−1 N −1 ln N + λ , x ∈ Dh , ν = ν(6.56) . Thus, for the solution of the difference scheme (6.52), (6.54) the estimate ¡ ¢ν |u(x) − z(x)| ≤ M N −1 ln N , x ∈ Dh , (6.58) holds, where ν = 8−1 . Theorem 6.3.5 Let the condition (6.14) hold, and for the components of the solution to the boundary value problem (6.1), (6.4) in the representations (6.27), (6.24) (or, in the representations (6.27), where u(x) is replaced by uλ (x), and (6.24)), let the estimates of Theorem 6.2.2 (Theorem 6.2.3), where K = 3, be satisfied. Then the difference scheme (6.52), (6.54), under the condition that the operator Λ(6.52) is ε-uniformly monotone, converges ε-uniformly. For the discrete solutions, (i) in the case of the a priori estimates (6.26), one has the estimate (6.56), (ii) in the case of the estimates (6.29) and the estimates (6.26), (6.30) for the components of the function uλ (x), replacing u(x) by uλ (x) in (6.26), then the estimate (6.58) is valid, (iii) when the estimate (6.33) is fulfilled then one has the estimate (6.55).

6.4


We now impose conditions on the data of the boundary value problems (6.1), (6.3) and (6.1), (6.4), and on the parameters of the meshes constructed, in order to guarantee the ε-uniform monotonicity of the difference schemes (6.35), (6.34) and (6.52), (6.51) and the ε-uniform convergence of their solutions. We use the grids (2.84) for the slab (6.3) and (3.64) for the parallelepiped (6.4) for appropriate values of their parameters. First, we consider the problem on the slab (6.3). To ensure the ε-uniform monotonicity of the difference schemes, we construct special piecewise-uniform meshes based on results of Subsection 2.4.3, Section 2.4, Chapter 2. In the case of the condition (6.5), the grid Dh condenses in a neighborhood of the set Γ1 , i.e., Γe(2.77) = Γ1 . Assume that the coefficients ask (x) of the operator L∗(2) satisfy the condition ask (x) = 0, x ∈ Γj , j ∈ J ∗ , s = 1 or k = 1, s 6= k,


(6.59)

186


on the boundary Γ. Here ∗ J ∗ = J ∗ (D) = J(2.77) (D; Γe ) = {j = 1}.

(6.60)

The condition (6.59) is necessary for the ε-uniform monotonicity of the difference scheme {(6.35), (6.36), (6.34)} provided that the difference scheme resolves the boundary value problem {(6.1), (6.3), (6.5)} ε-uniformly. Suppose that the coefficients ask (x) satisfy the condition (2.55) on the set Q(x∗ ) ⊆ D, and also the additional condition h min Q(x∗ ) r(x,Γj )≤m∗ j∈J ∗

n i X akk (x) − ν h−1 k −

max

Q(x∗ ) s=2 ∗ s6=k r(x,Γj )≤m j∈J ∗

|ask (x)| h−1 s > 0,

(6.61)

k = 2, . . . , n, where m∗ and ν are sufficiently small values. The conditions (2.55), (6.61), and (6.59) together allow us to construct grids Dh , piecewise-uniform in x1 , on which the operator Λ(6.36) is ε-uniformly monotone. Assume that the step-sizes h1 = h(2)1 and hs in the meshes ω 1 = ω 1(6.38) and ωs , where s = 2, . . . , n, satisfy the condition (2.55c), and also that the parameters of the mesh ω 1 satisfy the condition σ = σ l ≤ m∗(6.61) , −1 N∗1 ≡ σ h−1 (1) < M1 ν

(6.62) min [h−1 k ], ν = ν(6.61) , M1 = M1(2.67) ;

k=2,...,n

here N∗1 +1 is the number of nodes in the mesh ω 1 on the interval [d∗1 , d∗1 +σ]. Enforcing the conditions (6.59) and (2.55), (6.61), (6.62) in the case of the grid Dh(6.48) allows us to choose parameters of the piecewise-uniform mesh ω 1 and the uniform meshes ωs , where s = 2, . . . , n, guaranteeing the ε-uniform monotonicity of the operator Λ(6.36) . In the grid Dh(6.48) along the x1 -axis we use the piecewise-uniform mesh ω 1 = ω l1(6.63) (d∗1 , d∗1 ) ≡ ω 1(6.38) (d∗1 , d∗1 ).

(6.63)

The following Theorem is a special case of Theorem 2.4.6. Theorem 6.4.1 For the boundary value problem (6.1), (6.3), let the condition (6.5) hold, and let the coefficients ask (x) of the operator L∗(2) on the set D satisfy the relations (2.64), (6.59) and {(2.55), (6.61)}, where Q(x∗ ) ⊆ D. Then the grid ³ ¡ ¢ Dh = Dh Q(x∗ ), L∗(2) ; J ∗ = Dh Q(x∗ ), L∗(2) , ω 1 = ω 1(6.63) , (6.64) ωs = ωsu , s = 2, . . . , n,

under the conditions (2.55c), (6.62)

∗ in the case of the relations (6.59), {(2.55), (6.61)}; J ∗ = J(6.60)


´


187

is consistent on the set Q(x∗ ) with the monotonicity condition for the operator Λ∗(2) . Under the condition (2.71), one can choose the set Q in (6.64) as Q = D in order to ensure the monotonicity of the difference scheme (6.35), (6.36) on the grid (6.64), where Q = D. A corollary of Theorems 6.3.4 and 6.4.1 is the following result on the convergence of the scheme {(6.35), (6.36), (6.48)}. Theorem 6.4.2 Let ask ∈ C 1 (D), bs , b1s , c, c1 , f ∈ C(D), s, k = 1, . . . , n, and let the condition (6.5) be satisfied, the coefficients ask (x) satisfy the conditions (2.71), (6.59) and {(2.55), (6.61)}, where Q = D, and let for the solution of the problem (6.1), (6.3), the estimates of Theorem 6.2.1 for K = 3. Then the difference scheme (6.35), (6.36) on the grid Dh(6.64) , where Q = D and ω 1(6.63) = ω 1(6.49) , converges ε-uniformly at the rate O(N −1 ln N ): |u(x) − z(x)| ≤ M N −1 ln N,

x ∈ Dh .

Under the hypotheses of Theorem 6.4.2, for the interpolant z(x) = z (2.42) (x; z(·), Dh ),

x ∈ D,

where z(x), x ∈ Dh is the the solution of the difference scheme (6.35), (6.36) on the grid (6.64), the estimate holds |u(x) − z(x)| ≤ M N −1 ln N,

x ∈ D.

Next, we consider the problem on the parallelepiped (6.4). Here, when constructing piecewise-uniform meshes ensuring the monotonicity of the difference schemes, we use results of Subsection 2.4.3, Section 2.4, Chapter 2. In the case of the condition (6.14), the grid Dh condenses in a neighborhood S of the faces Γj , for j = 1, . . . , n, i.e., Γe(3.57) = j=1,...,n Γj . Assume that the coefficients ask (x) of the operator L∗(2) satisfy the condition ask (x) = 0, x ∈ Γj , j ∈ J ∗ , s = 1 or k = 1, s 6= k, on the boundary Γ. Here ¡ ¢ ¡ ¢ ∗ J ∗ = J ∗ D = J(3.57) D; Γe =

{j = 1, . . . , n}.

(6.65)

(6.66)

Suppose also that the coefficients ask (x) satisfy the condition (2.55), and, moreover, the additional condition h min ∗

Q(x ) T r(x, ji Γji )≤m∗ ∗ ji ∈J , i=1,...,r



max ∗

Q(x ) s=1 T ∗ s6=k,ji r(x, ji Γji )≤m ∗ ji ∈J , i=1,...,r

|ask (x)| h−1 s > 0,

188


k, j1 , . . . , jr = 1, . . . , n, k 6= j1 , . . . , jr , 1 ≤ r ≤ n − 2, ji ∈ J ∗ ,

(6.67)

∗

where m and ν are sufficiently small values. Assume that the step-sizes h(2)s of the piecewise-uniform meshes ω s satisfy the condition (2.55c), and also that the parameters of the meshes ω s satisfy the condition σs = σsl ≤ m∗(6.67) , n X s=1

(6.68)

N∗s < M1−1 ν min[h−1 k ], k = 1, . . . , n, ν = ν(6.67) , M1 = M1(3.52) , k

where N∗s + 1 is the number of nodes in the mesh ω s on [d∗s , d∗s + σs ]. In the grid Dh(6.54) we use the piecewise-uniform meshes ω s = ω ls(6.69) (d∗s , d∗s ) ≡ ω s(6.54) (d∗s , d∗s ), s = 1, . . . , n.

(6.69)

The following Theorem is a special case of Theorem 3.4.2. Theorem 6.4.3 Let ask ∈ C 1 (D), s 6= k, and let the coefficients ask (x) of the operator L∗(2) on the set D satisfy the relations (6.65), {(2.55), (6.67)}, where Q(x∗ ) ⊆ D. Then the grid ³ ´ ³ Dh = Dh Q(x∗ ), L∗(2) ; J ∗ = Dh Q(x∗ ), L∗(2) , ω s = ω s(6.69) , s = 1, . . . , n under the conditions (2.55c), (6.68)

(6.70)

∗ in the case of the relations {(2.55), (6.67)}; J ∗ = J(6.66)

´

is consistent on the set Q(x∗ ) with the monotonicity condition for the operator Λ∗(2) . Under the condition (2.71), one can choose the set Q = D in (6.70). A corollary of Theorems 6.3.5 and 6.4.3 is the following result on the convergence of the scheme (6.52), (6.54). Theorem 6.4.4 Let ask ∈ C 1 (D), bs , b1s , c, c1 , f ∈ C(D), s, k = 1, . . . , n, and let the condition (6.14) be satisfied, the coefficients ask (x) satisfy the conditions (2.71), (6.65) and {(2.55), (6.67)}, where Q = D, and let for the solution of the problem (6.1), (6.4) the estimates of Theorem 6.2.3 (the estimates of Theorem 6.2.2) be satisfied for K = 3. Then the difference scheme (6.52) on the grid Dh(6.70) , where Q = D, converges ε-uniformly. For the discrete solutions, the estimate holds |u(x) − z(x)| ≤ M (N −1 ln N )ν ,


x ∈ Dh ,

(6.71)


189

where ν = 1 in the case of the estimates (6.33), while ν = 4−1 in the case of the estimates (6.26) and ν = 8−1 in the case of the estimate (6.29) and the estimates (6.26), (6.30) for the components of the function uλ (x); here in (6.26) the function u(x) is replaced by uλ (x)). Under the hypotheses of Theorem 6.4.4, for the interpolant ¡ ¢ z(x) = z (2.42) x; z(·), Dh , x ∈ D, where z(x), x ∈ D is the solution of the difference scheme (6.52) on the grid (6.70) with Q = D, the estimate holds |u(x) − z(x)| ≤ M (N −1 ln N )ν ,


x ∈ D,

ν = ν(6.71) .

Chapter 7 Parabolic convection-diffusion equations

7.1

Problem formulation

In an n-dimensional domain D, we consider the Dirichlet problem for the parabolic convection-diffusion equation L u(x, t) = f (x, t),

(x, t) ∈ G,

u(x, t) = ϕ(x, t),

(x, t) ∈ S.

Here

S

G = G S,

G = D × (0, T ],

(7.1)

(7.2)

the operator L is defined by the relations L ≡ εL2 + L1 ; L2 ≡

n X

n

ask (x, t)

s,k=1

L1 ≡

n X s=1

X ∂2 ∂ + bs (x, t) − c(x, t); ∂xs ∂xk s=1 ∂xs

b1s (x, t)

∂ ∂ − c1 (x, t) − p(x, t) . ∂xs ∂t

The coefficients of the operators L2 and L1 satisfy the conditions a0

n X s=1

ξs2 ≤

n X

ask (x, t)ξs ξk ≤ a0

s,k=1

p(x, t) ≥ p0 , c(x, t), c1 (x, t) ≥ 0,

n X s=1

ξs2 ,

n X

(b1s (x, t))2 ≥ b20 ,

(7.3)

s=1

(x, t) ∈ G, a0 , b0 , p0 > 0.

The coefficients and the right-hand side f on the set G are assumed to be sufficiently smooth, just as is the boundary function ϕ on the closures of the smooth parts of the boundary S, i.e., the data of the problem (7.1), (7.2), are assumed to be sufficiently smooth. We suppose that the problem data are bounded in the case of an unbounded domain D. In that case, the solution of the problem is assumed to be bounded.


192

Parabolic convection-diffusion equations

The positive direction of characteristics of the operator L1 is determined by the (n + 1)-dimensional vector B(x, t) B(x, t) = (−b11 (x, t), . . . , −b1n (x, t), p(x, t)),

(x, t) ∈ G.

Depending on the behaviour of the characteristics of the operator L1 in a L S neighborhood of the boundary S , where S = S L S0 (see (5.4a)), this lateral − + boundary is split into subsets. By S (or S ) we denote that part of the set L S through which the characteristics leave (enter) the domain G. Define a S set S 0 by the relation S 0 = S L \ {S − S + }. The set S 0 is formed by the characteristics of the operator L1 . Assume that S 0 = ∅. For small values of the parameter ε, a boundary layer appears in a neighborhood of the set S L− . We consider the boundary value problem (7.1), (7.2) on the slab D = {x : d∗1 ≤ x1 ≤ d∗1 , |xs | < ∞, s = 2, . . . , n},

(7.4)

or on a rectangular parallelepiped D = {x : d∗s ≤ xs ≤ d∗s , s = 1, . . . , n}.

(7.5)

For the boundary value problem (7.1), (7.2), it is required to construct a difference scheme that converges ε-uniformly.

7.2

Estimates of the problem solution on a slab

In this section we use the spaces that were introduced in ¡Chapter 5, with ¢ the only difference that for the interior bounds, the value d (x, t), (x0 , t0 ) is ¡ ¢ ¡ ¢1/2 here defined by the relation d (x, t), (x0 , t0 ) = |x − x0 |2 + ε |t − t0 | . Assume that the data of the boundary value problem (7.1), (7.2) satisfy the condition ask , bs , b1s , c, c1 , p, f ∈ C l0 ,l0 /2 (G), l0 = l + α, l ≥ 0, α ∈ (0, 1). Then u ∈ C l1 ,l1 /2 (G), l1 = l0 + 2. Under the additional condition T

ϕ ∈ C α1 ,α1 /2 (S0 ) C α1 ,α1 /2 (S j ), j = 1, . . . , J, where α1 ∈ (0, 1), and J = 2 in the case of the slab while J = 2n in the case of the parallelepiped, we have T

u ∈ C α2 ,α2 /2 (G) C l1 ,l1 /2 (G), α2 ∈ (0, α1 ). Here Sj = Γj × (0, T ], Γj = Γj(3.6) , j = 1, . . . , J.


Estimates of the problem solution on a slab Let

193

T

ϕ ∈ C l1 ,l1 /2 (S0 ) C l1 ,l1 /2 (S j ), j = 1, . . . , J.

Taking into account interior a priori estimates and the estimates up to the smooth parts of the boundary (derived in the variables ξ, τ where ξs = ε−1 xs , τ = ε−1 t, and s = 1, . . . , n), we find ¯ ¯ ¯ ¯ £ ¡ ¢¤ ¡ ¢¤ ∂ k+k0 ¯ ¯ u(x, t) ¯ k1 ¯ ≤ M ε−k + r−k x, Γ [ε−k0 + εk0 r−2k0 x, Γ , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ (x, t) ∈ G,

x∈ / Γ for k + k0 > 0, 0 ≤ k + 2k0 ≤ K,

where K = l + 2. Let us discuss estimates of the solution of the problem (7.1) on the slab (7.4). Assume that the data of the problem satisfy the condition ask , bs , b1s , c, c1 , p, f ∈ C l1 ,l1 /2 (G), T

ϕ ∈ C l1 +2,l1 /2+1 (S0 ) C l1 +2,l1 /2+1 (S j ), j = 1, 2,

l1 = l + α,

l ≥ 0,

(7.6)

α ∈ (0, 1).

c , one has the condition Assume also that on the set S0c , where S0c = S0(5.4)

The data of the problem (7.1), (7.4) satisfy on S0c a compatibility condition for the derivatives in t up to order K0 ,

(7.7)

where K0 = [l1 /2]i + 1. Under these conditions, we have u ∈ C l1 +2,l1 /2+1 (G) [67, 37]. Taking into account the a priori estimates up to the boundary (derived in the variables ξ, τ ), we obtain the estimate ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ u(x, t) ¯ k1 ¯ ≤ M ε−(k+k0 ) , (x, t) ∈ G, 0 ≤ k + 2k0 ≤ K, k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ where K = l + 2. Further, we need more accurate estimates. Assume that the condition b11 (x, t) ≥ b0 , is fulfilled. Thus,

S L = S L−

(x, t) ∈ G S

(7.8)

S L+ ,

where S L− = S1 and S L+ = S2 are the left and right parts of the boundary S L . Write the solution of the problem as the sum of the functions u(x, t) = U (x, t) + V (x, t),


(x, t) ∈ G,

(7.9)

194


where U (x, t) and V (x, t) are the regular and singular parts of the solution. e The function U (x, t) is the restriction to G of the function U e (x, t), (x, t) ∈ G , e S where G = Ge S e , Ge = De ×(0, T ] is the extension of the domain G beyond the boundary S L− , De = {x : −∞ < x1 < d∗1 , |xs | < ∞, s = 2, . . . , n}. e

The function U e (x, t), (x, t) ∈ G is the bounded solution of the problem Le U e (x, t) = f e (x, t),

(x, t) ∈ Ge ,

(7.10)

U e (x, t) = ϕ e (x, t), (x, t) ∈ S e .

The function ϕ e (x, t) is sufficiently smooth on the boundary S e , and it coS incides with the function ϕ(x, t) on the set S L+ S0 . The function V (x, t), (x, t) ∈ G is the solution of the problem L V (x, t) = 0,

(x, t) ∈ G,

V (x, t) = ϕV (x, t) ≡ ϕ(x, t) − U (x, t), (x, t) ∈ S. S

The function V (x, t) vanishes on the set S L+ S0 . In the case of the conditions (7.6), (7.7), one has u, U, V ∈ C l1 +2,l1 /2+1 (G), and for the solution of the problem and its components in the representation (7.9), the following estimates are valid: ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ u(x, t) (7.11) ¯ k1 ¯ ≤ M ε−(k+k0 ) , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ U (x, t) ¯ k1 ¯ ≤ M ε−(k+k0 ) , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ ¯ ¯ ¯ ¯ ¡ ¢ ∂ k+k0 ¯ ¯ V (x, t) ¯ k1 ¯ ≤ M ε−(k+k0 ) exp − m ε−1 r(x, Γ) , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ (x, t) ∈ G,

0 ≤ k + 2k0 ≤ K,

where K = l(7.6) + 2, and m is an arbitrary constant in the interval (0, m0 ) £ ¤ 1 m0 = min a−1 11 (x, t) b1 (x, t) . G

Let us refine the estimates (7.11). Let the problem data satisfy the condition ask , bs , b1s , c, c1 , p, f ∈ C l2 (G), l2

T

l2

ϕ ∈ C (S0 ) C (S 2 ),

l2 = l + α,

(7.12) l ≥ 2,

α ∈ (0, 1).

One can represent the function U (x, t) by the formal expansion U (x, t) = U0 (x, t) + ε U1 (x, t) + v1 (x, t) = U (1) (x, t) + v1 (x, t), (x, t) ∈ G,



195

where U (1) (x, t) and v1 (x, t) are the main and remainder terms of the regular component of the solution. The functions U0 (x, t), U1 (x, t), v1 (x, t) are restrictions to the set G of the functions U0e (x, t), U1e (x, t), v1e (x, t) that are solutions of the problems Le1 U0e (x, t) = f e (x, t),

e

(x, t) ∈ G \ {S eL+

U0e (x, t) = ϕe (x, t),

(x, t) ∈ S eL+

Le1 U1e (x, t) = −Le2 U0e (x, t), U1e (x, t) = 0,

S

S

S0e }

S0e ; e

(x, t) ∈ G \ {S eL+ (x, t) ∈ S

Le v1e (x, t) = −ε2 Le2 U1e (x, t),

eL+ S

S

S0e },

(7.14)

S0e ;

(x, t) ∈ Ge ,

v1e (x, t) = ϕe (x, t) − U (1)e (x, t),

(7.13)

(7.15)

(x, t) ∈ S e .

Given sufficient smoothness of the data of the problem (7.1), (7.4), the solutions of the hyperbolic problems (7.13), (7.14) and the component U (1) (x, t) are sufficiently smooth provided that the data of the problem (7.13), (7.14) satisfy suitable compatibility conditions on the set S +c , i.e., on the intersection of the faces S0 and S L+ . Similar compatibility conditions for the hyperbolic system that defines the reduced problem will ensure the smoothness of the component U (1) (x, t) also in the case of the problem (7.1) on the parallelepiped (7.5). It is appropriate here to give compatibility conditions for the hyperbolic system (7.13), (7.14) in the case of the problem on the parallelepiped under the assumption that b1s (x, t) ≥ b0 > 0,

(x, t) ∈ G,

s = 1, . . . , n.

(7.16)

L+ S

Set S + = S S0 . Let S +c be the edges of S + , i.e., S +c is formed by taking pairwise intersection of the faces in S + . More precisely, taking (7.16) into account, S + comprises the faces S j , j ∈ J0+ , where J0+ = {j = 0, 1 + n, . . . , 2n}; these are the faces S0 and all faces S j from the set S L+ . S Set S + = q S q , q ∈ J0+ . Let the functions u1 (x, t) and u2 (x, t) be solutions of the problems L1 u1 (x, t) = f (x, t), u1 (x, t) = ϕ(x, t), L1 u2 (x, t) = f 2 (x, t), u2 (x, t) = ϕ2 (x, t),

(x, t) ∈ G \ S + ,

(7.17)

(x, t) ∈ S + ; (x, t) ∈ G \ S + ,

(7.18)

(x, t) ∈ S + ;

where f 2 (x, t) = −L2 u1 (x, t), x ∈ G and ϕ2 (x, t) = 0, (x, t) ∈ S + . Assuming sufficient smoothness of the data of the problem (7.1), (7.5), on each of the faces S j+n ⊂ S L+ , where j = 1, . . . , n, the following derivatives are defined: ∂ k+k0 ∂xk11

kj−1 . . . ∂xj−1


k

j+1 ∂xj+1 . . . ∂xknn ∂tk0

ui (x, t) =

196

Parabolic convection-diffusion equations =

∂ k+k0 kj−1 ∂xk11 . . . ∂xj−1

kj+1 ∂xj+1

. . . ∂xknn

∂tk0

ϕi (x, t),

(x, t) ∈ S j+n ,

k = k1 + . . . + kj−1 + kj+1 + . . . + kn , and on the face S0 the following derivatives are defined: ∂xk11

∂k ∂k ui (x, t) = ϕi (x, t), kn k1 . . . ∂xn ∂x1 . . . ∂xknn

(x, t) ∈ S0 , i = 1, 2.

By virtue of the equations (7.17), (7.18), we can compute the derivatives ∂ k+k0 k

∂xk11 . . . ∂xj j . . . ∂xknn ∂tk0

ui (x, t),

(x, t) ∈ S j+n , S j+n ⊂ S L+ ,

∂ k+k0 ui (x, t), ∂xk11 . . . ∂xknn ∂tk0

(x, t) ∈ S0 .

In the case of the problem (7.1) on the parallelepiped (7.5), we say that the data of the problem (7.17), (7.18), i.e., the coefficients of the operators L1 , L2 and the functions f (x, t), (x, t) ∈ G and ϕ(x, t), (x, t) ∈ S + , satisfy on the set S +c compatibility conditions for the derivatives of the component U (1) (x, t) up to order K if the derivatives ∂ k+k0 ui (x, t), ∂xk11 . . . ∂xknn ∂tk0

0 ≤ k + k0 ≤ K + 2(1 − i),

i = 1, 2

are continuous on S +c . This ensures the continuity on S +c of the derivatives in x and t of the function U (1) (x, t) up to order K, T In the case of the problem (7.1) on the slab (7.4), S +c = S0 S 2 ; we assume that For the data of the problem (7.13), (7.14) on the set S +c compatibility conditions are fulfilled for the derivatives U

(1)

(7.19)

(x, t) up to order l − 2, where l = l(7.12) .

Under the assumptions given above, the functions U0 (x, t), U1 (x, t), and U (1) (x, t) satisfy the condition U0 ∈ C K+2+α (G),

U1 , U (1) ∈ C K+α (G),

K ≥ 0,

α ∈ (0, 1);

moreover, the function U (1) (x, t) satisfies the estimate ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ (1) U (x, t) ¯ k1 ¯ ≤ M, (x, t) ∈ G, 0 ≤ k + k0 ≤ K, (7.20) ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ where K = l(7.12) − 2.



197

For l ≥ 4, the function v1 (x, t) satisfies the estimate |v1 (x, t)| ≤ M ε2 ,

(x, t) ∈ G.

(7.21)

In the case of the conditions (7.6), (7.7), (7.12), and (7.19), where l(7.6) ≥ l(7.12) − 2,

l(7.12) ≥ 4,

(7.22)

we have v1 ∈ C l3 ,l3 /2 (G), where l3 = l(7.12) − 2 + α. For the function v1 (x, t), we obtain the estimate ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ v (x, t) ¯ k1 ¯ ≤ M ε2−k−k0 , (x, t) ∈ G, 0 ≤ k + 2 k0 ≤ K. ¯ ∂x1 . . . ∂xknn ∂tk0 1 ¯ Taking into account this estimate and (7.20), for the component U (x, t), we obtain the estimate ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ U (x, t) (7.23) ¯ k1 ¯ ≤ M 1 + ε2−k−k0 , (x, t) ∈ G, ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ 0 ≤ k + 2 k0 ≤ K, where K = l(7.12) − 2. By virtue of (7.23), the function ϕV (x, t), (x, t) ∈ S L− , satisfies the estimate ¯ ¯ ¯ ¯ £ ¤ ∂ k+k0 ¯ ¯ ϕ (x, t) ¯ k2 ¯ ≤ M 1 + ε2−k−k0 , (x, t) ∈ S L− , V k n k ¯ ∂x2 . . . ∂xn ∂t 0 ¯ 0 ≤ k + 2 k0 ≤ K, k = k2 + . . . + kn . From this estimate we obtain ¯ ¯ ¯ ¯ ¤ £ ¤£ ∂ k+k0 ¯ ¯ V (x, t) ≤ M ε−k1 + ε1−k 1 + ε1−k0 × (7.24) ¯ k1 ¯ k n ¯ ∂x1 . . . ∂xn ∂tk0 ¯ ¡ ¢ × exp − m ε−1 r(x, Γ1 ) , (x, t) ∈ G, 0 ≤ k + 2 k0 ≤ K, where K = l(7.12) − 2, m = m(7.11) . If the conditions (7.7), (7.19) are not satisfied, then the functions f (x, t), (x, t) ∈ G and ϕ(x, t), (x, t) ∈ S are approximated by smooth functions f λ (x, t), (x, t) ∈ G and ϕλ (x, t), (x, t) ∈ S, for which the conditions (7.7),


198


(7.19) are fulfilled and also one has the estimates ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ λ f (x, t) ¯ k1 ¯ ≤ M (1 + λ1−k−k0 ), ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ |f λ (x, t) − f (x, t)| ≤ M λ, (x, t) ∈ G; ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ λ ϕ (x, t) ¯ k2 ¯ ≤ M (1 + λ1−k−k0 ), k ¯ ∂x2 . . . ∂xnn ∂tk0 ¯

L

(x, t) ∈ S ,

k2 + . . . + kn = k, ¯ ¯ ¯ ¯ ∂k ¯ ¯ λ ϕ (x, t) ¯ k1 ¯ ≤ M (1 + λ1−k ), (x, t) ∈ S0 , ¯ ∂x1 . . . ∂xknn ¯ ¯ λ ¯ ¯ϕ (x, t) − ϕ(x, t)¯ ≤ M λ, (x, t) ∈ S, where the parameter λ takes an arbitrary value from the interval (0, 1). Let uλ (x, t) be the solution of the problem (7.1), (7.4), where f (x, t) = f λ (x, t), (x, t) ∈ G,

ϕ(x, t) = ϕλ (x, t), (x, t) ∈ S.

The function uλ (x, t) satisfies the inequality |u(x, t) − uλ (x, t)| ≤ M λ,

(x, t) ∈ G.

(7.25)

For the components U (x, t) and V (x, t) in the representation (7.9), where u(x, t) is replaced by uλ (x, t), the estimates (7.23), (7.24) are fulfilled with a constant M that depends on λ: M(7.23) , M(7.24) = M0 λ−1−k−2k0 ,

0 ≤ k + 2k0 ≤ K,

(7.26)

where the constant M0 is independent of λ. The following theorem holds. Theorem 7.2.1 For the data of the boundary value problem (7.1), (7.4) let the conditions (7.8) and also (7.6), (7.12) and (7.22) for l(7.12) ≥ K +2, where K ≥ 2, be fulfilled. Then the function u(x, t), i.e., the solution of the problem (7.1), (7.4), can be approximated by the function uλ (x, t); moreover, for the function uλ (x, t) and its components U (x, t) and V (x, t) in the representation (7.9), where u(x, t) is replaced by uλ (x, t), the inequality (7.25) and the estimates (7.23), (7.24), and (7.26) hold. Under the additional conditions (7.7) and (7.19), the estimates (7.23) and (7.24) hold for the components U (x, t) and V (x, t) in the representation (7.9) of the function u(x, t).


Estimates of the problem solution on a parallelepiped

7.3

199


Let us consider the problem (7.1), (7.5) under the condition (7.16), assuming that the problem data are sufficiently smooth. In this case, Sj ⊂ S L− , for j = 1, . . . , n, and Sj ⊂ S L+ , for j = n + 1, . . . , 2n. Assume that for the data of the problem (7.1), (7.5), one has the conditions ask , bs , b1s , c, c1 , p, f ∈ C l1 , l1 /2 (G), T

ϕ ∈ C l1 +2, l1 /2+1 (S0 ) C l1 +2, l1 /2+1 (S j ), l1 = l + α,

l ≥ 0,

j = 1, . . . , 2n,

(7.27)

α ∈ (0, 1).

Under these condition, one has T

u ∈ C α1 , α1 /2 (G) C l1 +2, l1 /2+1 (G),

α1 ∈ (0, 1).

Taking into account the a priori estimates up to the smooth parts of the boundary S, we find ¯ ¯ ¯ ¯ £ ¡ ¢¤£ ¡ ¢¤ ∂ k+k0 ¯ ¯ u(x, t) ¯ k1 ¯ ≤ M ε−k + r−k x, Γ ε−k0 + εk0 r−2k0 x, Γ , ¯ ∂x1 . . . ∂xknn ∂tk0 ¯ (x, t) ∈ G,

x∈ / Γ for k + k0 > 0,

0 ≤ k + 2k0 ≤ K,

(7.28)

where K = l(7.27) + 2. Furthermore, the function u(x, t) satisfies the estimate ¯ ¯ ¡ ¢ ¯u(x, t) − ϕ(x∗ , t)¯ ≤ M ε−1 r x, Γ , (x, t) ∈ G, (7.29) where x∗ = x∗(5.6) (x, Γ) is a point on Γ nearest to the point x ∈ D, (x, t) ∈ G. Write the problem solution as the sum u(x, t) = U (x, t) + V (x, t),

(x, t) ∈ G,

(7.30)

where U (x, t) and V (x, t) are the regular and singular parts of the solution of the problem. The function U (x, t), (x, t) ∈ G, is the restriction to the set G of the e function U e (x, t), (x, t) ∈ G , that is the solution of the problem Le U e (x, t) = f e (x, t),

(x, t) ∈ Ge ,

(7.31a)

U e (x, t) = ϕe (x, t),

(x, t) ∈ S e .

(7.31b)

The domain Ge is an extension of the domain G beyond the boundary S L− , Ge = De × (0, T ], De = D(n+1,...,2n)(3.6) . The coefficients and the righthand side in (7.31a) are smooth extensions of those in (7.1), preserving their properties. The function ϕe (x, t), (x, t) ∈ S e , is smooth on each of the


200


faces Sje , and it coincides with the function ϕ(x, t) on the set S0 function V (x, t), (x, t) ∈ G, is the solution of the problem L V (x, t) = 0,

S

S L+ . The

(x, t) ∈ G,

V (x, t) = ϕV (x, t) ≡ ϕ(x, t) − U (x, t), The function V (x, t) vanishes on the set S L+ Let the problem data satisfy the condition

S

(7.32)

(x, t) ∈ S.

S0 .

ask , bs , b1s , c, c1 , p, f ∈ C l2 (G), T

ϕ ∈ C l2 (S0 ) C l2 (S j ), l2 = l + α,

l ≥ 2,

j = n + 1, . . . , 2n,

(7.33)

α ∈ (0, 1).

We write the function U (x, t) as a formal expansion U (x, t) = U (1) (x, t) + v1 (x, t),

(x, t) ∈ G,

(7.34)

where U (1) (x, t) = U0 (x, t) + ε U1 (x, t). The functions U0 (x, t), U1 (x, t), and v1 (x, t) are restrictions to the set G of the functions U0e (x, t), U1e (x, t), and v1e (x, t) that are solutions of the problems similar to (7.13), (7.14), (7.15): Le1 U0e (x, t) = f e (x, t), U0e (x, t) = ϕe (x, t),

e

(x, t) ∈ G \ S e+ ,

(7.35)

(x, t) ∈ S e+ ;

Le1 U1e (x, t) = −Le2 U0e (x, t), U1e (x, t) = 0,

e

(x, t) ∈ G \ S e+ ,

(7.36)

(x, t) ∈ S e+ ;

Le v1e (x, t) = −ε2 Le2 U1e (x, t),

(x, t) ∈ G e ,

v1e (x, t) = ϕe (x, t) − U (1)e (x, t),

(x, t) ∈ S e ,

S

where S e+ = S eL+ S0e , Le1 = L1(7.1) , Le2 = L2(7.1) for (x, t) ∈ G. Let the boundary value problems (7.35) and (7.36) satisfy the condition For the data of the problem (7.35) and (7.36) on the set S +c , compatibility conditions are fulfilled for the derivatives U

(1)

(7.37)

(x, t) up to order l − 2, where l = l(7.33) .

Then U0 ∈ C K+2+α (G), U1 , U (1) ∈ C K+α (G), α ∈ (0, 1), and the estimate holds ¯ ¯ ¯ ¯ ∂ k+k0 ¯ ¯ (1) U (x, t) ¯ k1 ¯ ≤ M, 0 ≤ k + k0 ≤ K, k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ where K = l(7.33) − 2. For l(7.33) ≥ 4, we have |v1 (x, t)| ≤ M ε2 ,


(x, t) ∈ G.


201

In the case of the conditions (7.27), (7.33), and (7.37), where l(7.27) ≥ l(7.33) − 2,

l(7.33) ≥ 4,

(7.38)

T

one has v1 ∈ C α1 ,α1 /2 (G) C l3 ,l3 /2 (G), where l3 = l(7.33) − 2 + α. For the function v1 (x, t), we obtain the estimate ¯ ¯ ¯ ¯ £ ¡ ¢¤£ ¡ ¢¤ ∂ k+k0 ¯ ¯ v (x, t) ≤ M ε2 ε−k + r−k x, Γ ε−k0 + εk0 r−2k0 x, Γ , ¯ k1 ¯ 1 k n ¯ ∂x1 . . . ∂xn ∂tk0 ¯ (x, t) ∈ G,

x∈ / Γ for k + k0 > 0,

0 ≤ k + 2k0 ≤ K,

(7.39)

where K = l(7.33) − 2. The estimates of the derivatives of the functions U (1) (x, t) and v1 (x, t) allow us to write down estimates of the derivatives of the component U (x, t) in (7.30). When considering the function V (x, t), we assume that the conditions (7.27), (7.33), (7.37), and (7.38) hold. Write the function V (x, t) as the sum of the functions V (x, t) = V0 (x, t) + v2 (x, t),

(x, t) ∈ G.

Here V0 (x, t) and v2 (x, t) are solutions of the problems L V0 (x, t) = 0,

(x, t) ∈ G,

V0 (x, t) = ϕV0 (x, t) ≡ ϕ(x, t) − U (1) (x, t), L v2 (x, t) = 0,

(x, t) ∈ S;

(x, t) ∈ G,

v2 (x, t) = −U (x, t) + U (1) (x, t),

(x, t) ∈ S.

For the function v2 (x, t), we have the estimate ¯ ¯ ¯ ¯ £ ¡ ¢¤£ −k0 ¡ ¢¤ ∂ k+k0 ¯ ¯ 2 −k −k k0 −2k0 v (x, t) ≤ M ε ε + r x, Γ ε + ε r x, Γ , ¯ k1 ¯ 2 k ¯ ∂x1 . . . ∂xnn ∂tk0 ¯ (x, t) ∈ G,

x∈ / Γ for k + k0 > 0,

0 ≤ k + 2k0 ≤ K,

(7.40)

where K = K(7.39) . One can represent the boundary layer function V0 (x, t) by the sum of more simple boundary layers X X V0 (x, t) = V(j) (x, t) + V(j...r) (x, t), (x, t) ∈ G. (7.41) j=1,...,n


j,..., r=1,...,n j

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