Multiscale Wavelet Methods for Partial Differential Equations
Multiscale Wavelet Methods for Partial Differential Equations
Wavelet Analysis and Its Applications The subject of wavelet analysis has recently drawn a great deal of attention from mathematical scientists in various disciplines. It is creating a common link between mathematicians, physicists, and electrical engineers. This book series will consist of both monographs and edited volumes on the theory and applications of this rapidly developing subject. Its objective is to meet the needs of academic, industrial, and governmental researchers, as well as to provide instructional material for teaching at both the undergraduate and graduate levels. This is the sixth book of the series. While wavelet techniques and algorithms have proved to be very powerful in several areas of applications such as signal processing and image compression, the wavelet approach to solving complex problems governed by physical models such as partial differential equations (PDEs) has to compete with the well-established and already very effective methods. There is, however, a common link between the multiresolution approximation structure in wavelet analysis and multilevel and multigrid techniques in numerical solutions of P DEs. This volume is aimed at bridging these two most fruitful approaches and is designed to update current developments in PDEs that are somewhat related to the wavelet approach. The series editor wishes to congratulate the editors of this volume for an outstanding job in selecting the most relevant chapters and in carefully editing the volume. He would also like to thank the authors for their very fine contributions.
Multiscale Wavelet Methods for Partial Differential Equations Edited by Wolfgang Dahmen Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen, Germany
Andrew J. Kurdila Department of Aerospace Engineering Texas A&M University College Station, Texas, U.S.A.
Peter Oswald Institute for Algorithms and Scientific Computing, GMD Sankt Augustin, Germany
ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto
Copyright
9 1997 by Academic Press 0
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Library of Congress Cataloging-in-Publication Data Multiscale wavelet methods for partial differential equations / edited by Wolfgang Dahmen, Andrew Kurdila, Peter Oswald. p. cm. m (Wavelet analysis and its applications ; v. 6) Includes bibliographical references and index. ISBN 0-12-200675-5 (alk. paper) 1. Differential equations, PartialmNumerical solutions. 2. Wavelets (Mathematics) I. Dahmen, Wolfgang. II. Kurdila, Andrew. III. Oswald, Peter. IV. Series. 97-12672 QC20.7.D5M83 1997 CIP 515'.2433mDC21
Printed in the United States of America 97 98 99 00 01 IP 9 8 7 6 5 4 3 2 1
Contents Preface .............................................................. Contributors ........................................................ I. F E M - L i k e
Multilevel
Preconditioning
.........................
Multilevel Solvers for Elliptic Problems on Domains . . . . . . . . . . . . . . . . . . . Peter Oswald Wavelet-Like Methods in the Design of Efficient Multilevel Preconditioners for Elliptic P DEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii xi 1
3
59
Panayot S. Vassilevski and Junping Wang II. F a s t W a v e l e t A l g o r i t h m s : C o m p r e s s i o n a n d A d a p t i v i t y
.. 107
An Adaptive Collocation Method based on Interpolating Wavelets ... 109
Silvia Bertoluzza An Adaptive Pseudo-Wavelet Approach for Solving Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gregory Beylkin and James M. Keiser A Dynamical Adaptive Concept Based on Wavelet Packet Best Bases: Application to Convection Diffusion Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal Joly, Yvon Maday, and Valdrie Pettier Nonlinear Approximation and Adaptive Techniques for Solving Elliptic Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
199
237
Stephan Dahlke, Wolfgang Dahmen, and Ronald A. De Vore I I I . W a v e l e t S o l v e r s for I n t e g r a l E q u a t i o n s
...................
Fully Discrete Multiscale Galerkin BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
285 287
Tobias yon Petersdorff and Christoph Schwab Wavelet Multilevel Solvers for Linear Ill-Posed Problems Stabilized by Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Andreas Rieder
347
Contents
vi IV. S o f t w a r e T o o l s a n d N u m e r i c a l E x p e r i m e n t s
..............
381
Towards Object Oriented Software Tools for Numerical Multiscale Methods for PDEs using Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Titus Barsch, Angela Kunoth, and Karsten Urban
383
Scaling Function and Wavelet Preconditioners for Second Order Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jeonghwan Ko, Andrew J. Kurdila, and Peter Oswald
413
V. M u l t i s c a l e I n t e r a c t i o n a n d A p p l i c a t i o n s t o T u r b u l e n c e . . . 4 3 9 Local Models and Large Scale Statistics of the Kuramoto-Sivashinsky Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
Juan Elezgaray, Cal Berkooz, Harry Dankowicz, Philip Holmes, and Mark Myers Theoretical Dimension and the Complexity of Simulated Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
473
Mladen V. Wickerhauser, Marie Farge, and Eric Goirand V I . W a v e l e t A n a l y s i s of P a r t i a l D i f f e r e n t i a l O p e r a t o r s . . . . . . . 4 9 3 Analysis of Second Order Elliptic Operators Without Boundary Conditions and With VMO or H61derian Coefficients . . . . . . . . . . . . . . . .
495
Jean-Marc Angeletti, Sylvain Mazet, and Philippe Tchamitchian Some Directional Elliptic Regularity For Domains With Cusps . . . . . . . 541
Matthias Holschneider Subject Index .....................................................
567
Preface Wavelet methods are by now well-established as a novel and successful mathematical tool with applications in Signal Analysis and Image Processing, Theoretical Physics and Mathematics itself. Meanwhile several books and edited volumes have appeared, in particular, in this series Wavelet Analysis and Its Applications, addressing this fast growing field of reasearch. More recently the intriguing aspects of these concepts have stirred considerable interest in applying them also to more traditional areas of numerical analysis and engineering applications. The high-level contributions to the Special Session on Wavelet Galerkin Methods at the 31st Annual Meeting of the Society of Engineering Science held in College Station, TX, in October 1994 as well as the Special Session on Meshless and Wavelet Methods at the 3rd U.S. National Congress in Computational Mechanics held in Dallas in June 1995 and the partly very controversial panel discussions on these meetings indicated a strong need of an up-to-date publication on current developments and further prospects of this methodology, specifically in the area of partial differential equations (PDEs). It is fair to say that wavelet analysis as it stands has by far not yet reached a steady state in this field. As promising as many of the underlying concepts are, it would be naive to expect their immediate practical success in complex, real-life applications governed by PDEs. Current wavelet methods and corresponding software tools are still primarily confined to model problems. While these new ideas may also actually trigger new developments in the context of well-established multilevel techniques, they will ultimately have to compete with the discretization methods and existing software packages for PDEs. This volume aims at contributing to the progress in this direction. We do not claim to reflect an exhaustive account of the state of the art. However, we have made an effort to address several aspects which we feel are important and typical in connection with solving partial differential equations and which bear potential for further progress. Key ideas such as sparsity of wavelet representations of operators, economic vii
viii
Preface
representations of functions with localized singularities or strongly scale dependent behavior, the availability of new libraries of flexible localized basis functions (wavelets and wavelet packets) as well as resulting fast multiscale algorithms will be presented in the context of linear and nonlinear operator equations. To support a first orientation we have grouped the material into six chapters which are, however, interrelated in many respects. The following comments might serve as a brief guide. The first chapter may be viewed as a bridge to finite element based multilevel preconditioning and multigrid techniques. Recent development has shown that the multiscale space decomposition framework typical for wavelets provides significant insight into the understanding of algorithms like hierarchical bases or BPX schemes. On the other hand, multigrid technology adds further algorithmical variety and fuels intertwining of these concepts. Oswald focusses on frame based multilevel Schwarz preconditioners for elliptic boundary value problems on bounded domains in IRd. The main goal is to preserve as much as possible the algorithmic advantages of scale- and shiff-invariant discretizations typical for the classical wavelet setting by introducing local modifications only near the boundary. As for finite element schemes, adaptive nested refinement can be incorporated. Vassilevski and Wang start from the hierachical basis method by Yserentant for linear finite element discretizations (which is asymptotically nonoptimal), and propose an improvement based on approximately computing wavelet-like complement basis functions. They also discuss the multiplicative algorithms corresponding to these space decompositions which provides a link to multigrid V-cycle solvers. Chapter 2 offers information on principal features of wavelet based discretizations centering upon sparse representations of operators and functions as well as resulting adaptivity concepts. Bertoluzza highlights the algorithmic benefits of interpolatory Deslaurier-Dubuc wavelets for adaptive collocation methods. The method is illustrated on several linear problems including dominating convection, but might be attractive for nonlinear problems as well. Beylkin and Keiser present a systematic algorithmic study of a class of periodic nonlinear evolution equations covering, for instance, the viscous Burgers equation and Korteweg-de-Vries equation. Their main objective is to produce a scheme which solves such problems for a given tolerance at a cost which remains proportional to the number of significant wavelet coefficients of the solution. Essential ingredients are sparse operator representations in the so-called nonstandard form and the fast evaluation of nonlinear terms. Joly, Maday, and Perrier apply wavelet packets and compression techniques to the adaptive treatment of nonlinear evolution equations. In particular, a best basis concept based on cardinal entropy is introduced and its practical implementation
Preface
ix
for time-dependent PDEs is discussed. The numerical examples are concerned again with the Burgers equation with small viscosity, and simple convection-diffusion problems. Dahlke, Dahmen, and DeVore address the issue of adaptivity from a primary analytical viewpoint. The goal is to interrelate the concepts of nonlinear approximation, Besov regularity, and wavelet based adaptive techniques for stationary elliptic problems covering integral as well as differential operators. In particular adaptive refinement strategies are shown to converge without additional a priori assumptions on the solution. In Chapter 3, two papers on integral equations closely related to the material of the previous chapters are included. Von Petersdorff and Schwab propose a wavelet based, fully discrete Galerkin scheme for a zero-order elliptic boundary integral equation. They combine the wavelet compression of the intergral operator with a carefully designed adaptive quadrature scheme which ensures the same asymptotical complexity in the computation of the compressed matrix as the solver. This is a central step towards efficient practical implementations. The paper by Rieder is devoted to additive and multiplicative wavelet solvers of Tikhonov regularized ill-posed problems. The techniques are similar in spirit to those in Chapter 1. The paper by Barsch, Kunoth and Urban in Chapter 4 surveys a software toolbox under development which aims at providing an experimental platform for wavelet discretizations of PDEs and integral equations. Some emphasis is put on the use of C + + for treating multidimensional multiscale data structures and algorithms. Ko, Kurdila and Oswald present a comparative study of several multilevel preconditioners for a second order model problem on simple domains. In particular, schemes based on finite elements, Daubechies and AFIF scaling functions and wavelets are compared. Chapter 5 is different in nature. Rather than the efficiency of algorithms, the main concern here is to employ wavelets as an adequate tool for analyzing and simulating multiscale interaction/separation in flows. In Elezgaray, Berkooz, Dankowicz, Holmes and Myers, local models for the study of coherent structures of solutions of the Kuramoto-Sivashinsky equation on large intervals are investigated. The use of periodic wavelets is proposed and compared with the traditional Fourier approximations. In the paper by Wickerhauser, Farge and Goirand, the complexity of fully developed turbulent flows is investigated with the aid of concepts like theoretical dimension, best bases and wavelet packets. Numerical experiments for the Burgers equation and two-dimensional Navier-Stokes flow are presented which suggest a strong interrelation of the theoretical dimension and the number of coherent structures in two-dimensional viscous turbulent flows. Chapter 6 is devoted to the use of wavelets primarily as a tool for analysing differential operators. Angeletti, Mazet and Tchamitchian study
x
Preface
second order differential operators in divergence and non-divergence forms with diffusion tensors of rather weak regularity on ]Rd. One of the main results is concerned with the boundedness of Galerkin projection operators in Lp-Sobolev norms. Holschneider presents a wavelet based microlocal analysis of local regularity spaces. As an application, the regularity of elliptic differential operators on domains with cusps is treated. This volume is comprised of both invited and contributed chapters. All contributions, whether of survey character or containing primarily original material, were refereed according to their respective goal. We wish to thank all authors and reviewers for their most valuable contributions, in particular, for their patience and cooperation. We are indebted to Margaret and Charles Chui for their diligent and kind assistance during the editorial process.
Aachen, Germany College Station, Texas Sankt Augustin, Germany June, 1997
Wolfgang Dahmen Andrew J. Kurdila Peter Oswald
Contributors Numbers in parentheses indicate where the authors' contributions begin.
J. M. ANGELETTI (495), Laboratoire de Mathdmatiques Fondamentales et Appliqudes, Facultd des Sciences et Techniques de Saint-Jdrdme, 13397 Marseille Cedex 20, France, et LATP, CNRS, URA 225 [jean-marc. angeletti@math, u-3mrs.fr] TITUS B ARSCH (383), Institut fffr Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [barsch @igpm. rwt h-aachen, de] GAL BERKOOZ (441), BEAM Technologies, Ithaca, N Y 15850 [gal@cam. cornell.edu] SILVIA BERTOLUZZA (109), I.A.N.-C.N.R., v. Abbiategrasso 209, 27100 Pavia, Italy [
[email protected]. pv. cnr.it] GREGORY BEYLKIN (137), Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, U.S.A. [
[email protected]] WOLFGANG DAHMEN (237), Institut fffr Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [dahmen @igprn. rwt h-aachen, de] STEPHAN DAHLKE (237), Institut fffr Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [dahlke@igprn. rwth-aachen, de]
xi
xii
Contributors
HARRY DANKOWICZ (441), Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden [
[email protected]] RONALD A. DEVORE (237), Department of Mathematics, University of South Carolina, Columbia, S.C. 29208, U.S.A. [
[email protected]] JUAN ELEZGARAY (441), CRPP-CNRS, Av. Schwietzer, 33600 Pessac, F~ance [
[email protected]. fr] PHILIP HOLMES (441), PACM, Fine Hall, Princeton University, Princeton, NJ 08544-1000, U.S.A. [
[email protected]] MARIE FARGE (473), LMD-CNRS, Ecole Normal Superieure, 24 Rue Lhomond, F-75231 Paris, France [
[email protected]] ERIC GOIRAND (473), LMD-CNRS, Ecole Normal Superieure, 24 Rue Lhomond, F-75231 Paris, France [goirand~lmd.ens.fr] MATTHIAS HOLSCHNEIDER(541), CNRS CPT Luminy, Case 907, F-13288 Marseille, France [
[email protected]] PASCAL JOLY (199), Laboratoire d'Analyse Numdrique, Tour 55-65, 5~me dtage, Universitd Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France
[email protected]] JAMES M. KEISER (137), 659 Main St., Apt. B, Laurel, MD 20707-4067, U.S.A. [
[email protected]] JEONGHWAN KO (413), Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, U.S.A. [
[email protected]] ANDREW J. KURDILA (413), Department of Aerospace Engineering, Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A. [
[email protected]] ANGELA KUNOTH (383), Institut fffr Geometrie und Praktische Mathematik, RWTH Aachen, 52056 Aachen, Germany [kunot h @igpm. rwt h-aachen, de]
Contributors
xiii
YVON MADAY (199), ASCI, UPR 9029, Bat. 506, Universitd Paris Sud, 91405 Orsay Cedex, France g~ Laboratoire d'Analyse Numdrique, Tour 55-65, 5@me dtage, Universitd Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, Prance [
[email protected] ussieu, fr] S. MAZET (495), Laboratoire de Mathdmatiques Fondamentales et Appliqudes, Facultd des Sciences et Techniques de Saint-Jdr6me, 13397 Marseille Cedex 20, Prance, et LATP, CNRS, URA 225 [
[email protected]] MARK MYERS (441) PETER OSWALD (3, 413), Institute for Algorithms and Scientific Computing, G M D - German National Research Center for Information Technology, D-53754 Sankt Augustin, Germany [
[email protected]] VALt~RIE PERRIER (199), Laboratoire d'Analyse, Gdomdtrie et Applications, URA 742, Universitd Paris Nord, 93430 Villetaneuse, & Laboratoire de Mdtdorologie Dynamique, Ecole Normale Supdrieure, 24, rue Lhomond, 75231 Paris Cedex 05, Prance [
[email protected]] TOBIAS
PETERSDORFF (287), Department of Mathematics, University of Maryland College Park, College Park, MD 20742, U.S.A. [
[email protected]]
VON
ANDREAS RIEDER (347), Fachbereich Mathematik, Universita't des Saar1andes, Postfach 15 11 50, 66041 Saarbrubken, Germany [andreas @num. uni-sb, de] CHRISTOPH SCHWAB (287), Seminar ffir Angewandte Mathematik, Eidgeno'ssische Technische Hochschule Zffrich, Ra'mistrasse 101, CH8092 Zffrich, Switzerland [
[email protected]] P. TCHAMITCHIAN (495), Laboratoire de Mathdmatiques Fondamentales et Appliqudes, Facultd des Sciences et Techniques de SaintJdr6me, 13397 Marseille Cedex 20, Prance, et LATP, CNRS, URA 225 [tch am p hi @mat h. u-3mrs, fr] KARSTEN URBAN (383), Institut ffir Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [urban@igpm. rwth-aachen, de]
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Contributors
PANAYOT S. VASSILEVSKI (59), Center of Informatics and Computing Technology, Bulgarian Academy of Sciences, "Acad. G. Bontchev" street, Block 25 A, 1113 Sofia, Bulgaria [
[email protected]] JUNPING WANG (59), Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071, U.S.A. [
[email protected]] MLADEN VICTOR WICKERHAUSER (473), Department of Mathematics, Campus Box 1146, One Brookings Drive, Washington University, Saint Louis, Missouri 63130, U.S.A. [
[email protected]]
I0 FEM-Like Multilevel Preconditioning
This Page Intentionally Left Blank
M u l t i l e v e l Solvers for Elliptic P r o b l e m s
on Domains
Peter Oswald
A b s t r a c t . We study to which extent the geometric multilevel approach based on dyadic scales of shift-invariant subspaces on ]Rd can be used to produce accurate discrete solutions of elliptic boundary value problems of positive order on nonrectangular domains. We also deal with the construction of optimal preconditioners, including the case of nested refinement. Sufficient geometric conditions on a domain are given such that a robust and asymptotically optimal algorithm can be expected. In contrast to other approaches which emphasize biorthogonal wavelet decompositions, we are satisfied with a simpler frame concept which incorporates recent experience with finite element multilevel solvers.
w
Introduction
For the numerical solution of elliptic boundary value problems for partial differential equations, multilevel methods have gained popularity over the last decade. This is mainly due to their nearly optimal complexity for a number of model problems. In m a n y practical cases, they are based on multiresolution scale of nested finite-dimensional subspaces
Vo cv~ c . . . c ~
c...
(1.1)
of a Hilbert space V serving as the energy space for the given variational problem. The scale (1.1) is used to produce stable subspace splittings v j - v0 + v~ + . . . + v j M u l t i s c ~ l e W a v e l e t M e t h o d s for P D E s Wolfg~ng D~hmen,
Andrew
J. K u r d i l ~ , ~ n d P e t e r O s w M d ( e d s . ) , p p . 3 - 5 8 .
C o p y r i g h t ( ~ 1 9 9 7 by A c a d e m i c P r e s s , I n c . All r i g h t s of r e p r o d u c t i o n in ~ny f o r m reserved. ISBN 0-12-200675-5
(1.2)
4
P. Oswald
and to design fast iterative solvers related to such splittings for the discretized variational problem associated with a computational discretization space Vj (or a properly defined subspace V] C Vj). For 2mth order elliptic boundary value problems in Sobolev spaces, considerable progress has been achieved in the theoretical understanding of multilevel and multigrid methods as well as of other subspace correction methods for finite element discretizations (see [65, 67, 11, 60]). We survey some of these results in Section 2. The underlying theory also applies to various wavelet discretizations; see, e.g., [26, 44, 45, 29] for some papers that deal with wavelet solvers for elliptic problems and are related to our approach. Roughly speaking, in these algorithms suitable "detail spaces" Wj C t~ are constructed, together with their algebraic bases, such that t~ - t~_l+Wj provides a stable splitting of V) into "low frequency" ( ~ - 1 ) and "high frequency" (Wj) parts. Using this two-level decomposition recursively, (1.2) is replaced by vj
= VoSW~ $ . . .4wj
.
(1.3)
One can use this splitting indirectly (i.e., the original problem is discretized with respect to a standard basis in Vj, and the wavelet decomposition behind (1.3) implicitly defines the structure of the multilevel preconditioner) or directly. In the latter case, the discretization is performed with respect to the wavelet basis and is automatically well conditioned. To achieve asymptotically optimal work estimates one has to use compression arguments. For elliptic problems of order 2m, and k~ with locally supported basis functions, the first approach is often preferred since the Vj-discretization matrix is automatically sparse and available from standard engineering codes. The direct use of the wavelet decomposition (1.3) is promising for those situations in which the Vj-discretization is not a priori sparse, e.g., for integral equations. In both cases, the explicit introduction of detail spaces Wj is the crucial step, and may add some theoretical and practical difficulties. For example, most of the popular examples of wavelet spaces (see [30, 16]) are derived in a one-dimensional, shift-invariant setting on IR. Multivariate examples on IRd are mostly obtained by tensor-product techniques. Adaptations to bounded intervals and domains have been studied in, e.g., [2, 17, 19, 18]. However, up to now there has been no comprehensive study of the practical potential of discretizations using multilevel structures based on shift-invariance and dyadic dilation (modulo boundary modifications) in the case of general, nonrectangular geometries. It is not completely clear to the author what will be left from the powerful wavelet machinery if the basic algebraic assumptions (invariance with respect to integer shifts and (dyadic) dilation) are significantly relaxed.
Multilevel Solvers for Elliptic Problems
5
In our opinion, the departure from these assumptions is unavoidable for many engineering problems. A simple example is 2ruth order elliptic PDEs with rapidly varying coefficients that exhibit a large ratio of ellipticity constants and are therefore far from a generic Hm-problem. Principal difficulties are to be expected if nonsymmetric equations with dominant low order hyperbolic parts, such as convection-diffusion equations, or nonlinear problems are to be studied. In our opinion, this robustness aspect is one of the target problems for future investigation, i.e., the adaptation of the multilevel concept to a class of operator equations (or even to an individual equation) still remains a decisive issue for practical implementations and engineering applications. An indicator of this tendency is analogous efforts within the FEM community and the renewed interest in the algebraic multigrid method and related algorithms. There is another observation that dampens expectations concerning the practical use of wavelet solvers, even for standard symmetric elliptic boundary value problems. Numerical experiments [39, 49] show that for generic HI-problems, i.e., for second order elliptic equations, wavelet and prewavelet discretizations perform slightly worse than traditional finite element preconditioners associated with (1.2). While condition numbers of L2-problems (and sometimes also of HS-problems with negative s) are usually improved and become uniformly bounded if J --+ c~, the preconditioning effect in the Hi-norm is reduced by a constant factor. This poses the problem of determining more carefully problem classes in which the use of wavelet preconditioners based on (1.3) is justified compared to simpler methods based only on the use of scaling functions and related to (1.2). Also, one may argue whether new wavelet families (Daubechies orthogonal wavelets [30], AFIF elements [51], etc.) are generally useful and are able to compete with traditional finite element and spline constructions from this more practical viewpoint. The reader should not expect an answer to these more philosophical questions. In Sections 3-5, we concentrate on studying the influence of the domain geometry on the optimality of multilevel preconditioners resulting from a standard multiresolution analysis (these sections are sometimes rather technical and represent the original part of this paper). The concept is based on sequences {Vj} of subspaces of a fixed sequence {Vj} of subspaces in some HS(IR d) which "live" on a uniform structure generated by shift-invariance principles and dyadic dilation as usual. These auxiliary subspaces Vj are spans of scaling functions (or, in the finite element terminology, nodal basis functions) of levels _< j such that a canonical HS(lRd)-elliptic Galerkin discretization can be solved efficiently, e.g., by preconditioned iterative methods, with the preconditioner inherited from the generating system, or frame, consisting of scaling functions (see Section 4). Thus, we essentially stay with splittings of the type (1.2).
6
P. Oswald
The connection of the auxiliary problems in ~. with the originally given HS(f~)-elliptic problem, with natural boundary conditions, on a generic bounded domain f~ C lRd is established in Section 3 where sufficiently rich subspaces ~ , n C V)In will be constructed. The construction consists of a local boundary modification which is similar to constructions outlined in other papers on wavelets on intervals and domains, too, but is relatively simple. A drawback is that, in contrast to { ~ } , the sequence V),n is not monotone, which requires additional considerations when adaptivity is an issue. Information between V),~ and k~ is exchanged by appropriate restriction (R j) and extension (E j) operators. To obtain uniform condition number and work estimates, certain geometric conditions on f~ arise in a natural way. Asymptotically, they hold for domains with boundaries satisfying a uniform Lipschitz condition. For details, see Sections 3 and 4. In Section 5 several extensions will be considered in less depth. We have decided to detail the exposition in Sections 3 and 4 to tensor-product spline spaces t~. In Subsection 5.1, we discuss the conditions on generating functions r r and the necessary modifications such that the conclusions of the theory of the previous sections still hold. In Subsection 5.2 modifications for problems with essential boundary conditions will be considered. This case is somewhat more difficult to handle (compare [44]), and we do not have a satisfactory proposal for d _> 3 at present. For d - 2, the basic idea is to enforce additional refinement near the boundary which leads to a modified construction of ~ , n (resp.
yj).
Adaptivity by nested refinement (or, in our terminology, nested basis function selection) is dealt with in Subsection 5.3. We share the more naive viewpoint of most of the adaptive finite element codes, and use local a posteriori estimators based on a sort of local higher regularity or superconvergence assumption (which is hard to justify theoretically but leads to reasonable results in practical computations). The author admits that there is a lot of closely related work, and that some of the ideas are straightforward and have appeared, in one or another form, in other papers, too. For example, the construction of Cohen, Dahmen, DeVore [18] of a biorthogonal wavelet system for Sobolev spaces, though complicated in technical details, looks theoretically much more powerful and is based on a clever boundary modification as well. There is a lot of activity on solving large linear systems arising from finite element discretizations on engineering (so-called unstructured) grids or of obstacle problems, where an embedding into a regular structure has been one of the options. See the recent papers [54, 55, 66, 5, 42] as well as [43, 35, 48]. Some of these investigations originate from the domain embedding or fictitious domain methods for finite difference discretizations, where boundary modifications, extension, and restriction operators have
Multilevel Solvers for Elliptic Problems
7
been used for a long time, also in connection with multigrid techniques. We refer to [40, 52, 36, 8, 54]. Finally, we wish to mention one more time that this is a paper on geometric multilevel methods; i.e., the approximating subspaces as well as their multilevel splittings are constructed for a generic linear, symmetric, uniformly elliptic HS-problem (s > 0). No anisotropies, behavior of coefficient functions, physical background, or other specifics of the boundary value problem have been used. It is tempting but rather difficult to further extend this work to a more operator-adapted setting (see, e.g., [46, 22, 23]), and to attack the robustness aspect at large. w
Stable subspace splittings and iterative methods
This is a short introduction to a class of approximation and solution methods for operator equations in Hilbert spaces which is based on the concepts of multilevel scales and stable subspace splittings. The solvers for the approximate problems fall into the class of iterative subspace correction methods. Particular examples are domain decomposition and multigrid algorithms, orthogonal, Riesz basis, and frame decomposition and reconstruction techniques, and others. The framework typically incorporates adaptivity (with respect to individual features of the solution of the operator equation) in a natural way. We refer to the books and surveys by Bramble [11], Chan and Matthew [15], Dahmen [25], Hackbusch [41], Oswald [60], Xu [65], and Yserentant [67], where different aspects and the history of the subject have been illuminated. We start with the notion of stable subspace splittings for Hilbert spaces. Assume that V is a (finite-dimensional or separable) Hilbert space, with scalar product (-, ")v and norm I1" IIv. Let {Vj } be an at most countable collection of closed subspaces of V such that V - E j Tv~, in the sense that for each u E V there is at least one V-converging representation
J Let us further assume that the symmetric bilinear forms a(., .) (resp. bj(., .)) are scalar products on V (resp. t~ for all j), such that the spaces are Hilbert when equipped with these alternative scalar products. We use the notation {V; a} (resp. {1~; bj}) to indicate this assumption. In particular,
cllull,5 _< Ilulla < Cilull,5
VuEV,
(2.1)
where I[ulla = v/a(u, u) denotes the energy norm in {V; a}. The two positive constants 0 < c,C < ~ in (2.1), are sometimes called ellipticity constants of a(.,.) with respect to V, and do not depend on u E V. In
8
P. Oswald
the following, we use the symbol ~ for two-sided inequalities such as (2.1), while A _ B (resp. A _ B) stands for a one-sided inequality A < C . B (resp. A >_ c. B). Thus, A ~. B is the same as A ~ B ~ A. The constants 0 < c, C < oe, are assumed to be generic, if not stated otherwise; they do not depend on the arguments of the expressions A and B. The subspace splitting {V; a} - E { V j ; bj}
(2.2)
J
is called stable if IIlulll~bj -
inf E bj(uj uj) x a(u, u) ,,jevj . , , = ~ j ,,j j
(2.3)
One of the simple consequences of this definition is the fact that the operator equation 7:'u-r
P - ERj~
'
r162
J
(2.4)
J
where Rj 9 ~ ~ V denotes the natural injection, and :I) 9 V ----, I~ t5
9 bj(Tju, vj) - a(u, vj) 9
V Yj E
(vj)
t5
(2.5)
for all j, is an equivalent formulation of the variational problem of determining u E V such that a(u, v) - r
Vv E V
(0 E V*) .
(2.6)
The problem (2.4) is called the additive Schwarz formulation of the variational problem (2.6) associated with the subspace splitting (2.2). Moreover, the additive Schwarz operator P acting in V is symmetric, positive definite with respect to a(., .), with spectral condition number defined by
~('P)- )~max(~O) )~min(~) '
(2.7)
where )~max('P) - -
sup
a(Pu, u) -
inf
a(Pu, u) -
uEV "a(u,u)=l
sup
O#uEV
a(u, u)
[]]U[l[~bi}
and ~min(~O)
--
u6_V "a(u,u)=l
inf
O~uEV
a(u, u)
[[[U[[[~bj}
Multilevel Solvers for Elliptic Problems
9
As a consequence, the operator equation (2.4) is well conditioned if the norm equivalence (2.3) holds with tight constants. This is the key to preconditioning using subspace splittings. Before we discuss the typical algorithms associated with finite-dimensional versions of (2.2) we wish to add a few comments. First of all, the inclusion of infinite-dimensional spaces, and countable splittings is very appropriate if we come to our multiscale applications. As a rule, if Cx9
j=O
is stable then the finite splittings J
{Vj; a} - ~--~{Vj ; bj}
j=o
are uniformly stable for J ~ cr and vice versa. This is a desirable feature for algorithmical reasons" the convergence rates of the solvers do not degenerate if better resolution is needed. In addition, this "asymptotical" viewpoint links us with the rich mathematical theory of Fourier analysis and function space decompositions. For instance, any complete orthogonal system but also any Riesz basis or frame in V leads to stable subspace splittings. On the other hand, the practical performance of the approach depends on a "nonasymptotical" range of small to moderate J, and is heavily influenced by the constants in (2.3). Thus, constructions which do not lead to stable splittings in the infinite-dimensional setting may well be successful in some applications. The hierarchical basis method of Yserentant designed for HI-elliptic problems on two-dimensional domains may serve as an illustration. However, especially for methods working in an adaptive refinement environment where larger J are much more likely, the stability of the multiscale splitting in V becomes crucial. A second remark is on the typical method of proof of the stability condition (2.3) which has been put into its present abstract form by J. Xu in his thesis (see [65]). The lower bound, i.e., I]lu]]]{bj} ~ a(u, u), is usually formulated as a separate condition; proofs combine (depending on the context) methods of approximation theory and elliptic regularity arguments. The upper bound is often replaced by assuming so-called slrengthened Cauchy-Schwarz inequalities controlling the interaction between the different subspaces in the splitting. One formulation is to require
a(uj, uk) 2 < 7y,kbj(uj, uj)bk(uk, uk)
V uj E t~, uk E Vk .
(2.8)
Then the finiteness of the largest eigenvalue )~m~x(F) of the matrix F = ((Tj,k)) is a sufficient condition for the upper estimate. It is convenient to
P. Oswald
10 assume that
7j,j = 1
(2.9)
in (2.8) for j - k, which amounts to an appropriate scaling. The inequalities (2.8) and the condition on F are also important for the more sophisticated multiplicative algorithms for which stability alone will not give the desired optimal result; see below. With slight modifications, this concept is also surveyed in [67, 38]. A refined theory serving the needs of multigrid applications, and including a discussion of all kinds of perturbations typical for practical implementations, is given in [11]. See also [41, Sections 10-11] for an introduction and the link to multigrid. Finally, the approach of [60, 25] emphasizes the close connection of the stability assumption for multiscale splittings with results on scales of approximation spaces. For a survey on applications to domain decomposition methods (not necessarily of multilevel type), see [15]. Thirdly, it is possible to further generalize the concept by removing the assumption V~ C V. This is important for several reasons. In the multilevel context, we have in mind situations in which the monotonicity condition Vj C Vj+I is violated. The formal cure is the introduction of a suitable set of mappings Rj : t~ ---+V (replacing the natural injections) such that R - E j ~ j 9~)vj ~ y is onto, and ]][ull] 2
-
inf
{bj,Rj} -- ujEVj " u = E j R j u j
y~. bj(uj, uj) ~ a(u, u) j
Vu e V
The latter condition replaces (2.3), and guarantees that V' - E j RjTj' preserves the above-mentioned properties of P. Condition (2.5) is replaced by
bj(Tj'u, vj)
-
-
a(u, njvj)
V vj E t~ ,
where Tj' : V ---, }). All these modifications can be subsumed in a simple Hilbert space lemma, called fictitious space lemma, which was first used in connection with fictitious domain and domain decomposition methods by Nepomnyaschikh [54]; see [60, Theorem 17] or [66]. As a last remark we note that the abstract formulation of the stability concept in the form of the two-sided inequality (2.3) allows us to better understand simple transformations of one splitting into another, which are useful especially for practical purposes and add flexibility. In [60, Section 4.1] and [38], refinement, clustering, and selection have been discussed. Just to give an example, selection is typical for adaptivity applications and can be characterized as follows" For each j we select a subspace Vj* C (both extremes Vj* - ~ and Vj" - {0} are allowed!), and form the new, selected space V* - ~ j Vj*. Due to the definition of the triple-bar norm,
Multilevel Solvers for Elliptic Problems
11
establishing the stability of the new splitting {V*; a } - ~-'~.{Vj*;bj} , J
requires only the verification of the lower estimate, while the upper estimate is preserved from (2.3), with the same or a better constant. A further simplification comes for the case of splittings into direct sums of subspaces where the infimum in the definition of the triple-bar norm can be removed: in this case any selection leads again to a stable splitting, with the same or a better condition. Splittings into one-dimensional subspaces are of particular interest. Let Vj be generated by the (nontrivial) element fj E V. It turns out that then the stability condition (2.3) is equivalent to the frame property of the normalized system 1
v/bj(fi, fi) fj in {V; a} (or, equivalently, in V equipped with the original scalar product). Recall that a system {gj } in a Hilbert space V is called frame if
J
(see [30, Section 3.2] for the definition and properties of frames). Indeed, one easily computes that
a(u, fj) ~ u - ~_ _ ~j-(]j : -f; ) f j ,
(2.10)
which together with the stability of (2.2) implies
a(u, u) "~ a(7)u, u) -- ~ la(u, ]j)l 2 . J Alternatively, look at [30, Proposition 3.2.4] and compare with (2.3). Riesz bases, which are, by definition, minimal frames, are particularly attractive since they lead to stable direct sum splittings of V. Another reason for the interest in frames or Riesz bases is that, based on their stability property, one can derive many other computationally interesting stable splittings. Examples are provided in [60, Section 4] and [58] for finite element applications. The explicit formulas for 7) (2.10) resp. for the subspace mappings 7) and the ej associated with the given functional (I) in the right-hand side of
Tj " u C V,
, Tju-
a(u, fj) vJtJJ,JJC-/-7:)-)fJ e Vj
O(fj) ( r = bj77-7 )
)
(2.11)
12
P. Oswald
are useful to derive matrix representations suitable for the implementation of the algorithms explained next. Let us now briefly outline the basic algorithms associated with a subspace splitting (2.2). For this purpose we make the natural assumption that all spaces involved are finite-dimensional and that their number is finite: ./
(2.12)
(V;a}j=O
Note that stability itself is obviously guaranteed; the question is the size of ~(P). The iteration step of the additive algorithm A is given by J
un+l -- un -~- ~ E
rj(un) ~
= Rj(r
(2.13)
- Tju).
j=0
The same amount of work is formally needed to perform one iteration of the multiplicative algorithm M V0
---
It n
vJ+ 1
=
~tn+l
--
Vj .+. w r j _ j ( v j ) , vJ+l
j=0,...,J
.
(2.14)
The role of the relaxation parameter w is analogous to the classical iterative methods (as a matter of fact, (2.13) corresponds to the extrapolated Jacobi (resp. . Richardson) iteration while (2.14) generalizes SOR). Note that these are stationary linear iterative schemes (in the usual terminology; see [41]); the iteration operators for the two algorithms are MA = I d - w P
,
MM = ( I d - w R o T o ) ( I d - w R i T 1 ) . . . ( I d - w T j )
.
The convergence theory (which is trivial for the additive algorithm) is covered in [65, 67, 11, 38, 60]. We quote the following result from [38], or [60, Theorem 18]. T h e o r e m 1. Assume that V is finite-dimensional and that the algorithms A and M are defined with respect to (2.12). (i) The additive algorithm A converges for 0 < w < 2/Amax(P). The optimal convergence rate is achieved for ~* = 2/(Amax(P)-bAmin (P)), and equals PA *
-
2 min I [ M A l l . - 1-. 0 1. For the preconditioner which results from the Riesz basis H0 u
u...u
one would need to put
B/1
-
-
IjBTllf
,
[~j
-
-
diag{a(r
~)j,i)}i=l,...,mj
, j > 1,
in the recursion (2.20). Here, the nj • mj matrix Ij describes the natural embedding Wj ~ t~, and contains as entries the mask coefficients in the expressions nj
Cj,i' -- ~ aj,iCj,i 9 i=1 This, and the precomputation of the diagonal matrix/~j-1, are the places where the choice of the Riesz basis adds to the arithmetical complexity of the preconditioning operation. Other advantages (e.g., better stability estimates or robustness properties) should compensate for this drawback. Some examples of "cheap" finite element Riesz bases are available, see [50] for a survey. w 3.1
S u b s p a c e s for b o u n d e d d o m a i n s
C o n s t r u c t i o n of Vj,~
Throughout the paper, we use the following notation. Let ~ E IRd be a bounded open d-dimensional domain, and 0fl its boundary. We assume that ~ possesses the extension property for the scale of Sobolev spaces H s (for it to hold, the uniform cone condition would be sufficient, see [64, 1]). Let the Euclidean space I~ d be partitioned into cubes of sidelength 1 such that the origin is a vertex of one of the cubes. The collection of all these socalled 0-cubes will be denoted by TO0 (partition of level 0). The partitions Tdj of level j >_ 1 into j-cubes of sidelength 2 -j will be obtained from TO0 by dyadic dilation. Let
t~ -
S~ (Tij ) N L2(IR d)
(3.1.1)
be the L2-subspaces of tensor-product splines of degree k and smoothness r with respect to 7~j where 0 < r < k - 1. Obviously, { ~ } is an increasing
17
Multilevel Solvers for Elliptic Problems
sequence of subspaces of the Sobolev spaces Hs(]Rd), 0 < s < r + 3/2. Alternatively, the V) could be defined by dyadic dilation from V0" Vj = {u(2J.) 9 u E V0}. Note that ~ locally contains all algebraic polynomials of degree < k. We fix the local and L2-stable basis of tensor-product B-splines {r in ~ , see [63]. This basis has the remarkable property of local linear independence" If uj E V~ vanishes on a j-cube [] then cj,i - 0 for coefficients in the B-spline representation corresponding to all basis functions Cj,i which do not vanish identically on []. For our convenience, we introduce the notation wQ for the set of indices i such that Cj,i does not vanish on the j-cube []. Thus, local linear independence is equivalent to cj,iCj,i(x) - O, x E []
==~
cj,i - 0, i E w 9 9
(3.1.2)
i
A consequence of the local linear independence property is the existence of well-localized biorthogonal functions" For any j-cube [] C supp Cj,i (or, in other words, for any i E wo) there is a function r/j,i E L ~ (IRd) supported on [] such that V i, i' .
o r]j,iCj,i, dx -
(3.1.3)
As is obvious from the translation-dilation invariance of all constructions, the rlj,i can be obtained as scaled translates of dilates of a finite number of functions associated with the unit cube u0 - [0, 1]d. The j-cube associated with rlj,i will be denoted by 9 It will be fixed depending on the specific setting. If no explicit choice is made, then any j-cube in supp Cj,i will serve. We introduce some modified basis functions which will be used for the boundary modification below. Consider the finite-dimensional space X0 - V0iDo (which in this specific case coincides with all polynomials of coordinate-wise degree < k). It contains all monomials x ~, ic~l < k, the set of which can be complemented by some other functions to yield a basis in X0. Let {r (i E WOo) denote this basis, and {r/ 9149the corresponding biorthogonal system in X0, i.e., /o
dx - hi,i, ,
r
i,i' E WOo 9
(3.1.4)
o
The same notation r will be used for the extensions to S[: (JRd) obtained as follows: For the monomials x a, lal < k, there is a unique representation x ~ -
el i
x
18
P. Oswald
while for the complementing basis functions the minimal extension is used, i.e., the B-spline coefficients of the extension vanish for r with i ~ Who (coefficients with i E WOo are uniquely determined by the spline values on Do as follows from (3.1.2)). This construction is illustrated for the bilinear case ( d - 2, k - 1, r - 0) in Figure la-d. The upper row shows the nodal values at the integer points near Do of the extended Cno,i corresponding to the monomials 1, Xl, x2, and one arbitrarily fixed complementing basis function, respectively, while the second row depicts the nodal values of the bilinear functions on Do defining r/no,i. By translation and dyadic dilation we obtain systems {r and {r/n,~} (i E wn) for any j-cube D and all j > 0. To be definite, and in order to preserve the biorthogonality relation (3.1.4), we apply scaling (by a factor 2jd) only to the y-functions. In the final construction, suitable restrictions of the extended functions Cn,i will be used, see below. .0 .
.
.
1
1
1
-I
0
1
1
1
I
0
0
1
1
1
I
-1
0
I
0
0
0
0
1
0
0
I
1
1
-I
0
I
-1
-1
-1
0
0
0
0
.
0
0 0
"0
1
i 0
0
_0
o
I0
-14
-6
6
6
-18
-2
I0
-6
6
-6
18
a)
b)
c)
d)
181 -18
-8
4
16
-8 e)
Figure 1. Bilinear elements: Nodal values for r and y-functions. From now on, we assume that generic constants C, c , . . . (also those occurring in ~ and ~ relations) may depend on k, r, s, and ~ but are independent of other parameters, especially of j, i, l, and the functions involved. For each j > 0, we define the sets F~j C ~ C ~ as unions of j-cubes: ~j-U{DET~j
9[ ] C ~ } ,
~--U{DET~j
9DNgt#O}.
(3.1.5)
We require the following geometric property of ~" (G1) For each j-cube D E fl~ and any l - 0, 1 , . . . , j, there is at least one /-cube [:1~ C ~j at a distance < C2 -z from D. The constant C is assumed to be independent of D, j > 0, and 1. Roughly speaking, this condition means that the domain has a sufficiently "fat" interior and a regular boundary. In particular, (G1) implies F~0 ~ 0. Though restrictive, (G1) seems to be rather natural if robustness of a
Multilevel Solvers for Elliptic Problems
19
geometric multilevel method is expected. Asymptotically (i.e., if required only for j0 _ 1 _< j and some sufficiently large j0), the above condition is satisfied for domains with a Lipschitz boundary (resp. the uniform cone condition). By wj (resp. cOwj) we denote the sets of all indices i such that the support of the tensor-product B-spline ej,i intersects ~j (resp. intersects f~ but not ~j). These are the sets of interior and boundary indices of level j. Let Cj be a family of j-cubes rn C ~j near the boundary of f~ satisfying the following properties: 9 No function
ej,i
contains two different cubes from Cj in its support.
There is a constant C such that for each i E cOwj there exists a cube []i E Cj at a distance _~ C2 -j from the support of ej,i. The second property implies the existence of a partition of the set of boundary indices cOwj into small sets cOwo, [] E Cj, such that for i E cOwo the distance condition is satisfied with this particular [] (we will assume that 0wo is nonempty, otherwise the corresponding cube [] can be excluded from Cj). The existence of the families Cj easily follows from (G1) for l - j; the constant C depends on the constant in (G1) and k, r. Analogously, the set of interior indices wj decomposes into the pairwise disjoint sets w 9 (• C Cj), and a possibly larger "remainder" set ~j - ~j \ uoecj ~o .
Figure 2 schematically illustrates the definitions of ~j, ~$, and Cj for the bilinear case. The j-cubes in Cj are given by hatching. The numbers at nodal points associated with w 9U cgwo indicate the number of the corresponding [] C Cj, and the unnumbered points correspond to indices from wj. Since for i C wj, the support of ej,i contains at least one j-cube [] C ~j (which is not in Cj !) we can fix ~j,i such that its support clj,i is in supp ej,i n ~j. Figure le shows the nodal values of the biorthogonal function for j - 0. The general case follows by scaling with a factor 2jd. We come to the description of the boundary modification. Roughly speaking, only basis functions ej,i with i E w 9 for j-cubes from Cj will be changed. For notational convenience, define the restriction operation M~ associated to an arbitrarily given set w of indices of level j by
uj - ~
cj,iej,i E S~(Tij), i
~ M~ouj - ~
cj,iej,i e S~(Tij) .
(3.1.6)
iEw
If w is finite, the mapping is clearly into I~. The restriction of M~ onto Vj is an L2-bounded operator, due to the L2-stability of the B-spline basis. For each [] E Cj, we replace the associated set {r 9 i E r by a set
P. Oswald
20
10
10
10 ~_ _
I
2
10
9
10 1
1
2
3 2
3 3
8
L
7
8
~
~
8
8
3
7
7
~
~
7
6
6
~
5
6
f
7
4
, , 4N4 N / N /
6 7
3
3
9
N9 9 N N
8
2
6
5
~ ~ "
-
55
6
6
4
5
Figure 2. Boundary cubes and nodes for bilinear elements. of "new" basis functions {r 3,$ 9 i E wo} which coincides up to index ordering with {M~ou0~0or (i.e., with the restrictions of the dilated and translated extensions r defined above to some neighborhood of n, compare Figures 1 and 2). The biorthogonal functions r/~,i are identified with rio,i, accordingly. Thus, our construction ensures that
IICj,~IIL~(R~) "< 1, II~j,illL~(Oj,,) ~ IIr
___ 1,
2jd ,
II~/~,illL~(O) ~ 2jd ,
!
i E wj,
(3.1.7)
i E wo,
hold for all o E Cj, and j > 0. In addition, the biorthogonality conditions are preserved. The main advantage of our construction is the relatively simple, cube-oriented local basis exchange which still guarantees local reproduction of polynomials of (total) degree < k in the spaces Vj,a to be defined next. Let (3.1.8) ~ , a - span{r " i E wj } , where
r
I Cj,ila r
if
if
iEwj
!
i Ewo,
oer
21
Multilevel Solvers for Elliptic Problems
The space Vj,a is a subspace of Vj la, the restriction of ~ to f~ (as a rule, the inclusion is proper). More precisely, functions from ~ , a are uniquely determined by their values on fly, and extended to f~ in a specific way. This can be seen from the definition of the biorthogonal system by r/j,i,~which
ensures
I
rlj,i r/~,i
if if
!
i E wj , iEwm, []ECj,
Dj, i -- supp rlj,i C f2j to hold, and
u - Qju - E
Aj,i(u)r
f Aj,i(u) - 1=
iEwj
~Tj,i,audx,
(3.1.9)
doj,,
for all u E Vj,a since o
rlj,i,flCj,i,,a dx - hi,i, ,
i, i' E wj .
(3.1.10)
j,i
The spaces Vj,~ still have the same approximation power (with respect to the Sobolev scale) as V)I~ since the boundary modification allows for local reproduction of polynomials of total degree < k. Indeed, the quasiinterpolant operator (3.1.9) is well defined for functions u E L I ( ~ j ) and maps into ~ , a . By construction, Qj is a projector onto V),a. Also, if u coincides on fly with a spline function from V) or from S~(Tij) then its values are preserved on ~j. By a standard argument, we can prove
[IQjulIL,(~)
d
II~IIL~(~j),
1 _ p _< ~ .
(3.1.11)
To this end, observe that on each ra N f~ at most a fixed number of terms in (3.1.9) does not vanish. This gives
Ilqj l'IL~(Ona)
I
-
i E w j 9O n s u p p C j , , , n # 0
for all j-cubes D. The generic term in this sum is the Lp-norm of a function of the form fo, r/u dx. r where r 7/satisfy (3.1.7), and D' C f~j is a j-cube at distance < c2 - j from D. This implies
II/ , and
IIQjull (on )-
_ 0). A more general example for the bilinear case and j - 3 is given in Figure 4a while 4b shows a domain with a slit where condition (G2) would be violated if the construction were continued for j ~ exp.
30
P. Oswald
II II
,
I
I
E I
I
I
ii I
I
iii
ii'i'
m
a)
b) Figure 4. Illustrations for (G2).
Now we define appropriate/-cubes t3z,i for all possible 1 and i which enter the construction of the quasi-interpolant operators {Pz}, as explained in Subsection 3.2 (see (3.2.10), (3.2.9)). If/>__ j, the only condition is [::ll,i E supp Cz,i 9
(4.1.3)
If I < j we have three cases9 if supp Cz,i C (2z+1 then only (4.1.3) has to be observed, 9 for i with supp Cz,i f)f~* r 0 there is at least one l-cube (denoted by t::lz,i) in the intersection of supp Cz,i and f~*, 9 finally, for the remaining i ~ wz we fix as [::iz,i an/-cube outside f~0 which satisfies (4.1.3) or, if this is impossible, is closest to supp Cz,i (according to the first two cases and (G2), this distance cannot exceed
c2-z).
By Lemma 2, for 0 < s < r + 3/2, the second norm equivalence in (3.2.7) holds. In particular, it applies to all fij E ~ C ~. By construction, we have Pzuj - uj for all l >_j and uj E ~ . Hence, J
" llP0 - -
l
IL~(R') + ~ 22t'll(Pz - P'-~)aJ 121L,(rt') /=1
9
(4.1.4)
31
M u l t i l e v e l Solvers f o r Elliptic P r o b l e m s
Now we observe that according to our specific choices of the cubes rnL,i, for l < j the functionals At,i(fij) given by (3.2.9) vanish if the index i corresponds to the last case. Moreover, on the set f2~* we have Pzfij - fij. Indeed, let us fix I. By definition of the index sets ahz,, each fij E Vj can be split into two parts J uj = i "supp Cz,iClrlo
11=I+1 iEt~l
9
,j
~,
r
where Wl+l has support in QI+I- For i corresponding to the second case (where Dz,i C supp Cz,i N Q~* -f~0\f~l+l), we get by examining the values of the linear functionals Al,i(') Az,i(vz) - ( cz,i 0
if supp Ct,i C f20 otherwise,
)~l,i(Wl+l) -- 0
Since there is no contribution from i corresponding to the third case, we have UJ -- Pl~tJ =
E
Cl,ir
"Jr"Wl+l 9
i "supp r
This shows the coincidence of Plfij and fij on f~}**. In turn, this implies that the difference Pl?-lj -- Vl- l Uj
--
( U j -- P l - 1 u j ) -
:
( W l - W l + l ) -t-
(ttj -
Pl ttj )
E
Cl-l,ir
i 9supp Cz- t, i C ~z
E
Cl,ir
i "supp r
i ECoz
belongs to the subspace -span{r
" i E Cot} .
(4.1.5)
The latter produce, according to (3.2.1), a subspace splitting of I)j, for which we have just proved the following analog of Lemma 2.
32
P. Oswald ..,,,
Lemma 4. Assume that the construction of {az}, {&t} (see (3.2.3)) from the sequence f~ satisfies the above conditions, especially (G2). Let the spaces f/j and Q be defined by (3.2.1) and (4.1.5), respectively. Then, for 0 < s < r + 3/2, we have the norm equivalence J
II~Jll~.(~,,) ~ (111~.~111;)~ -
2 Z 2~J'll'i'llz,~(R") (4.1.6)
inf
'-';'~, '~,=~-o ~, z=o
for all fij E Vj. The constants in (4.1.6) depend on the constants in the norm equivalencies of Lemma 2 and in (G2).
One direction of the two-sided estimate follows from
II~jlI~.(R,) _-_ IIRTAjxjll 2
by using the surjectivity of Rj). By the above spectral equivalence of Cj and .~-1, we have 2
((Aj RjCj RTAjxj, xj)) - ((Cjflj, flj)) ~_. ((~j-l~)j, ~)j))_ ( ( A j X x j , x j ) ) ,
(4.2.3)
where ~lj - R TAjxj. Since Rj is onto, for each xj there is at least one s such that xj = Rj ~j. Thus,
( ( A j X - l xj, xj)) - ((7tjflj, ~j)) 0. Its value may significantly influence the quality of error estimation. Note that the above splitting of the global error estimate into local components does not necessarily reflect the true local error of fly. However, the use of such local error indicators to monitor the refinement process is a well-established numerical practice. In a second step, those cubes Q are refined for which e 9 exhibits a large error contribution. Error equidistribution is a common approach, but many other issues might also influence the decision to refine a given cube. After this, one usually cares about keeping the refinement structure sufficiently regular. In our framework, this could lead to additional refinement in order to stay close to the geometric assumptions (G1)', (G2) (which is needed to guarantee an efficient multilevel scheme in the next solution step),^ and (G3)'. This enables us to set up the new cube structures {~j} (resp. {fl~}), and so on.
Multilevel Solvers for Elliptic Problems
53
Acknowledgments. The author thanks his colleagues at the Institute for Algorithms and Scientific Computation at GMD, Sankt Augustin, Germany, and the Department of Mathematics, Texas A&M University, College Station, USA, where large parts of this paper have been prepared, for their support. References [1] Adams, R. A., Sobolev Spaces, Academic Press, Boston, MA, 1978. [2] Auscher, P., Wavelets with boundary conditions on the interval, in: [16], pp. 217-236. [3] Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. User's Guide 6.0, Frontiers Appl. Math., vol. 7, SIAM, Philadelphia, PA, 1990. [4] Bank, R. E. and R. K. Smith, A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30 (1993), 921-935. [5] Bank, R. E. and J. Xu, An algorithm for coarsening unstructured meshes, Numer. Math. 73 (1996), 1-36. [6] Beck, R., B. Erdmann and R. Roitzsch, KASKADE User's Guide, Report TR-95-11, ZIB, Berlin, 1995. [7] Bespalov, A. N. and Yu. A. Kuznetsov, RF cavity computations using the domain decomposition and fictitious component methods, Russian J. Numer. Anal. Math. Modelling 2 (1987), 259-278. [8] Bespalov, A. N., Yu. A. Kuznetsov, O. Pironneau, and M.-G. Vallet, Fictitious domains with separable preconditioners versus unstructured adapted meshes, Impact Comput. Sci. Engrg. 4 (1992), 217249. [9] de Boor, C., K. Hhllig, and S. Riemenschneider, Box Splines, Springer, New York, 1993. [10] Bornemann, F. and H. Yserentant, A basic norm equivalence for the theory of multilevel methods, Numer. Math. 64 (1993), 455-476. [11] Bramble, J. H., Multigrid Methods, Pitman Res. Notes Math. Ser., vol. 294, Longman Sci.& Tech., Harlow, 1993. [12] Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Cutup. 55 (1990), 1-22.
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P. Oswald
[13] Bramble, 3. H., J. E. Pasciak and X. Zhang, Two-level preconditioners for 2m'th order elliptic finite element problems. Preprint, Texas A&M University, 1996. [14] Brenner, S., Preconditioning complicated fem's by simple fem's, SIAM J. Sci. Comput. (1996), to appear. [15] Chan, T. and T. Mathew, Domain decomposition algorithms, in Acta Numerica 94, Cambridge Univ. Press, New York, 1994, pp. 61-143. [16] Chui, C. K. (ed.), Wavelets: A Tutorial in Theory and Applications, Academic Press, Boston, 1992. [17] Chui, C. K. and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, vol. 9, D. Braess, L. L. Schumaker (eds.), Birkh~iuser, Basel, 1992, pp. 53-75. [18] Cohen, A., W. Dahmen, and R. A. DeVore, Multiscale decompositions on bounded domains, IGPM-Report Nr. 113, RWTH Aachen, May 1995. [19] Cohen, A., I. Daubechies, B. Jawerth, and P. Vial, Multiresolution analysis, wavelets and fast algorithms on the interval, C. R. Acad. Sci. Paris Ser. I Math. 316 (1993), 417-421. [20] Dahlke, S., W. Dahmen, and R. A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, this volume.
[21] Dahlke, S., W. Dahmen, R. Hochmuth and R. Schneider, Stable multiscale bases and local error estimation for elliptic equations, IGPMReport Nr. 124, RWTH Aachen, January 1996. [22] Dahlke, S. and I. Weinreich, Wavelet-Galerkin-methods: An adapted biorthogonal wavelet basis, Constr. Approx. 9 (1993), 237-262. [23] Dahlke, S. and I. Weinreich, Wavelet bases adapted to pseudodifferential operators, Appl. Comput. Harmon. Anal. 1 (1994), 267283. [24] Dahmen, W., Stability of multiscale transformations, J. Fourier Anal. Appl. 2 (1996), 341-361. [25] Dahmen, W., Multiscale analysis, approximation, and interpolation spaces, in Approximation Theory VIII, vol. 2, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 1995, pp. 47-88.
Multilevel Solvers for Elliptic Problems
55
[26] Dahmen, W. and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315-344. [27] Dahmen, W., A. Kunoth, and K. Urban, A wavelet Galerkin method for the Stokes equations, Computing 56 (1996), 259-301. [28] Dahmen, W. and C. Micchelli, Biorthogonal wavelet expansions, IGPM-Report Nr. 114 , RWTH Aachen, May 1995. [29] Dahmen, W., S. PrSssdorf, and R. Schneider, Multiscale methods for pseudodifferential equations, in Recent Advances in Wavelet Analys/s, L. L. Schumaker and G. Webb (eds.), Academic Press, New York, 1994, pp. 191-235. [30] Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Conf. Ser. in Appl. Math., # 61, SIAM, Philadelphia, 1992. [31] Daubechies, I., Using Fredholm determinants to estimate the smoothness of refinable functions, in Approximation Theory VIII, vol. 2, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 1995, pp. 89-112. [32] DeVore, R. A., B. Jawerth, and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), 737-785. [33] DeVore, R. A. and B. J. Lucier, Wavelets, Acta Numerica 92, Cambridge Univ. Press, New York, 1992, 1-56. [34] DeVore, R. A. and V. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988), 297-314. [35] Erdmann, B., R. H. W. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems in three space dimensions, Preprint SC 93-8, ZIB, Berlin, 1993. [36] Finogenov, S. A. and Yu. A. Kuznetsov, Two-stage fictitious components method for solving the Dirichlet boundary value problem, Russian J. Numer. Anal. Math. Modelling 3 (1988), 301-323. [37] Griebel, M., Multilevelmethoden als Iterationsverfahren fiber Erzeugendensystemen, Teubner Skr. Numer., Teubner, Stuttgart, 1994. [38] Griebel, M. and P. Oswald, Remarks on the abstract theory of additive and multiplicative Schwarz methods, Numer. Math. 70 (1995), 163-180.
56
P. Oswald
[39] Griebel, M. and P. Oswald, Tensor-product-type subspace splittings and multilevel iterative methods for anisotropic problems, Adv. Cornput. Math. 4 (1995), 171-206. [40] Hackbusch, W., On the multi-grid method applied to difference equations, Computing 20 (1978), 291-306. [41] Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations, Springer, New York, 1994. [42] Hackbusch, W. and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated microstructures, Bericht Nr. 9505, University Kiel, Germany, 1995. [43] Hoppe, R. H. W. and R. Kornhuber, Adaptive multi-level methods for obstacle problems, SIAM J. Numer. Anal. 31 (1994), 301-323. [44] Jaffard, S., Wavelet methods for fast resolution of elliptic equations, SIAM J. Numer. Anal. 29 (1992), 965-986. [45] Jaffard, S. and P. Laurencot, Orthogonal wavelets, analysis of operators, and applications to numerical analysis, in [16], pp. 543-601. [46] J awerth, B. and W. Sweldens, Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations, In: Proc. Sixth Copper Mountain Conf. on Multigrid Methods, A. D. Melson, T. A. Manteuffel, and S. F. McCormick (eds.), NASA Conf. Publ. 3224, 1993, pp. 259-273. [47] Jia, R.-Q. and C. Micchelli, Using the refinement equation for the construction of prewavelets II: Powers of two, in Curves and Surfaces,J. P. Laurent, A. LeMehaute, L. L. Schumaker (eds.), Academic Press, New York, 1991, pp. 209-246. [48] Kornhuber, R. and H. Yserentant, Multilevel methods for elliptic problems on domains not resolved by the coarse grid, in Proc. 7th Conf. Domain Decomposition Meth., D. Keyes, J. Xu (eds.), Cont. Math., vol. 180, AMS, Providence, 1994, pp. 87-98. [49] Ko, J., A. Kurdila, and P. Oswald, Scaling function and wavelet preconditioners for second order elliptic problems, this volume. [50] Lorentz, R. and P. Oswald, Constructing economical Riesz bases for Sobolev spaces, GMD-Bericht Nr. 993, GMD, Sankt Augustin, Germany, May 1996. [51] Massopust, P. R., Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, San Diego, 1994.
Multilevel Solvers for Elliptic Problems
57
[52] Marchuk, G. I., Yu. A. Kuznetsov, and A. M. Matsokin, Fictitious components domain and domain decomposition methods, Russian J. Numer. Anal. Math. Modelling 1 (1986), 3-35. [53] Meyer, Y., Wavelets and Operators, Cambr. Stud. Adv. Math., vol. 37, Cambridge Univ. Press, Cambridge, 1992. [54] Nepomnyaschikh, S. V., Method of splitting into subspaces for solving elliptic boundary value problems in complex-form domains, Russian J. Numer. Anal. Math. Modelling 6 (1991), 151-168. [55] Nepomnyaschikh, S. V., Preconditioning operators on unstructured grids, Technical University Chemnitz, Germany, 1995, preprint. [56] Oswald, P., On the degree of nonlinear spline approximation in BesovSobolev spaces, J. Approx. Theory 61 (1990), 131-157. [57] Oswald, P., On function spaces related to finite element approximation theory. Z. Anal. Anwendungen 9 (1990), 43-64 [58] Oswald, P., Stable subspace splittings for Sobolev spaces and some domain decomposition algorithms, in Proc. 7th Conf. Domain Decomposition Meth., D. Keyes, J. Xu (eds.), Contemp. Math., vol. 180, AMS, Providence, RI, 1994, pp. 87-98.
[59] Oswald, P., On the convergence rate of SOR: a worst case estimate, Computing 52 (1994), 245-255.
[60] Oswald, P., Multilevel Finite Element Approximation. Theory ~ Applications, Teubner Skr. Numer., Teubner, Stuttgart, 1994.
[61] Oswald, P., Preconditioners for nonconforming elements, Math. Comp. 65 (1996), 923-941.
[62] Riide, U., Mathematical and Computational Techniques for Multilevel Adaptive Methods, Frontiers Appl. Math., vol. 13, SIAM, Philadelphia, PA, 1993.
[63] Schumaker, L. L., Spline Functions: Basic Theory, Wiley, New York, 1981. [64] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, Dt. Verlag Wiss., Berlin, 1978; North-Holland, Amsterdam, 1978. [65] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), 581-613.
58
P. Oswald
[66] Xu, J., The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing 56 (1996), 215-235. [67] Yserentant, H., Old and new convergence proofs for multigrid methods, in Acta Numerica 93, Cambridge Univ. Press, New York, 1993, 285-326. [68] Zhang, X., Multilevel additive Schwarz methods, Numer. Math. 63 (1992), 521-539. Peter Oswald
Institute for Algorithms and Scientific Computing GMD- German National Research Center for Information Technology D-53754 Sankt Augustin, Germany
[email protected] W a v e l e t - L i k e M e t h o d s in t h e D e s i g n of Efficient M u l t i l e v e l P r e c o n d i t i o n e r s for E l l i p t i c P D E s
Panayot S. Vassilevski and Junping Wang
A b s t r a c t . This article introduces a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L~-projections onto finite element spaces of coarser levels. It is proved that the corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners for model elliptic problems. In practical computation, one action of the proposed preconditioner is of optimal order algebraically. Numerical results are included to illustrate the efficiency and stability of the new preconditioning methods.
w
Introduction
In this paper we are concerned with the construction of efficient numerical methods for matrix problems arising from finite element methods for elliptic partial differential equations. In practical computations, the standard nodal basis for the finite element space is often chosen as the computational basis and the resulting matrices are ill-conditioned. Our objective is to seek a substitution for the standard nodal basis so that the stiffness matrix arising from the new basis is well conditioned. A computationally feasible basis should possess the following properties: (a) the basis functions must be computable and have local support; (b) the resulting stiffness matrix is sparse and well conditioned. This paper will introduce a wavelet-like method proposed by the authors in [41] which can be employed to construct a new basis with the above mentioned features for the finite element application to elliptic problems. Multiscale Wolfgang Copyright
Wavelet Dahmen,
Methods
for P D E s
Andrew
g. K u r d i l a ,
59 and Peter Oswald
(~)1997 by A c a d e m i c P r e s s , Inc.
All rights of reproduction in any form reserved. ISBN 0-12-200675-5
(eds.), pp. 59-105.
60
P. Vassilevski and J. Wang
The method has a very close relation with the multiresolution (or multiscale) decompositions exploited especially in the wavelet literature (e.g., [17, 14]). In our approach, the requirements on the L2-orthogonality and the existence of a single generating function r (more precisely, r generates an orthogonal basis by dilation and translation as in {r k. - i ) , i E zd}) commonly imposed in the wavelet literature (see Mallat [26] for details) are relaxed in order to have a computationally feasible basis. The basic idea lies in approximating the wavelets without deteriorating their stability, yielding a stable Riesz basis for the finite element space under consideration. Attempts in the search of a stable Riesz basis with some restrictions, either on the mesh or on the analysis, have been made by Griebel and Oswald [20], Kotyczka and Oswald [24], and Stevenson [35, 36]. For a recent comparative study on the construction of economical Riesz bases for Sobolev spaces we refer to Lorentz and Oswald [25]. Our method is general and provides a satisfactory answer for most elliptic equations. The method is based on modifying the existing (unstable) hierarchical basis by using operators which are approximations of the L 2projections onto coarse finite element spaces. A similar approach was used by J affard [22] in seeking a wavelet basis for finite element spaces. In [22], the construction starts with the standard nodal basis functions which are transformed to an orthogonal multiresolution basis based on an explicit orthogonalization procedure exploiting wavelets in each subspace Wj of ~ . Here Wj is the L2-orthogonal complement of the finite element space Vj_ 1 in the next finer finite element space k). The latter procedure generally leads to basis functions that are not locally supported but have good decay rates and hence allow for locally-supported approximations. The method proposed by Vassilevski and Wang [41] can be regarded as an approximation of the wavelet basis in [22]; the approximate-wavelet basis functions are locally-supported and, therefore, computationally feasible. There are two alternatives in the implementation of the approximatewavelet basis in practical computation. In the first approach, one first computes an explicit form for each new basis function and then assembles the global stiffness matrix corresponding to the new basis. The major drawback for this approach is the difficulty encountered while assembling the global stiffness matrix; the robustness corresponding to the standard nodal basis is no longer retained for the new basis. In the second approach, one makes use of the standard nodal basis as the computational one and constructs a preconditioner for the stiffness matrix by using the new basis functions. This is possible because most of the iterative algorithms such as the CG method require only the action of the stiffness matrix on vectors. Thus, the technique of basis changing can be employed to compute the action of the matrix corresponding to the new basis. The second approach can be viewed as a "black-box fix" of the conditioning of the standard nodal
Wavelet-Like Multilevel Preconditioning
61
basis stiffness matrices. Details can be found in Section 8. The organization of the present paper is as follows. Section 2 contains some preliminaries and the problem formulation in an abstract Hilbert space setting. The importance of having a stable Riesz basis and its relation to the conditioning of discrete matrix problems, especially those from the finite element discretization, is the topic of Section 3. Specific examples of second order elliptic PDEs are presented in Section 4. Section 5 discusses the basic idea in constructing multilevel direct space decompositions. The hierarchical basis of Yserentant [44] is reviewed in Section 6. Section 7 contains a detailed discussion of the wavelet-like method in constructing a stable basis. The implementation issues, including two basic preconditioning schemes (namely the additive and multiplicative algorithms) and the construction of the approximate L2-projections, are discussed in Section 8. Finally, some numerical results are presented in Section 9 for convectiondiffusion problems, to confirm the theory developed in previous sections. w
P r e l i m i n a r i e s a n d basic p r o b l e m s
This preliminary section introduces the basic problem we address in this paper. Let H be a Hilbert space equipped with the inner product (., .) such that H C V C H', (2.1) where H is dense in another Hilbert space V with inner product (., ")0, and H ~ is the dual of H with respect to the pairing of V defined by the inner product (., ')0. Assume that the imbedding from H to V is continuous. More precisely, there exists a constant ~0> 0 satisfying
Ilullo
,ilull
v
u E/-jr.
(2.2)
Here I1" II- v/U, )indicates the norm in the Hilbert space H and i1" II0 = x / ( , .)0 that of V. Let a(., .) be a bounded bilinear form defined on H x H satisfying the following inf-sup condition of Ladyzhenskaya, Babugka, and Brezzi:
sup
Ilwll
>/31lv[i -
V v E H,
(2.3)
where/3 > 0 is a fixed constant. We are interested in approximate solutions of the following problem: Given f E H ~, find u E H satisfying
a(u, r
f(r
V r E H.
(2.4)
62
P. Vassilevski and J. Wang
It is well known that if a(., .) is symmetric and H-coercive (i.e., there exists a positive c~ satisfying a(v,v) >_ c~[[vl]2 for all v E H), then the problem (2.4) is equivalent to the minimization problem for the quadratic functional if(v) - la(v, v) - f(v). To approximate (2.4), let {gk}~=l be a sequence of finite dimensional subspaces of H satisfying the following approximation property: For any v E H one has lim
(2.5)
inf I I v - r
k---* oa CE Hk
The well-known Galerkin method for (2.4) seeks uk E Hk such that
a(uk, Ok) -- f(Ok) V Ok 6 Hk. (2.6) In order to have a well-posed discrete problem (2.6), we assume the following discrete inf-sup condition" There exists a constant /3k > 0 such that sup a(v, w) > ~k[iv[ [
V v E Hk.
(2.7)
It is known that if (2.7) holds true, then the Galerkin problem (2.6) has a
unique solution in the subspace Hk. Interested readers are referred to [18] and [11] for more details. In practical computation, we usually formulate a matrix problem for (2.6) by choosing a suitable basis for the subspace Hk. More precisely, let {r . i -- 1 , 2 , . . . , n k } be a computational basis of Hk. Expand the approximate solution u(k) in terms of this basis, yielding rtk
F_,
(2.s)
i--1
Let u (k) - (Cl, c 2 , . . . , cnk) T be the coordinates of u (k). It is not hard to see that the vector u (k) is given as the solution of the following linear system" A(k)u(k) _ f(k),
(2.9)
where the right-hand side vector f(k) is defined as follows: f(k) _ (bl, b2,..., b,~) T
with bi - f(r
The matrix is given by A (k) - {a(r k), r
(2.10)
One may view A (k) as
a linear operator on Hk defined, for any v - ~ vi
E Hk, by
i=1 nk
A(k)v- E i=1
nk
cir
where c i - E j=l
a(r k)' r
(2.11)
Wavelet-Like Multilevel Preconditioning
63
Of main interest in this paper, we study techniques for solving the linear system (2.9) by iterative methods. It is known that the conditioning of the matrices A (k) is of great importance in practical computation. Let us see how the condition number of A (k) is related to the choice of the basis (I)(k) - {elk)} i=1 '~k and the discrete inf-sup condition (2.7) For any v E Hk, denote in the bold face the coordinates of v with respect to the basis (I)(k). Namely, the vectors v (Vl, v2, 9 vnk )T and v are related as follows" _
..
_
,
rtk
v-
~
vir k).
(2.12)
i--1
Introduce a new inner-product in Hk by using the basis {r
i=1 as follows:
nk
k,0 - ~
viwi - v . w .
(2.13)
i--1
Denote by I]vl]k,0 - x/V v, the norm induced by the new inner-product. Recall that the condition number of the matrix A (k) is defined by ~ ( A ( k ) ) - IiA(k)ll IlA(k)-~ll ,
(2.14)
where [[A(k)l[ indicates the norm of the matrix A (k) with respect to the new norm I1" []k,0 in the vector space Hk. The following result can be checked easily. T h e o r e m 1. Assume that there are positive constants Ok,1 and Qk,2 satisfying ~ok,l[[v[]k,o _< IIA(k)vlik,o O. The conditioning estimate in Theorem 3 contains two important factors:
a(k) ~k and a~-'~" The first one depends on the norm of the given bilinear form and the stability constant/3k from the discrete Jar-sup condition. In practical computations, the space Hk must be constructed so that it ensures the boundedness of f3k from below by some fixed/3* > 0. This is the case for the model problems and their discretization spaces to be considered in Section 4. The second factor is basis-dependent. More precisely, it is a characterization of the difference between the discrete coefficient norm I1" llk,0 and the continuous norm I1" II. Stable Riesz bases are important because the corresponding discretization matrices are well conditioned. Thus, simple iterative methods such as the conjugate gradient (CG) can be successfully applied to solve the matrix problem from the Galerkin discretization with a geometric rate of convergence. As is well known, the convergence factor 1 where x - n (A(k) ) -< ~#k o(k) is bounded by ~/~-v/-E+I, o~k). For nonsymmetric problems, one could apply the CG method to the normal equation or the GMRES method applied to A (k). For symmetric and indefinite problems, the MINRES (minimum residual) algorithm would be a good choice. The convergence rate is no worse than that of the CG-method applied to A (k)2. There is another practical criterion in the choice of the basis {(~Ik)}n~l . It is of great practical importance to represent the matrix entries of A (k) as sparsely as possible. This is trivially achieved (assuming a(., .) is symmetric and positive definite) if the basis is a(., .)-orthogonal. Thus, the matrix A (k) admits diagonal forms and only nk entries need to be stored. Such a situation is too special and rarely happens in practice. In general, we assume that the basis is computationally feasible in the sense that the basis function are computable and the corresponding matrix A (k) is sparse (the number of nontrivial entries in the matrix is of order O(nk)). This is the case in practice for the standard nodal bases of finite element spaces Hk if the bilinear form arises from partial differential equations. However, this choice will not make a stable Riesz basis for most of the PDEs. The following section contains a detailed discussion of this aspect. w
Model problems
Here we consider some model problems of (2.4) in partial differential equations. Boundary value problems for the second order elliptic equations are
P. Vassilevski and J. Wang
68
of major consideration in this discussion. Finite element methods will be applied to approximate the solution defined on an open bounded domain gt in 11%d with d = 2 or 3. 4.1
S e c o n d - o r d e r elliptic e q u a t i o n s
Consider the homogeneous Dirichlet boundary value problem for the following second-order elliptic equation:
L(u) -- -~7 . (a(x)~Tu) + b(x). ~7u + c(x)u = f(x),
x E f2,
(4.1.1)
where a = a(x) is a symmetric and positive definite matrix with bounded and measurable entries, b = b(x) and c = c(x) are given bounded functions, and f = f(x) is a function in H-l(ft). Note that we do not intend to consider problems which are convectiondominated. Let Hl(f~) be the standard Sobolev space equipped with the norm:
I1 11 - (llullo + II ullo
V u E Hl(ft).
(4.1.2)
Here ll" {i0 stands for the L2-norm. Let H01(ft) be the closed subspace of Hi(f2) consisting of functions with vanishing boundary values. The following relation is well known: H~(ft) C L2(ft)C H-I(f2).
(4.1.3)
A weak form for the Dirichlet problem of (4.1.1) seeks u E H~(~) satisfying b(u, v ) - f(v) V v E H~(~), (4.1.4) where
b(u, v) - / f ~ (a(z)Vu . Vv + b ( z ) . Vu v + c(z)uv) dx
(4.1.5)
and f(v) is the action of the linear functional f on v. Assume that the problem (4.1.4) has a unique solution. Then the inf-sup condition (2.3)is satisfied for the bilinear form b(., .) defined on H01(f2) • H01(f2). Let us approximate (4.1.4) by using the Galerkin method with continuous piecewise polynomials. If Sh denotes the finite element space associated with a prescribed triangulation of ft with mesh size h, then the Galerkin approximation is given as the solution of the following problem: Find Uh E Sh satisfying
b(uh, r
f(r
V r E Sh.
(4.1.6)
69
Wavelet-Like Multilevel Preconditioning
It has been shown that the discrete problem (4.1.6) has a unique solution when the mesh size h is sufficiently small. Thus, the discrete inf-sup condition (2.7) is satisfied for this problem. Details can be found from [32, 33]. Choose the standard nodal basis as the computational basis for the finite element space Sh. Let {r }~=1 be the set of nodal basis functions and Ah be the corresponding discrete matrix (also called the global stiffness matrix). The condition number of Ah can be estimated by using Theorem 3. To this end, let us establish Inequality (3.10) for the standard nodal basis. For any v E Sh, let tl
v - ~
with vi - v(xi),
vir
(4.1.7)
i=1
where xi are the interior nodal points of the finite element partition. Relation (3.10) is equivalent to the following: n
~lllvll21~ ~ v~ ~ llvll ,
v v e Sh
(4.1.8)
i=1
for some positive constants &l and #2. It is not hard to see that n
,lvll02
~ h-
(419)
i=1
It follows that &2 - O(h-d). Also, by the standard inverse inequality one sees that #1 is bounded from below by a constant proportional to h 2-d. Thus, from Theorem 3, the condition number of Ah is bounded from above by Ch -2 for some constant C; the lower bound for its condition number is also bounded from below by some Ch -2.
4.2
Stokes equations
Consider the problem which seeks u e [H~(a)] d and p e L 2 ( a ) s u c h that -Au+Vp V.u u
= = =
f, 0, 0,
inf,, in f~, on Oft,
(4.2.1)
where f E [L2(f~)] d is a given vector-valued function and 0f~ denotes the boundary of f~. A weak form of the problem (4.2.1) involves the following bilinear form: A(u, p; v, w) = (Vu, Vv)0 - ( V . v , p)0 - ( V . u , w)0
(4.2.2)
P. Vassilevski and J. Wang
70
defined on W • W with 142 - [H0i(fl)] d • L20(f~). Here L~(~)is the closed subspace of L2(fl) consisting of functions with vanishing mean value. The weak problem seeks (u, p) E )42 satisfying A(u, p; v, w) - (f, v)
V (v, w) E W.
(4.2.3)
The inf-sup condition is satisfied for the bilinear form defined in (4.2.2). Details can be found from [18, 11]. To apply the finite element method to (4.2.3), we employ the HoodTaylor element [21] which satisfies the discrete inf-sup condition (with a mild restriction on the triangulation). The Hood-Taylor element is a combination of continuous piecewise linear functions for the pressure variable p and continuous piecewise quadratic functions for the velocity variable u. Denote by Wh = Xh x Sh the corresponding finite element space, where Xh contains continuous piecewise quadratic functions and Sh contains continuous piecewise linear functions for the pressure variable. The finite element approximation (Uh, Ph ) E Wh satisfies .A(uh, ph ;',,, w) - (f, v)
V (v, w ) E Wh.
(4.2.4)
If the standard nodal basis is selected in formulating a matrix problem for (4.2.4), then the condition number of the global stiffness matrix can be estimated by using Theorem 3. More precisely, let {r be the that of Sh, as discussed in the standard nodal basis of Xh and {r previous section. For any (v, w) E Xh • Sh, let tl
w - E wir
(4.2.5)
i=1
and
m
v - E
vjCj.
(4.2.6)
j=l
Using the inverse inequality and the Poincar~ inequality one can derive the following relations" n
2
(4.2.7)
v~ _< c2h-d]JvlJ 21,
(4.2.8)
i=1
and
m
ci h2-dl]vi] ~ _< ~ j=l
where ci and c2 are two absolute constants. Thus, we have from Theorem 3 that the condition number of the global stiffness matrix is bounded by Ch -2.
Wavelet-Like Multilevel Preconditioning
71
We emphasize that the poor conditioning for the Stokes problem is caused by Relation (4.2.8), where the Hi-norm of v was approximated by a discrete norm stable in L 2 only. The equivalence (4.2.7) indicates that the standard nodal basis is a good choice for the pressure variable in the Stokes problem. Therefore, attention should be focused on stabilizing the velocity component in the Stokes equation. A direct wavelet approach to the Stokes problem has been developed by Dahmen et al. [16]. One could also use block-diagonal preconditioners for the saddle-point discretization matrices A (k) with one block corresponding to Laplace-like preconditioners for the velocity component and a second block corresponding to massmatrix preconditioners for the pressure unknown in the MINRES method. Details of this approach can be found in Rusten and Winther [31] and Silvester and Wathen [34]. 4.3
Mixed methods
Here we consider the mixed method for the second order elliptic equation (4.1.1). For simplicity, assume that b = 0, c _= 0, and the following Dirichlet boundary condition u =-g on 0f~ (4.3.1) is imposed on the solution. Let (
, V . v C L2(f~)},
H(div; f ~ ) - ~v" v G which is equipped with the following norm: []Vl[H(div; a) -Let c~(x)=
(Iv[ 2 + [V. vl2)dx
a-l(x) be the inverse of the coefficient matrix a = a(x) and .A(q, u; v, w) = (a(x)q, v)o - (V- v, P)o - (V -q, w)o
be a bilinear form defined on W x W, where W = H(div; f~) x L2(fl). Then, a mixed weak form for (4.1.1) with Boundary condition (4.3.1)seeks (q, u) E W satisfying A(q, u; v, w) = (g, v . n)or~ - (f, w)o
V (v, w ) E W,
(4.3.2)
where (., ")0~ denotes the inner product in L2(O~). The inf-sup condition (2.3) can be verified for the bilinear form A(.; .). Furthermore, finite element spaces satisfying the discrete inf-sup condition (2.7) are available for this bilinear form. Details can be found in the book by Brezzi and Fortin [11]. If the standard nodal basis for the finite element
P. Vassilevski and J. Wang
72
space of Raviart and Thomas is employed in practical computation, the condition number of the global stiffness matrix is known to be proportional to h -2. The question is how to stabilize the nodal basis. Observe that in this application, one needs to construct a basis for the mixed finite element space (called Wh) so that the discrete norm is equivalent to the following norm: /l(v; w)ll~v -Ilvll~(div; ~) + IIwli~. One sees from above that the standard nodal basis is a good choice for the pressure unknown. Thus, the difficulty lies in stabilizing the flux component with the norm in H(div; ft). We do not yet have a positive answer yet for this stabilization, but the approach of ttelmholtz decomposition for vectors might provide a partial answer for problems of two-space variables. The method decomposes each vector-valued function v as follows: v - curl r -b Vr
(4.3.3)
where r and r are functions in Hl(f~). A discrete version of (4.3.3) can be studied in order to apply it to the mixed method. Results for the standard conforming and non-conforming finite elements should be investigated first in this direction. We point out that this approach may have some difficulty for problems of three-space variables. For multilevel methods relying on the above Helmholtz decomposition, see Vassilevski and Wang [40] and Arnold, Falk, and Winther [1]. In the following sections we will devote ourselves to modifying the nodal bases in the application to second order elliptic problems. Our goal is to construct some stable Riesz bases that are computationally feasible for elliptic equations. More precisely, the basis should be so constructed that the resulting discretization matrices are both well conditioned and sparse. w
M u l t i l e v e l direct d e c o m p o s i t i o n s
To construct a computationally stable basis for the second-order elliptic and the Stokes equations, we take advantage of the fact that the weak problem (2.4) is discretized on a sequence of finite element subspaces. In particular, a sequence of nested subspaces may be possible in practical computations. Our objective in this section is to exploit ways of constructing stable basis by using the information from each approximating subspace. 5.1
T h e basic idea
The basic idea comes from the fact that an L 2 orthonormal basis of wavelets is also H 1-stable in applications to partial differential equations. There-
Wavelet-Like Multilevel Preconditioning
73
fore, wavelet bases are good candidates in formulating matrix problems for (2.6). Since the conventional wavelet bases have complicated structures which limit their application in the numerical methods, we shall focus our attention on approximations of wavelet bases. Below we present a detailed discussion. Assume that we have a sequence of nested subspaces {l~ }~=0 satisfying
VoCV~C...cvkc....
(5.1.1)
Each vector space V) shall be referred to as a coarse subspace of Vk when j < k. In applications, they are finite element spaces consisting of continuous piecewise polynomials over a sequence of finite element partitions for the domain f~. Upon viewing Vj as a subspace of L2(f~), one has (:x:}
U V~ - L2(a),
(5.1.2)
j=0
where the closure was taken in the strong topology induced by the L2-norm. We assume that V0 is a very coarse subspace of L2(ft) whose dimension is a small number. For every j > 1, define Wj to be the L2-orthogonal complement of Vj-1 in I~. We have Vj -
Vj _ 1 (~ W j
(5.1.3)
and
Wi_I_Wj if i C j ,
(5.1.4)
where we have assumed that W0 - V0. It follows that k
(5.1.5)
j=0
where all these subspaces are orthogonal. By virtue of (5.1.2) and (5.1.5), this implies C~
j=l
which is a decomposition of L2(f~) into mutually orthogonal subspaces. A wavelet basis for L2(f~) can be constructed if one is able to find an orthonormal basis for each subspace Wj. Such a basis would be ideal in preconditioning the discrete problem (2.6), if it is computationally feasible. In practice, it is very hard to find an L2-orthonormal wavelet basis which is also computable. Therefore, the orthogonality requirement in the
74
P. Vassilevski and J. Wang
decomposition (5.1.5) shall be relaxed to allow only a direct decomposition of the following form" - Voe
e Vl
v:,
(5.1.6)
where each Vj1 is a complement of ~ _ 1 in Vj such that the corresponding two-level decomposition is direct. But in order to attain the stability of the wavelet basis, it is crucial to have some approximate orthogonality among the subspaces Vj1. 5.2
A general approach
A general method for deriving the hierarchical complement of each ~ _ 1 in V) is based on the existence of some computationally feasible projections 7rj from C, a dense subspace of the Hilbert space H, onto Vj. In particular, we assume that C D UI>,VI and zrjr = r for any r E Y). Thus, one has 7rjTri - 7ri for j >_ i ifV~ C V). With Vj1 - ( I - T r j _ l ) ~ , one has the following two-level direct decomposition: (5.2.1)
Definition 2. (Multilevel Hierarchical Basis) For j - O, 1 , . . . , k, let {r 1 , . . . , nj } be a c o m p u t a t i o n a l l y feasible basis o f l ~ . A s s u m e that {r
ii --
1 , . . . , n j _ l } U{ r ), i - nj_ 1 "~- 1 , . . . , nj } forms a basis of k~ . A multilevel hierarchical basis for Vk is defined as follows: k
(I) k -- U { ( / - - 7 r j _ l ) r j=O
j)
i--nj
1-~" 1
nj}
(5.2.2)
Here we have assumed that 7r_ 1 - 0 and n-1 - O. We now discuss the stability of the multilevel hierarchical basis. The following result sets a guideline for the selection of the operators 7rj.
Theorem 4. A necessary condition for ~k to be a stable Riesz basis o f Vk is that the projection operators 7rr be uniformly bounded on Vk with respect to r and k for any r 2-a). R e m a r k 2: Corollary 1 indicates that in order to have the L2-stability of the deviations, one has to assume a level dependence on the tolerance r. More precisely, there exists a 7"o > 0 such that if v < r0J -1, then k
E s--1
[}e~-1[[02 -< C[Iv[[~)
for all v e Vk.
(7.2.7)
Wavelet-Like Multilevel Preconditioning
81
L e m m a 2. Let V~ - ( I - Mk_I)Vt:(1), with V(1) - (Ik - Ik-1)Vk, be the modified hierarchical subspace of level k for any given L2-bounded operator Mk-1. Then, there are positive constants Cl and c2 independent of k such that
c:11r for any r
_ (I-
_< IIr
_< c=11r
r=0,1,
(7.2.8)
Mk_l)r 1 E V1, (/)1 E V(1). Here I1" I1: ~ . , . d ~ tot th~
. o t t o i . th~ Sobo1~v ~ p ~
Hg(~) ~.d
II Iio denotes the L2(f2)-norm.
Proof: The following strengthened Cauchy inequality is valuable: There exists a constant 7 E (0, 1), independent of the mesh size or the level index k such that
(Vr 1, V(~) < "y(Vr 1, Vr
89(V(~, V~) 89 ,
(7.2.9)
for all (~1 E Yk(1) and r E Yk-1. In fact, we shall make use of the following version of (7.2.9). For any r
e V(1) and r e V~-I, one has
(:7(rI + r V(r 1 + 8)) >_ (i - 72)(Vr I,Vr
(7.2.10)
A derivation of (7.2.9) and (7.2.10) can be found from Bank and Dupont [5] or Axelsson and Gustafsson [3]. We first establish (7.2.8) for the case r = 1. With r = - M k _ : r 1 we see from (7.2.10) that
(1
-
~=)11r
< i1r
Thus, the inequality on the left-hand side of (7.2.8) is valid with cl = i - 7 2. To derive the part on the right-hand side, we use the standard inverse inequality to obtain
2 where we have used the L2-boundedness of the linear operator Mk-1. Observe now'that since r E V(1), there exists a constant C such that
IIr ~ ~ Ch~llr
(7.2.11)
It follows that Itr _< ciir for some constant C. This completes the proof of (7.2.8) for r - 1. Similar arguments can be applied to verify the case r = O.
P. Vassilevski and J. Wang
82 Lemma 3. F o r a n y r 1 - ( I - M k _ l ) r w i t h r X1 e V(1) -- (Ik - I k - 1 ) V k ,
1E
V 1 a n d ~o1 - ( I - M k _ l ) X
1 E V~,
del~ne
(A~kl)~l, ~pl) _ a(~l, ~pl) _ / a ( x ) ~ l .
~1.
fl Here, the bilinear form a(., .) is equivalent to the H I - i n n e r product. there are positive constants ri such that
vlh~-2[lr
2 _< (A~kl)r r
r2h~-2llr
Then
2
Since a(.,.) is equivalent to the H~-inner product, there are two
Proof:
positive c o n s t a n t s ~l and ~2 such t h a t
~111~)1[[2 < (A~)$I r m
~
--
h11r
l"
Using the n o r m equivalence (7.2.8), e s t i m a t e (7.2.11), and the inverse inequality, we o b t a i n (with possibly different constants 7"1 and r2),
nh~-2[1r
_< (A~)r
r
(7.2.12)
r2h-~2llr
T h e above inequalities conclude the l e m m a . L e m m a 4. Given v, let v( k)' - (Trk - 7rk-1)v. There exists a sutticiently s m a l l constant 7"0 > 0 such that if the approximate projections Q~ satisfy (7.1.1) with r e (0, 7"0) (see (7.2.4) a n d (7.2.5)), then
J
Ilvll
h;- llv
k=0
II02.
(7.2.13)
Proof." Let v E V. S t a r t i n g with v (J) - v, for s - J down to 1, one defines V(S-- 1) __
( I , _ 1 + Q ~ - I (I~ - I~_ 1)v(~) - 7r._ 1 V. T h e n the d e c o m p o s i t i o n v
in t e r m s of entries in Vs1 - (I -- Q~-I) V(1) _ (i - Q~_ 1)(18 - 18_1 ) v reads as
d v - v (~ + Z
v(/)l'
v(')l - v(S) - v ( s - l ) .
(7.2.14)
j=l F r o m the r e p r e s e n t a t i o n v (~)~ - v (~) - v ( s - l ) - (Q~ - Q , - 1 ) + es - es-1, one arrives at the e s t i m a t e
J k=0
_ 1. The polynomial ~rm also satisfies 7rm(0) - 1 and 0 _< 7r,~(t) < 1 for t E [a, fl], where the latter interval contains the spectrum of the mass matrix Gk-1. Since Gk-1 is
91
Wavelet-Like Multilevel Preconditioning
well conditioned, one can choose the interval [a,/3] independent of k. Thus, the polynomial degree m can be chosen to be mesh-independent so that a given prescribed accuracy r > 0 in (7.1.1) is guaranteed. More precisely, given a tolerance r > 0, one can choose m = re(r) satisfying
i]G~_I (G-kll - O-k:l) Ik-lGkv]l = [[G~_lgrm(Gk-1)Gklllk-lGkv[I
ilQ~_,v-e~-,,ilo
-
< -
=
max
7rm(tlllG-~)lI~-lGkvll
m~x
~',,,(t)ilek-,vllo.
te[o,,~']
tED,#]
Here we have used identity (8.3.2) and the properties of ~rm. The last estimate implies the validity of (7.1.1) with r > max 7rm (t). - tED,~] A simple choice of ~rm(t) is the truncated series m-1
(1 - 7r.~(t))t -1 - p . ~ - l ( t )
- fl-1 E (1 - fit) k,
(8.3.4)
k=0
which yields G~-_I1 - p m - l ( G k - 1 ) . from the following expansion:
We remark that (8.3.4) was obtained
OO
1 - tf1-1E(1
-tfl-1)
k
t E [a fl] ~
9
k=O
With the above choice of the polynomial ~m(t), we have ~m(t) -
1 - tp~_l(t)
- tZ - ~ ~
(1 - Z - ~ t ) k - (1 - Z - ' t ) ~ .
k>m
It follows that max
~m(t)-
1-
In general by careful selection of ~m, we have max 7rm(t) < Cq m for some ' te[~,Z] constants C > 0 and q C (0, 1), both independent of k. Since the restriction on r was that r be sufficiently small, one must have m - O(log r - l ) .
(8.3.5)
The requirement (8.3.5) obviously imposes a very mild restriction on m. In practice, one expects to use reasonably small m (e.g., m - 1, 2). This
P. Vassilevski and J. Wang
92
observation is confirmed by the numerical experiments performed in Vassilevski and Wang [42]. We show in Figure 1 a typical plot of a nodal basis function of V(1), and in Figure 2, we show its approximate-wavelet modification for m - 2. The conjugate gradient method was employed to provide polynomial approximations for the solution of the mass-matrix problem (8.3.2). 8.4
M a t r i x f o r m u l a t i o n s of t h e A W M - H B p r e c o n d i t i o n e r s
We now turn to the description of the multiplicative and additive AWM-HB methods in a matrix-vector form. Let us first derive matrix representations for the operators A~ ), A ~ ), and A(~) introduced in (8.1.1)and (8.1.2). In what follows of this section, capital letters without overhats will denote matrices corresponding to the standard nodal basis of the underlined finite element space. For example, A (k) denotes the standard nodal basis stiffness matrix with entries ra~(k) ~9i , l(k),~ q~j )~,,xj~fk. For any v E Vk and its nodal coefficient vector v, we decompose v as follows:
where w2 E Vk- 1 is uniquely determined as w2 - Ik- 1v + Q~_ 1(Ik - Ik- 1)v. Our goal is to find a vector representation for components of v. Since the above decomposition is direct, it is clear that there are vectors 91 and v2 satisfying
v-(I-I:-lG-~l-lI:-lGk)[ 91 ]0
}Afk-1}A/' \A/'k-1 k + i:_192.
[,1]
(8.4.1)
The vectors 91 and 92 represent the components of our wavelet-modified two-level HB coefficient vector ~ -
"~2
of v.
Now, consider the following problem A v - d,
(8.4.2)
which is in the standard nodal basis matrix-vector form. We transform it into the approximate wavelet modified two-level HB by testing (8.4.2)
(I-I~_IG~llI~-IGk)""[ @1
[ and I~ 1@2 for 0 J arbitrary @1 and @2. By doing so, we get the following two-by-two block system for the approximate wavelet modified two-level HB components of (denoted by Vl and ~r with the two components
k
,1
Wavelet-Like Multilevel Preconditioning
93
where ^
A~I)-[I [I
0]
(
) ( ~" )[I] i - a ~ 2 _ ~ a ; ~ l I 2 -1 A(~) i-i2_~a-i!1~2-~a~ o
O] .(I-
GkI~_IG-~I_II~ -1)
A(k)I~_l,
A~kl) - I~-IA (k) I - I~_~a;~_lI~-lak
o
^ k 1 _ A(k-1) A~k2)I k - 1A(k )Ik_ Note that having computed Vl and v2, the solution v of (8.4.2) can be recovered by using the formula (8.4.1), i.e., V -- g l V l nu Y2"r
where r,
-
I-ILIO;_~II~-*c~
Y~
= I~_1.
,-
~=
o
'
We have, v-Y~,
, Y-[Y1, Y,], Y1-Y1 (~), y~-y~(~)
The transformed right-hand side vectors of (8.4.3) read similarly as follows: dl d2
-
[I O] (I-GkI~_l(l-~llI~ / ~ - l d - yTd.
-1)
d-
yTd,
Therefore, the multiplicative AWM-HB preconditioner B (k) from Definition 3, starting with B(~ = A (~ takes the following block-matrix form:
~ ][ 0 1
844,
The preconditioner B (k) is related to/~(k) in the same way as A (k) to ~(k). More precisely, one has
~(k)_
[y~,y2]rB(k)[yl,y2] ' B(k)-' _ [yl,y2]~(~)-'[yl,y2]r"
We will show below that the inverse actions of B (k) can be computed only via the actions of A (k), Y1, Y2, and yT, yT in addition to the inverse actions o f / ~ ).
'
94
P. Vassilevski and J. Wang
We point out that (8.4.4) has precisely the same form as the algebraic multilevel method studied by Vassilevski [37] (see also Axelsson and Vassilevski [4] and Vassilevski [38]). Observe that, in (8.4.4) , Bii ^(k) is an appropriately scaled approximation of A^~]). We have shown that A~])is well conditioned (see Lemma 3). Thus, it is possible to utilize some simple polynomial approximation B~k1) for A(i]) in the implementation. However, in order to take into account any possible jumps in the coefficient of the differential operator, it would be preferable to compute the diagonal part of .4~]). This is computationally feasible since the basis functions of V~ - ( I - Q~_i)V (i) have reasonably narrow support if m is not too large, which should be the case in practice. Nevertheless, one can employ in actual implementation the CG method to compute reliable approximate actions of ~ ] ) - 1 . A l g o r i t h m 1 (Computing inverse actions of B (k)) The inverse actions of B (k) are computed by solving the system B(k)w = d, with the change of basis w w
-Yi~
dl d2
-Y~d, = Y~d,
^
Y@. Namely, by selling
+Y2~2- [Y~, Y2] ~2
w -- B(k)-ld is computed via the solution of B(k)~r -- ~l as follows: Forward recurrence: 1. compute zi - ~il ~(k)-I ^di, 2. change the basis; z.e., compute z - Yizi, " ^ ~;(k ).. A(k)z), 3. compute d2 "- d 2 - ~2i zi - y T ( d ~. compute ~ r 2 - B(k-i)-l~t2, 5. change the basis, i.e., compute v -
Y2@2.
Backward recurrence: 1. update the fine-grid residual, i.e., compute
di^ "- di^ _ .~(k)~,122 - y T ( d - A(k)Y2@2) - y T ( d - A(k)v),
95
Wavelet-Like Multilevel Preconditioning 2. compute ~r 1 -3. get the solution by w
-
Y l ' ~ r 1 + Y2'~r2 -
Y1'~"1
+ V.
End. Note that the above algorithm requires only the actions of the standard stiffness matrix A (k), the actions of the transformation matrices Y1 and }I2 and their transpositions yT and yT, the inverse action of/3~), and some suitable approximations to the well conditioned matrices .4~). The actions of y - 1 are not required in the algorithm. We now formulate the solution procedure for one preconditioning step using the multiplicative AWM-HB preconditioner B - B(J). A l g o r i t h m 2 (Multiplicative A WM-HB preconditioning) Given the problem Bv- d Initiate d(J) - d. F o r w a r d r e c u r r e n c e . For k -
d down to 1 perform:
1. Compute ~k)_
Ix
O]
(I- GkI~_IG;llI~-1) d (k),
2. Solve 1 __ d~ ^ k), B^(k)~r ll
3. Transform basis
W--(I-- Ik-lGkllik-l~k) [ ~W1
}d~fk_l}'/~fk\'Afk-1
~. Coarse-grid defect restriction d(k-1)
__
= 5. Set k -
k-1.
i~-ld(k) -- ]-i21 7(k)^Wl / ~ - l ( d ( k ) - A(k)w),
If k > O go to (1), else,
6. Solve on the coarsest level A(O)x(O)_ d(O).
P. Vassilevski and J. Wang
96 Backward
recurrence.
I. Interpolate result: Set k := k + 1 and compute x(k)_ i~_lx(k-1),
2. Update fine-grid residual:
._
a~)
Z(~)v(k-~)
-- ~l~k)_[I O](I-Gke2_lS"~llI~-l)A(k)x (k) 3. Solve B1 l^(k)~ 1 _ ~ k ) ,
~. Change the basis 0
5. Finally, set x (k) = x (k) + w.
6. Set k := k + 1. If k < J go to Step (1), else s e t v = x (J).
End. Similarly, one preconditioning solution step for the additive A W M - H B preconditioner D - D (J) takes the following form: 3 (Additive A WM-HB Preconditioning) Given the problem Dv=d
Algorithm
Initiate d (J) -- d. Forward
recurrence.
For k = J down to 1 perform:
1. Compute
~k)_ IX 0] (I--GkX~_lVkll Ik-1) d (k),
97
Wavelet-Like Multilevel Preconditioning 2. Solve
B1 l^(k),~r1 __ ~k)
3. Transform basis
X(k)
--(I--Ikk_lS;!1I~-lGk)[ Wl] 0
\Nk-1 }&-I
~. Coarse-grid defect restriction
d ( k - 1 ) _ i~-ld(k) 5. Set k - k - 1 .
I f k > O go to (1), else
6. Solve on the c o a r s e s t level
A(O)x(O) _ d(O). Backward recurrence. 1. Interpolate result: Set k "- k + 1 and compute
w-
I~_1 x(k-1),
2. Update at level k
x (k) - x (k) + w, 3. Set k "- k + 1. I l k < J go to Step (1), else set
v - x (J). End.
For both the additive and multiplicative preconditioners, it is readily seen that the above implementations require only actions of the stiffness matrices A (k), the mass matrices G (k), and the transformation matrices I~_ 1 and I kk-1 . The approximate inverse actions of A^ ~ ) can be computed via some inner iterative algorithms. Similarly, the action of G~-_I1 can be computed as approximate solutions of the corresponding mass-matrix problem using m steps of some simple iterative methods. Therefore, at each discretization level k, one performs a number of arithmetic operations proportional to the degrees of freedom at that level, denoted by nk. In the case of local mesh refinement, the corresponding operations involve only the stiffness and mass matrices computed for the subdomains where local
P. Vassilevski and J. Wang
98
refinement was made. Hence, even in the case of locally refined meshes, the cost of the AWM-HB methods is proportional to nj. The proportionality constant depends linearly on m = O(log r - l ) , but is independent of J (or h). Some numerical results for the AWM-HB preconditioners can be found from Vassilevski and Wang [42]. A performance comparison with the BPX method [10] and Stevenson's method [36] on more difficult elliptic problems in three dimensions, and in other applications such as interface domain decomposition preconditioning, is yet to be seen. w
Numerical experiments
In this section we present some numerical results to illustrate the efficiency of the method discussed in Section 7. Consider the boundary value problem of seeking u satisfying s
Vu u
-f - g
inFt, on c9~.
(9.1)
Here, / E L2(~), g E H 89 and b - [bib2] are given single-valued or vector-valued functions. We assume that all given functions are sufficiently smooth on their domains. For simplicity, we take ~ to be a square domain and g - 0 on c9~. If u is a solution of (9.1), then it solves the following problem: .A4cu - - S V . (b s
+ s
- - 6 V . (b f) + f
in ~,
(9.2)
subject to the boundary condition u - 0 on 0~. Here 5 > 0 is a parameter. The purpose of considering the problem (9.2) is to get the so-called streamline derivative o~, _ b. Uu in a variational formula for u More precisely, by testing (9.2) against any r E H ] ( ~ ) one obtains
b (u, r
-
r e ) + (r b. b. r e )
+ 6(_b.
b. r e ) (9.3)
-- (f6, r for all r E H~(fl). Here ]'6 - f 9.1
5U. (b f).
Galerkin discretization
The Galerkin method for the approximation of u is based on the variational problem (9.3). Let V - Vh be a C~ finite element space of piecewise polynomials corresponding to a quasiuniform triangulation Th of
Wavelet-Like Multilevel Preconditioning
99
f~. The Galerkin approximation is a function Uh E Vh such that
b~(~h, r
- ~(V~h, r e ) + (r -e5 E
V~) + 6(b. V~, b. r e )
f Auh(b.Vr
TETh T
(9.1.1)
= (f6, r for all r E Vh. For continuous piecewise linear functions, one has AUh -- 0 on each element. It follows that the discrete problem seeks Uh C Vh such that
~(v~h, r e ) + (r b. v~h) + 6(5. v~h, b. r e ) - (f~, r
(9.1.2)
for all r E Vh. The convection term is assumed to satisfy V.b_< 0
in ft.
(9.1.3)
For a convergence analysis of the streamline diffusion finite element approximation Uh, we refer to [23] and [2]. 9.2
Numerical tests
We choose the same test examples as in [2]. Namely, _b- [ ( 1 - x cos a) cos a(1 - y sin a) sin a], for various angles c~. Note that V. _b- - 1 . The right hand side f is chosen so that u - x(1 - x)y(1 - y) is the exact solution. Thus, the right-hand side function f is e-dependent. The stopping criterion is that the relative error of the residual be less than 10 - s in the discrete L2-norm. The objective is to test the number of iterations in the solution procedure by using the wavelet-like hierarchical basis. The matrix form of the discretized problem (9.1.2) reads as follows" Au-f. (9.2.1) Here A is a nonsymmetric matrix. For small e, it is very difficult to find a good preconditioner for A. It was seen in [2] that a block-ILU factorization method turns out to be very robust with respect to arbitrary positive e, though very little is known theoretically about this good performance. For any finite element function v, we use the bold face v to denote the vector with respect to the nodal basis and 9 the vector representation with respect to the modified hierarchical basis. In the implementation, the modified hierarchical basis is employed to provide a preconditioner for the global stiffness matrix A as follows. Let Y be the transformation from -~ to v such that v - Yg. The preconditioner is given by P - ( y y T ) - I with y T being the conjugate of Y. We discuss the computation of Y and y T .
100
P. Vassilevski and J. Wang
Let
and y(k)_ g-l"
(9.2.3)
Here, G (k) stands for the mass matrix at kth discretization level, I2_ 1 is the natural coarse-to-fine interpolation matrix from ( k - 1)th grid to the kth one a n d / ~ - i _ (/~-l)T" Also, (~(8)-1 is an approximate inverse of the mass matrix G(') 9 For example, a good choice for G (8)-1 w would be an approximate solution of G(~)x = w by polynomial iterative methods. In practice, the J acobi and conjugate gradient methods are good candidates. The numerical results in this section are based on the Jacobi iterative method with two iterations. A l g o r i t h m 4 For any given ~ - (9~J) ,''', Y 9 is computed as follows:
~1), ~(0))T,
the action v -
9 Set v(~ - 9(0), 9 Fork-1
toJ
do
9 v - v(J),
Algorithm 5 yTd
For any given d is computed as follows:
(d~ s) , . . - , d~1), d(~ T, the action a
-
. Set d(J) - d (~ 9 Fork-J
down t o l
do
~(k-i) 9 a-(a(J )
a(~ 9
Once the preconditioner P - ( y y T ) - I is known, one can solve the discrete problem (9.2.1) by using the generalized conjugate gradient method employed in [2]. We comment that this simple preconditioner may not work well for convection-dominated diffusion problems. This fact can be seen from the numerical results illustrated in Tables 1-4.
Wavelet-Like Multilevel Preconditioning
101
T a b l e 1. Iteration counts for h -1 = 64, a = 75 ~
iter
c-1 6 - 0.001 30
e-O.1 6 - 0.01
28
c-
c - 10 -2 6-0.1 42
10 -3 5 - 1.0 59
...........
c - - 10 -4 5--10
74
T a b l e 2. Iteration counts for h -1 --64, a = 105 ~
iter
c-1 5 - 0.001 31
c-O1 6 - 0.01 27
c-lO
2
c-lO -a 6 - 1.0
6 - 0.1 70
e - - 10 . 4 6--10
........
*
.
....
T a b l e 3. Iteration counts for h -1 = 128, c~ = 75 ~
iter
e-1 5 - 0.0025
c -0.1 5 - 0.025
32
29
....
c - 10 -2 5 -- 0.25 48
T a b l e 4. Iteration counts for h -1 = 128, a = 105 ~
iter
c-1 6 - 0.0025 32
c-O.1 5 - 0.025 29
c-
10 -2
6 - 0.25 71
In Tables 1-4, " , " is used to indicate a non-convergence in 100 iterations. It is clear t h a t the use of the wavelet-like hierarchical basis gives an efficient p r e c o n d i t i o n e r if the p r o b l e m is not c o n v e c t i o n - d o m i n a t e d . T h e present i m p l e m e n t a t i o n of the modified hierarchical basis is c o m p a rable with the additive p r e c o n d i t i o n i n g m e t h o d discussed in [42]. T h e inform a t i o n is also c o n t a i n e d in A l g o r i t h m 3 of this p a p e r with B"(k) l l _ Ch.~2ik , where Ik s t a n d s for the identity m a t r i x . Acknowledgments. T h e research of Vassilevski is s u p p o r t e d in p a r t by the U.S. NSF g r a n t INT-95-06184 and also in p a r t by B u l g a r i a n M i n i s t r y for E d u c a t i o n g r a n t M M - 4 1 5 , 1994. T h e research of W a n g is s u p p o r t e d in p a r t by the NSF g r a n t INT-93-09286. T h e a u t h o r s are also grateful to Dr.
P. Vassilevski and J. Wang
102
Oswald for his careful comments and remarks on the paper. References [1] Arnold, D. N., R. S. Falk, and R. Winther, Preconditioning in H(div) and applications, Math. Comp., to appear. [2] Axelsson, O., V. Eijkhout, B. Polman, and P. S. Vassilevski, Incomplete block-matrix factorization iterative methods for convectiondiffusion problems, BIT 29 (1989), 867-889.
[3]
Axelsson, O. and I. Gustafsson, Preconditioning and two-level multigrid methods of arbitrary degree of approximations, Math. Comp. 40 (1983), 219-242.
[4]
Axelsson, O. and P. S. Vassilevski, Algebraic multilevel preconditioning methods, II, SIAM J. Numer. Anal. 27 (1990), 1569-1590.
[5]
Bank, R. E. and T. Dupont, Analysis of a two-level scheme for solving finite element equations, TechnicM Report CNA-159, Center for Numerical Analysis, The University of Texas at Austin, 1980.
[6] Bramble, J. H., Multigrid Methods, Pitman Res. Notes Math. Ser., vol. 294, Longman, Harlow, U.K., 1995 (2nd Ed.). [7] Bramble, J. H. and J. E. Pasciak, Iterative techniques for time dependent Stokes problems, ISC Report, Texas A&M University, 1994, preprint.
[8]
Bramble, J. H., J. E. Pasciak, J. Wang and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), 1-21.
[9]
Bramble, J. H., J. E. Pasciak, J. Wang, and J. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 5 7 (1991), 23-45.
[10]
Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22.
[11] Brezzi, F. and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. [12] Carnicer, J. M., W. Dahmen, and J. M. Pefia, Local decompositions of refinable spaces and wavelets, Appl. Comput. Harmon. Anal. 3 (1996), 127-153.
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[13] Ciarlet, P., The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1977. [14] Chui, C. K., An Introduction to Wavelets, Academic Press, Boston, 1992. [15] Dahmen, W. and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315- 344. [16] Dahmen, W., A. Kunoth, and K. Urban, A wavelet Galerkin method for the Stokes equations, Computing 56 (1996), 259-301. [17] Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [18] Girault, V. and P.-A. Raviart, Finite Element Methods for NavierStokes Equations, Springer-Verlag, Berlin, 1986. [19] Griebel, M. and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), 163-180. [20] Griebel M. and P. Oswald, Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems, Adv. Comput. Math. 4 (1994), 171-206. [21] Hood, P. and C. Taylor, A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids 1 (1973), 73-100. [22] Jaffard, S., Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal. 29 (1992), 965-986. [23] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, Cambridge, 1994. [24] Kotyczka, U. and P. Oswald, Piecewise linear prewavelets of small support, in Approximation Theory VIII, vol. 2, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 1995, pp. 235-242. [25] Lorentz, R. and P. Oswald, Constructing economical Riesz bases for Sobolev spaces, presented at the 9th Domain Decomposition Conference held in Bergen, Norway, June 3-8, 1996. [26] Mallat, S., Multiresolution approximations and wavelet orthonormal bases of L2(IR), Trans. Amer. Math. Soc. 315 (1989), 69-88. [27] Maubach, J. M., Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995), 210-227.
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[28] Mitchell, W. F., Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Star. Comput. 13 (1992), 146-167. [29] Ong, M.-E. G., Hierarchical basis preconditioners in three dimensions, SIAM J. Sci. Comput., to appear. [30] Oswald, P., Multilevel Finite Element Approximation: Theory and Applications, Teubner Skr. Numer., Teubner, Stuttgart, 1994. [31] Rusten, T. and R. Winther, A preconditioned iterative method for saddle-point problems, SIAM J. Matrix Anal. Appl. 13 (1992), 887904. [32] Schatz, A. H., An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959-962. [33] Schatz, A. H. and J. Wang, Some new error estimates for RitzGalerkin methods with minimal regularity assumptions, Math. Comp. 65 (1996), 19-27. [34] Silvester, D. S. and A. J. Wathen, Fast iterative solution of stabilized Stokes systems, part II: using general block preconditioners, SIAM J. Numer. Anal. 31 (1994), 1-16. [35] Stevenson, R., Robustness of the additive and multiplicative frequency decomposition multilevel method, Computing 54 (1995), 331-346. [36] Stevenson, R., A robust hierarchical basis preconditioner on general meshes, Technical Report # 9533, Department of Mathematics, University of Nijmegen, Nijmegen, The Netherlands, 1995. [37] Vassilevski, P. S., Nearly optimal iterative methods for solving finite element elliptic equations based on the multilevel splitting of the matrix, Technical Report # 1989-09, Institute for Scientific Computation, University of Wyoming, Laramie, WY, USA, 1989. [38] Vassilevski, P. S., Hybrid V-cycle algebraic multilevel preconditioners, Math. Comp. 58 (1992), 489-512. [39] Vassilevski, P. S., On two ways of stabilizing the hierarchical basis multilevel method, SIAM Rev., to appear. [40] Vassilevski, P. S. and J. Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992), 235-248.
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[41] Vassilevski, P. S. and J. Wang, Stabilizing the hierarchical basis by approximate wavelets, I: Theory, Numer. Linear Algebra Appl. (1996), to appear. [42] Vassilevski, P. S. and J. Wang, Stabilizing the hierarchical basis by approximate wavelets, II: Implementation and numerical experiments, SIAM J. Sci. Comput. (1996), submitted. [43] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Reg. 34 (1992), 581-613. [44] Yserentant, H., On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), 379-412. [45] Yserentant, H., Old and new convergence proofs for multigrid algorithms, in Acta Numerica, Cambridge Univ. Press, New York, 1993, pp. 285--326.
Panayot S. Vassilevski Center of Informatics and Computing Technology Bulgarian Academy of Sciences "Acad. G. Bontchev" street, Block 25 A 1113 Sofia, Bulgaria p an ayot @iscbg.ac ad. bg Junping Wang Department of Mathematics University of Wyoming Laramie, Wyoming 82071
[email protected] This Page Intentionally Left Blank
II. Fast Wavelet Algorithms: Compression and Adaptivity
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An Adaptive Collocation Method B a s e d on I n t e r p o l a t i n g W a v e l e t s
Silvia Bertoluzza
A b s t r a c t . A wavelet collocation method for the adaptive solution of second order elliptic partial differential equations in dimension d is presented. The method is based on the use of the Deslaurier-Dubuc interpolating functions. The method is tested on an advection dominated advection-diffusion problem, and on a Laplace problem posed on a nonrectangular domain.
w
Introduction
The solutions of many differential equations are very smooth in a large part of the domain, but is not as smooth globally. In order to approximate them in an optimal way one needs to use high order methods, capable of taking advantage of the local smoothness of the solution by using a coarse grid in the greatest part of the domain, but capable, on the other hand of coping with singularities. Among the methods that display this potential one can count the methods based on wavelet bases and multiscale analysis. On one hand, such bases allow one to design methods of arbitrarily high order, and on the other hand they display a local behavior, allowing one to take into account the presence of singularities by locally refining the approximation space. Furthermore, such bases actually provide a singularity detection tool that has been successfully employed in designing adaptive schemes for the numerical solution of several classes of differential equations [9, 17, 27,28]. Multiscale Wavelet M e t h o d s for P D E s Wolfgang D a h m e n , A n d r e w J. Kurdila, and P e t e r Oswald (eds.), pp. 109-135. C o p y r i g h t O 1997 by A c a d e m i c Press, Inc. All rights of r e p r o d u c t i o n in any form reserved. ISBN 0-12-200675-5
109
110
S. Bertoluzza
The fundamental idea underlying such bases is that, in a multiscale analysis, the space ~ corresponding to a uniform discretization with step size h - 2-J can be obtained as the span of two different bases, the nodal basis of scaling functions {r and the hierarchical basis of wavelets {r m = j 0 , . . . , j - 1}. Such a basis is obtained by decomposing t~ as - Om=jo Win, where the detail or complement spaces ~ Vm being Wm are defined as Wm = ( P m + ~ - Pm)Vm+~, Prn : L2(IR) the L 2 orthogonal projection, or in the biorthogonal case an L 2 bounded projection operator. The change of basis between the two can be performed by applying the Fast Wavelet Transform (FWT), which takes O(2 j) operations. It is important to underline the known fact that when expressing a function in Vj in terms of the nodal basis, generally, no matter how smooth it is, all the coefficients will be needed in order to get a good approximation, while, when expressing the same function in the wavelet basis, in order to get an approximation of the same order, one would usually need only a subset of the coefficients (essentially the ones corresponding to those basis functions whose center is close to singularities). In other words, the nodal basis naturally corresponds to taking a uniform discretization and the wavelet basis to a nonuniform one. In the solution of PDEs one would like to work with the second basis, which is the one allowing high order approximation in smoothness regions and at the same time grid refinement near singularities. Unfortunately, working with such a nonuniform basis presents some difficulties. Let us consider for example the Galerkin approach. Suppose that we have an equation Au = f, and that we want to find the solution of the form Uh -- ~(rn,k)ehh UmkCrn~ (where Ah is a small subset of all the admissible indices (re, k)). In implementing a Galerkin scheme, the need arises in evaluating integrals of the form f A C j k r or of the form f A u h r We recall that computing such integrals by applying a classical quadrature rule (see [15])is not convenient. In fact, in order not to lose the accuracy properties of the scheme (which we assume to be of order M), one would need to choose a quadrature rule which is exact for polynomials of order 2 M - 2. In order to obtain an order M error estimate one needs however to assume that the approximation space is included in H M. On the other hand, it is well known that wavelets are usually much less regular than accurate ( i.e., Crn,~ ~ HM). This implies the need of using a lower order quadrature rule, with a much finer grid of quadrature nodes, which raises the cost of the whole procedure. This difficulty can be overcome by applying a result by W. Dahmen and C. Micchelli [21]. The values of the integrals f ACjkCjn can be computed by taking advantage of the autosimilarity properties of the functions r
Adaptive Collocation Method
111
and then a change of basis can be performed in order to retrieve the desired values. This procedure, however, requires one to express u (or r in terms of the nodal basis ej k. We are therefore obliged to perform this computation on a uniform fine grid. This difficulty is even stronger when we deal with nonlinear operators .T'(u). In fact, when linear expressions are concerned, one can assemble the stiffness matrix once and for all, while when dealing with nonlinearities, (especially the ones which are not of multilinear type, e.g., e ~ , for example), one has to go back to the uniform grid each time one computes f.~(U)r One possible way to overcome such a problem is to use a collocation approach, by which one totally avoids integration. In particular, we propose here a collocation method based on the multiscale decomposition relative to the Deslaurier-Dubuc interpolating functions [23, 24]. This is a class of compactly supported scaling functions, which will be denoted by 0, that satisfy a property of interpolation, i.e., 0(0) = 1,
O(n) = O, for n r O,
(1.1)
rather than the usual property of orthonormality (or biorthogonality). Instead of defining the complement spaces as (Pj+I - Pj)t~+l, in such a construction, which is described in some detail in Section 2, the complement spaces are defined as Wj -- (Lj+I - Lj)~/~+I,
(1.2)
where Lj is a Lagrange interpolation operator. Clearly such an operator is not bounded in L 2, i.e., this multiscale decomposition is somewhat out of the usual L 2 framework. Nevertheless, it turns out that most of the properties of the usual wavelet decomposition still hold, provided that the decomposed function is smooth enough. In particular, a property of smoothness characterization through wavelet coemcients still holds. It is possible to write an equivalent norm for H ~(IRd) in terms of the wavelet coefficients for all s such that H 8 C C u, i.e., in dimension d, s > d/2 (see Theorem 2.1 in the following). This property is perhaps the most relevant one from the point of view of applications to the numerical solution of PDEs, for preconditioning and adaptivity. Another interesting feature of this basis is the particular structure of the fast wavelet transform (which in this case takes the name of Interpolating Wavelet Transform). The algorithmical structure is the same as in the L -~ bounded case, but due to the particular properties of the scaling and wavelet functions, it exhibits some extra features. First of all, due to the interpolation property (1.1), the coefficients of the developement of a given functon f in Vj with respect to the scaling
S. Bertoluzza
112
functions basis are its values at the Lagrange interpolation nodes,
f E Vj,
f-
~f(k/2J)ojk,
(1.3)
k
i.e., the wavelet transform and its inverse allow one to go back and forth from the coefficient space to the physical space. Moreover, with this basis it is particularly simple to work with nonuniform discretizations. In fact, it is possible to assign to each wavelet function a corresponding dyadic point and vice versa. Suppose now that one needs to work with a nonuniform subset of the wavelet basis Bh = {r (m, n) E Ah}. A function f in span{Bh} is uniquely determined by its values at those points corresponding to the reduced set of basis functions. Given the values of f at such points, the coefficients of the expansion of f in the basis Bh are computed by applying a reduced version of the interpolating wavelet transform which works only on such points. Its matrix form is obtained from the matrix form of the full interpolating wavelet transform by simply taking a square submatrix. Moreover, such a submatrix can be assembled very easily, independently of the full interpolating wavelet transform matrix. The aim of this paper is to describe how such a basis can be used for the numerical solution of PDEs by means of a collocation method. Consider, for example the very simple problem: -u"+u=f
inlR.
(1.4)
The solution method proposed consists in first selecting a grid Gh of dyadic points and the corresponding subspace Uh C V), and then looking for Uh E Uh such that for all points p in Gh
- u ' ( p ) + u(p) = f(p).
(1.5)
We remark that in the case Uh = ~ , the method falls into the framework studied in [19] for the wider class of Petrov-Galerkin schemes, and has been extensively studied and tested in [8]. The method is stable, convergent, and good preconditioning techniques are available. In the case where Uh is much smaller than Vj, as we already stressed, the use of a collocation scheme allows us to work at the nonuniform grid level, and there is no need of performing any computation on a uniform grid. On the other hand, such a choice imposes the constraint of dealing with problems whose solution is a least C 2. In other words, the method we propose here is directed to the solution of problems that have such a minimal smoothness, but still have some kind of localized singularity (for instance in an higher order derivative). In particular, the method that we
Adaptive Collocation Method
113
propose aims at getting a high order approximation (possibly in high order norms) for such kinds of problems. There is then the problem of selecting a grid which is well suited to the solution of a given problem. This is usually done through the application of an adaptive scheme. Adaptive refinement based on different types of a posteriori error estimations has been studied recently in several papers [25, 31]. In particular, wavelet based adaptive schemes have been studied in different papers and in different approaches [17, 26, 27], which have in common the fact of taking advantage, in one way or another, of norm equivalence properties of the form (2.17). In the framework of Galerkin discretizations it is known that a hierarchical decomposition of the approximate solution provides an a posteriori error estimator [3, 4, 33]. Although such a theory does not apply to collocation schemes, one can nevertheless try to apply the same principle to such a case. In particular, in the framework of collocation with Deslaurier-Dubuc interpolating multiscale decomposition, the ideas of [27] have been tested on Burgers eqaution in [4]. Here we propose to use such techniques in the framework of the solutions of linear elliptic PDEs in dimension d. The paper is organized as follows. In Section 2 we describe the construction of the Deslaurier-Dubuc interpolating scaling function and the corresponding multiresolution decomposition. We consider both functions on IRd and on (0, 1) d. In Section 3 we review some of the definitions and properties related to the collocation method on a uniform grid. In Section 4 we describe the collocation method on nonuniform grids, while in Section 5 we introduce the adaptive strategy and the resulting adaptive collocation method, which in Section 6 we test on several examples in one and two dimensions. In particular we consider a problem whose solution presents boundary and internal layers, and a Laplace problem on a nonrectangular domain. This is treated by mapping the domain onto a square and solving the resulting problem on the square. The results of all tests are very promising and show that collocation with Deslaurier-Dubuc interpolating wavelets is a valid alternative to Galerkin approximation, especially in the case of adaptive discretization. w
Deslaurier-Dubuc interpolating wavelets
This section is devoted to the multiresolution framework in which the method will be stated. In the following, we will denote by t1" [Is,v,~ the WS'V(f~) norm, and by [[. [[8,~ the H~(D) norm. Let eL be the Daubechies compactly supported scaling function of order L [22]. Recall that eL satisfies, among other properties, (i) supp eL = [0, 2L + 1]. (ii) eL E W R/2,~ for some R > 0 (R is proportional to L).
S. Bertoluzza
114
(iii) eL is refinable with refinement equation of the form (in the Fourier domain' CL(~) - mo((/2)r (2.1) where m0 is a trigonometric polynomial of degree 2L + 1. (iv) eL is orthogonal to its integer translates. (v) Polynomials up to order L can be written as a linear combination of the integer translates of eL. A new scaling function 0 is defined as the autocorrelation of eL. Definition 1. The Deslaurier-Dubuc fundamental function 0 of order N - 2L + 1, is defined by
o(=) - s r162
1.5
Fig l a - Haar scaling function ......................................
0.5 0
-0.5
x)dy.
(2.2)
Fig. l b - D e s l a u r i e r - D u b u c - N = I 1.5
0.5
0
0.5
1
1.5
0
-1
0
1
Figure 1. The Haar function r and its autocorrelation. As a consequence of (i-v) the function 0 satisfies the following properties" 1) supp0-[-N,N]
and
0 E W R'~176
(2.3)
2) The function 0 is clearly refinable. In fact, it satisfies 0 - ICLI2, and hence (2.1) gives 0~(~)- Imo(~/2)12b'(~/2). (2.4) 3) As a direct consequence of the orthogonality of the translates of eL, the function 0 satisfies the following interpolation property:
o(n) -/,~ CL(V)r (Y- n)dy- ~,~o.
(2.5)
4) As a consequence of (v), polynomials up to order N can be written as linear combinations of the integer translates of 0.
Adaptive Collocation Method
115
R e m a r k . The function 0 was originally introduced by G. Deslaurier and S. Dubuc [23] as the limit function of an interpolatory subdivision scheme. Its relation with the minimal phase Daubechies orthonormal scaling functions was pointed out by G. Beylkin and N. Saito [11]. Based on such a function, we can build multiscale decompositions on IR and, by tensor product, on IRd, which we will denote respectively {V)(lR)} and {Vj(IRd)}. In o r d e r to work on (0, 1) d o n e can apply the technique of [15]. The basis functions interacting with the boundaries can be suitably modified in order to get a multiscale analysis {Vj ((0, 1)d)}. In all these cases there will be a Riesz basis for Vj (f~) which will satisfy an interpolation assumption, relatively to the grid-points ,~ - x~ - k/2J E Gj(f~), with Gj(f~) - z d / 2 j N-~. It will be convenient in the following to index the basis functions by the corresponding interpolation grid-point, that is we will use the notation .
lO (f~) - span{O~,, ,X C Gj(t2) - 2 - J z n t2}.
(2.6)
The above mentioned interpolation property will read: for all points )~, # C Gj(a)
1, C~
o,
r #.
(2.7)
It is therefore natural to introduce an interpolation operator Lj ~ Vj (~) defined by
Lj(f)-
~ f(A)oJ),. ~eaj(a)
(2.8)
In such a framework the complement spaces are introduced according to the interpolation operator
W j ( a ) - (Lj+I - Lj)Vj+I(~).
(2.9)
Such spaces admit Riesz bases of the form {Ca,
A C Jj(gt)},
with J j ( a ) -
aj+l(a)\aj(~)
(2.10)
which can be defined, for instance, by simply taking (see Figure 2)
- 0{
c aj+l(a)\aj(a).
(2.11)
R e m a r k 1. In dimension d _> 2 other choices for defining the basis functions CA for the complement spaces are possible. For example, one could define the basis starting with the one dimensional basis of the form (2.11), and use the usual construction of tensor product wavelets (the one that leads
S. Bertoluzza
116 Function theta and psi for N=5 !
!
1
............................
I
!
theta: solid line
i
i
i
!
/%
psi: dashed line 0.5
-5
-4
I
I
-3
-2
l
-1
I
0
.....
|,
I
1
2
I
3
I
4
5
Figure 2. Interpolating Scaling and Wavelet functions. in two dimensions to the three types of wavelet functions r162r162and r 1 6 2 see [28]). Also in such case there is a well-defined correspondence between basis functions for Wj and points of the grid Gj+I (12)\Gj(12). We can use such a basis in the following. For simplicity we will make the choice (2.11), which allows us to avoid some technicalities. Any continuous function f defined on ft for 12 - lRd and for 12 = (0, 1) d can be mapped into its interpolating wavelet transform (IWT), i.e., the sequence of its coefficients IWT(f)-
{{fjo,~, A e Gjo(ft)}, {e~,
)~ e A(12)}},
(2.12)
where A(f~) = Uj>joJj(f~ ) indicates the set of admissible dyadic points. Any function f E C~ may be reconstructed from its transform by means of f - - E fJo,AOi~ + E e),r (2,13) AEGj 0
Proposition
AEA(f~)
I. The following estimates hold: Jackson Inequality. Let f E H'(ft), with d/2 < s _ 0.
Adaptive Collocation Method
123
Let t = m a x { s - 2, 0}. A first grid can be selected by looking at the behavior of the data of the equation. Assume for simplicity that the coefficients and the boundary data are smooth. We select the first grid G~ -- Gjo tOA0 by requiring that the right-hand side f is well approximated on such grid, in the sense that
Ilf - Laoflit,o~ t. Then, by estimate (2.13) for J big enough, we will have I l f - La fllt,~ ~ e/2. For each j, j0 _< j _< J -
(5.2)
1, we can then retain the points ,~ E
Jj(~) whose corresponding coefficients in the expansion of f is bigger than e'/(2Jtj2), and discard the remaining points, obtaining in such a way a grid A0. It is easy to prove that the following estimate holds.
ILj f(x) - LAof(X)l e/2 is, for A E Jj(f]) implies
[If -- Lj fils,~,suppr
> ce.
(5.5)
In other words, for ~ E Jj(f2) the coefficient e~ measures the lack of approximation of f by Lj f in the neighborhood of the corresponding point. The size of the coefficients of the IWT is also an indicator of the local relevance of high frequency components in u, since the function r ~ E Jj(f2), is oscillating at frequencies around 2J. Both facts imply the need of locally refining the grid. On the other hand, a very small coefficient in the IWT tells us that the corresponding point is not relevant, that is the solution is well approximated already on a coarser grid. The point can then be removed without deteriorating the quality of approximation. We then fix two tolerances 5~ < 5a " one is used to decide whether a coefficient is "small" and the other whether it is "relevant." We remove the points )~ E Jj(~) corresponding to the wavelet coefficients smaller (in absolute value) than 5~/(2J'j 2) and add some neighboring points at the higher level in the neighborhood of the points ,~ E Jj (f2) corresponding to wavelet coefficients bigger that 5a/2 `j. On the new grid G1 that we obtain, we solve the problem by means of a collocation scheme and then we repeat the refining procedure. At the n-th step, given a grid a n and the relative approximate solution un with coefficients u~, n }, - {{ Cjo
e h.}},
(5.6)
we compute the next grid G "+1 by removing the useless points and refining where the approximation is bad. For each point ,~ - ( k l , . . . , kd)/2 j define a set Ux of neighboring points UA - - ~ N {X~ +1 , ] r
(2kl "~" rll, 9
2kd + rid), r l i - --1, 0, 1},
(5.7)
and let A . + I - {A'
> 5,/(2J*J2)}
u L/x, {~: 1~1>6./2J~}
G n+t - Gjo u An+l.
(5.8) (5.9)
The (n + 1)-th approximation space is defined as U ~ + I - VA,+,.
(5.10)
The (n + 1)-th approximate solution is computed by solving the collocation problem: find u '~+1 E U n+l verifying the boundary condition u~+l(A) g(A) at boundary points A E G n+l'b and the equation at interior points .Attn+l(,~) -- f(/~),
,'~ E a n+l'i.
(5.11)
125
Adaptive Collocation Method
We want to remark that one of the strengths of the proposed refining procedure is its extreme simplicity. In particular, it does not need extra computation of any quantities, the coefficients of the approximate solution themselves being the parameters driving the decision on whether to refine or not. Moreover, since the discretization is not based on any triangulation, adding or removing each point has no influence on the other points of the grid, since there is no conformity requirement to fulfill, which causes some difficulty when refining and derefining a finite element triangulation. In this paper we will not address the problem of solving the sequence of linear systems resulting from the adaptive procedure. We want to remark that in order to be able to apply the proposed strategy to real life problems, there is the need of proposing an efficient solver for the resulting linear system. The author is actually studying the problem of designing a suitable iterative solver, by exploring the possiblility of applying multilevel preconditioning techniques [12, 29] and algorithms of the cascadic multigrid type [13] to the collocation scheme proposed here. The aim of this paper is, however, to study the behavior of the collocation scheme and of the refining strategy from the point of view of approximation and of reliability of the grid selection procedure. w
Numerical tests
The method has been tested with the aim of assessing the behavior in terms of approximation, the sensitivity to the choice of the parameters co, ca, and er, and the reliability of the refinement procedure. For the time being the linear systems arising from the discretization on different grids of the equation considered have been solved by a direct method. Two types of tests have been performed. Quantitative tests in one dimension and qualitative tests in two dimensions. 6.1. O n e - d i m e n s i o n a l test p r o b l e m s We tested the proposed scheme on the following two boundary value problems in one dimension. Problem T1.
Find u such that
{
-ul'-f u(0)-a,
in(0, 1), u(1)-b,
for f, a, and b chosen in such a way that the true solutions of P1 are respectively Test 1. Ul
--
Iix a -X 2
- ~ x1 +
for x < .5, 2-!4 f o r x > . 5 .
(6.2)
S. Bertoluzza
126
Test 2. u2 -
{ sin(Trx) 6
~i=oCiX'
9
for x < .5, -
for x > .5,
(6.3)
where the coefficients ci are chosen in such a way that the function u is C 6 (but not C7).
Test 3.
(6.4) P r o b l e m T2.
Find u such that
"-cu"+ u'- 1 u(1)-
in (0, 1), 0,
(6.5/
for c - .01 and c - .001. The method has been tested on both problems for different values of the tolerances Ca and or. All the tests are performed with the DeslaurierDubuc interpolating wavelet of order 5. Depending on the different smoothness properties of the three different solutions, one can observe different behavior of the method. For Tests 1, 2, and 3, the error is considered as a function of the number of degrees of freedom. The function u l has a discontinuity in the third derivative at x - .5. Far from such point the solution is C ~ . Using a uniform discretization is not covenient, since the solution is not regular enough and the high order m e t h o d does not show up. However, by applying the proposed adaptive strategy the method takes advantage of the high regularity of the solution far from the point x - .5, while coping with the singularity by a local refinement. The results of the numerical test (Figure 4a) indicate that for such a problem the adaptive strategy (o 9 cr - 10 -4 , * 9 c~ - 10 -7, + 9 Cr - - 1 0 - 1 0 ; Ca -- 5Or), is clearly winning over the uniform discretization (displayed as a d,=,~ted line). For comparison we also display in a dashed line the results of using piecewise linear finite elements on a uniform discretization ill which the point x - .5 is a node (so that the singularity is treated in an optimal way), and with a dashdot line the results obtained by spectral methods. In Table 1, we display the convergence history for such a test. The number of points discarded (resp. added) at each grid refinement is displayed along with the L ~ error. The solution u2 of test problem 2 has a discontinuity in the seventh derivative, that is, it is everywhere as regular as the method demands in order to have optimal convergence. As a result, the best approximation is obtained by using a uniform discretization. Also the adaptive strategy, both without and with analysis of the right-hand-side (Figure 4b, co - c~ - *" c~ - 10 -7, o: c~ - 10 -13, +" co - c~ - 10 -7) eventually converges to the solution on a uniform mesh. We nevertheless remark that an analysis
Adaptive Collocation Method
127 Problem 1 . - T e s t 1.
10 -2
9
\ 10 -3
\
\
\
'
.
.
.
.
.
.
.
i
\ \
9
1 0 -4
\
tO
\
L~ 10 -5
".
\
\ \
10 -6
"''''....
I
"''''.....
I !
1 0 -7
"'... "''''''''r
10 .8 10 ~
i
|
,
i
= , I
,
10 2 10 3 Number of Degrees of Freedom F i g u r e 4a.
10 4
Results of Test 1.
Problem 1 . - Test 2.
10 .2
.
.
.
.
.
.
i
.
.
.
.
''
'
'~
'1
II
10 -4
+ il(
+
10 -6
O. .9
s
4-
lO -8 O. "'-
0.. . . . . . . . . .
1 0 -10
10 -12 _
1 0 -14
. . . . . . . . . . . . . . . .
101
I
|,,
i
,
i
,,
~
,
,
I
14 10 2 10 3 Number of Degrees of Freedom Figure
4b.
R e s u l t s of T e s t 2.
10 4
S. Bertoluzza
128
of the right-hand side allows us to start from the beginning with a better grid. Also the case comparison with piecewise linear finite elements (dashed line) and with spectral methods (solid line) is displayed. In particular the spectral method can exploit the regularity of the solution, and therefore it gives the best results. Table 1
Test 1- Convergence history of the adaptive procedure.
Tolerances
Nadd
Nrem
N. d.o.f.
L ~ Error
0
16
33
3.616E-5
12
20
41
9.617E-6
18
22
45
5.968
5~ = 5 . E - 7
0
16
33
3.616E-5
5~= 1 . E - 7
12
20
41
9.617E-6
18
16
39
9.892E-6
5a
"--
1.E- 7
5~= 1 . E - 7
1 . E - 10
0
16
33
3.616E-5
5~ = 1 . E - 10
6
26
53
9.607E-6
22
30
61
2.472E-6
24
36
73
6.269E-7
34
38
77
1.581E-7
40
36
73
3.965E-7
5 . E - 10
0
16
33
3.616E-5
5~= 1 . E - 10
6
22
49
9.609E-6
18
30
61
2.472E-6
24
28
65
6.278E-7
26
26
65
1.580E-7
5a =
5a =
It is well known that if a hierachical decomposition of an approximate solution to a PDE is to provide an error indicator, the discretization grid can not be chosen arbitrarily, but has to be coupled to the (unknown) solution in a suitable way. This is expressed by means of the saturation
Adaptive Collocation Method
129
assumption (see [2]). This also holds for the method proposed. Consider, for example, Test 3 (see Table 2). If the first grid Go is too coarse, the adaptive scheme converges in one iteration to a wrong solution. We want to stress that the use of the more refined error indicator of [17] allows one to avoid the need of making such an assumption, whose validity is naturally enforced by the chosen adaptive strategy, which, on the other hand, starts with a choice of a starting discretization providing a good approximation of the right-hand side. The resulting adaptive strategy is proved to be convergent. The numerical tests show that our adaptive collocation method exhibits a similar behavior. If the grid Go is chosen by an analysis of the right-hand side, as proposed in Section 5, the results of the numerical tests show that one gets a good solution as a result of the adaptive procedure. In other words, choosing Go according to Section 5 seems to guarantee also in this case that the grid Go is chosen properly and it is a good starting point for the scheme proposed. For fixed cr = 10 .8 and ca = 510 -s, we tested the scheme with different values of co. In Table 2, we report the number n of refinements performed, the number of points of the initial grid Go and of the final grid G,~-I, and the L 2 error on the final grid. Table 2
Test 3 - Results of the procedure after convergence.
N. points of Go
N. points of G . _ 1
L2 Error
1
17
17
0.707
10.
4
113
505
7.6E-9
.1
4
177
505
7.6E-9
1.E-3
2
245
505
7.6E-9
1.E-5
2
497
505
7.6E-9
1.E-7
2
497
505
7.6E-9
~50
n
No check
,
,
.........
As far as problem T2 is concerned, the structure itself of the equation is different. Here the problems do not derive from a lack of regularity of the data but on a degeneracy of the ellipticity of the operator considered. The operator -ud2/dx 2 + d/dx is elliptic, but its coercivity constant (u) is much smaller than the continuity constant (which is of order O(1)). As a consequence its solutions may develop boundary layers, which are
S. Bertoluzza
130
not detectable by an a priori analysis. The results of the numerical tests are displayed in Figure 5. The L 2 error versus the number of degrees of freedom during the successive refinements is plotted in log-log scale, for different values of the tolerances ea and cr ( o: cr - 10-4; *" er - 10-6; +" e~ - 10-s; Ca - 5Or ). The method gives good results by properly refining the grid in the vicinity of the boundary layer. Test n.4 - Viscosity = .01 1 0 -~
...........
" ".'.~11.
or '
1 0 -2
10 -3
.. 1 0 -'i
\
ill
\
e'l lO_S
-
9 X
.
\
. 9
..
10 -6 .
1 0 -7
10 -8
,
,
,
,
,
,
103
Number of Degrees of Freedom
Figure 5. 6.2
,
10 2
10 ~
Two-dimensional
Adveetion-diffusion
Results of Test 4.
tests problem
Among the problems in which the use of adaptive schemes is necessary, are advection-diffusion problems when the advection term is dominating. Consider for example the following model problem. P r o b l e m PA-D.
Find u such that
-uAu+fl. Vu-f
u - g
i n ( 0 , 1 ) 2, on 0(0, 1) 2.
(6.6)
It is well known that when u < < [/3] the solution of such a problem can develop boundary and internal layers, which need to be resolved. A straightforward application of a uniform dicretization is either too expensive or inaccurate, since Gibb's phenomena (see [34]) may arise if, at least in proximity of the layer, the discretization is not fine enough. Therefore an adaptive procedure is needed. Such problem is the two-dimensional problem corresponding to Test 1. The aim of this test is to check that in the twodimensional case the adaptive method proposed behaves qualitatively in
Adaptive Collocation Method Test 5. ,
131 Test 5. -
Visc=,(X)8 - jmax=6 - d.o.f.=lg71
§ § :::::::::::::::::::::::::::::::::::::t4§
~+nm,_g~
+
+ +
::;laRt;:g~;W,:: ;::+ +
o ~+
1.2,
+ + + + + + + + + + + + + + + + § § + + + § ++++~.++++++++
+ §
o.7
1,
*+ +§+ + + + + + + + + 1 - + + +. +. +.+.+ + + ++~+ . . . . . . . .
+++. +++ ++++§ ++++ § + + + + +++++++++ + .H-+++ § + ++++§ 0.6 +++ §
+
+
+,+
0.5 H, + + + + +
0.4
Vise=.008 - tmax---6 - d.o.f.=Ig71
9
+
+
+ + + + +
+
+
+
+
+
+
+
+
0.8, 0.6,
+
+
+ + + + +
+
+
t.
+ + + + §
+++++++++ ++-~+++ +
+ + , + + + +
+
+
+
0.4,
++++++
~t+:: +§§:,, :; :: ~: +::,* . . . . . . + ;,+:::++:; :1," * +
0.
o . ~
-0.2
.
1
0.4 0
0.1
0.2
0.3
0.4
0.5
F i g u r e 6.
0.6
0.7
0.8
0.g
0.2
,,, O, (1.5) [15], and equations having special soliton solutions, e.g., the Korteweg-de Vries equation ut + a u u x + / 3 u ~ = O, (1.6) where a and 3 are constant [1, 24]. Finally, a simple example of equation (1.1) is the classical diffusion (or heat) equation ut = v u x x ,
v > 0.
(1.7)
Although we do not address multi-dimensional problems in this paper, we note that the Navier-Stokes equations may also be written in the form (1.1). Consider u t + ~ 1 [U" VU -~- V(U" U)] -- V V 2 U - Vp,
(1.8)
div u - 0
(1.9)
where and p denotes the pressure. Applying the divergence operator to both sides of (1.8) and using (1.9), we obtain
A p = f(u),
(1.10)
where/(u) - - 8 9 [u. V u + V(u- u)] is a nonlinear function of u. Equation (1.1) is formally obtained by setting Lu-
vV2u,
(1.11)
and N u - -31 [u. Vu + V ( u - u ) ] - V ( A - I f ( u ) ) .
(1.12)
The term A - i f ( u ) is an integral operator which introduces a long range interaction and has a sparse representation in wavelet bases. A one-dimensional model that may be thought of as a prototype for the Navier-Stokes equation is ut = 7-l(u)u, (1.13) where ~(.) is the Hilbert transform (see [18]). The presence of the Hilbert transform in (1.13) introduces a long range interaction which models that found in the Navier-Stokes equations. Even though in this paper we develop algorithms for one-dimensional problems, we take special care that they generalize properly to several dimensions so that we can address these problems in the future.
140
G. Beylkin and J. Keiser
Several numerical techniques have been developed to compute numerical approximations to the solutions of equations such as (1.1). These techniques include finite-difference, pseudo-spectral and adaptive grid methods (see e.g. [19, 24]). An important step in solving equation (1.1) by any of these methods is the choice of time discretization. Standard explicit schemes (which are easiest to implement) may require prohibitively small time steps, usually because of diffusion terms in the evolution equation. On the other hand, implicit schemes allow for larger time steps but require solving a system of equations at each time step and, for this reason, are somewhat more difficult to implement in an efficient manner. In our approach [11] we have used new time discretization schemes for solving nonlinear evolution equations of the form (1.1), where s represents the linear and A/'(f(u)) the nonlinear terms of the equation, respectively. A distinctive feature of these new schemes is the exact evaluation of the contribution of the linear part. Namely, if the nonlinear part is zero, then the scheme reduces to the evaluation of the exponential function of the operator (or matrix)/: representing the linear part. We show in [12] that such schemes have very good stability properties and, in fact, describe explicit schemes with stability regions similar to those of typical implicit schemes used in, e.g., fluid dynamics applications. In this paper we simply use one such scheme. The main difficulty in computing solutions of equations like (1.1) is the resolution of shock-like structures. Straightforward refinement of a finitedifference scheme easily becomes computationally excessive. Specialized front-tracking or adaptive grid methods require some criteria to perform local grid refinement. Usually in such schemes these criteria are chosen in an ad hoc fashion (especially in multiple dimensions) and are generally based on the amplitudes or local gradients in the solution. Pseudo-spectral methods, as described in, e.g., [24], usually split the evolution equation into linear and nonlinear parts and update the solution by adding the linear contribution, calculated in the Fourier space, and the nonlinear contribution, calculated in the physical space. Pseudo-spectral schemes have the advantages that they are spectrally accurate, relatively straightforward to implement and easy to understand analytically. However, pseudo-spectral schemes have a disadvantage in that the linear and nonlinear contributions must be added in the same domain, either the physical space or the Fourier space. For equations which exhibit shocklike solutions such transformations between the domains are costly. The Fourier transform of such solutions possesses frequency contributions across the entire spectrum as the shock becomes more pronounced. The wavelet approach, described next, is comparable to spectral methods in their accuracy, whereas the automatic placement of significant wavelet coefficients in regions of large gradients parallels general adaptive grid approaches.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
141
Let the wavelet transform of the solution of (1.1) consist of Ns significant coefficients concentrated near any shock-like structures which may be present in the solution. We describe two adaptive algorithms that update the solution using O(Ns) operations, using only the significant wavelet coefficients. In other words, the resulting algorithmic complexity of our approach is proportional to the number of significant coefficients in the wavelet expansions of functions and operators. The algorithms we describe have the desirable features of specialized adaptive grid or front-tracking algorithms and pseudo-spectral methods. We also recall that in the wavelet system of coordinates, differential operators may be preconditioned by a diagonal matrix, see e.g. [7, 28, 20]. For a related approach used in finite elements, see e.g. [14]. In addition, a large class of operators, namely Calder6n-Zygmund and pseudo-differential operators, are sparse in wavelet bases. Therefore, efficient numerical algorithms can be designed using the wavelet representation of these operators. These observations make a good case for developing new numerical algorithms for computing in wavelet bases. The theoretical analysis of the functions and operators appearing in (1.1) by wavelet methods is well understood, [21, 16, 30, 36]. Additionally, there have been a number of investigations into the use of wavelet expansions for numerically computing solutions of differential equations, see e.g. [34, 29, 25]. In our approach we emphasize the adaptive aspects of computing the solution. Any wavelet-expansion approach to solving differential equations is essentially a projection method. In a projection method the goal is to use the fewest number of expansion coefficients to represent the solution since this leads to efficient numerical computations. We note that the number of coefficients required to represent a function expanded in a Fourier series (or similar expansions based on the eigenfunctions of a differential operator) depends on the most singular behavior of the function. Since we are interested in solutions of partial differential equations that have regions of smooth, nonoscillatory behavior interrupted by a number of well-defined localized shocks or shock-like structures, using a basis of the eigenfunctions of differential operators would require a large number of terms due to the singular regions. Alternately, a localized representation of the solution, typified by front-tracking or adaptive grid methods, may be employed in order to distinguish between smooth and shock-like behavior. In our approach the number of operations is proportional to the number of significant coefficients in the wavelet expansions of functions and operators and, thus, is similar to that of adaptive grid methods. The basic mechanism of refinement in wavelet-based algorithms is very simple. Due to the vanishing moments of wavelets, see e.g. [22], we know that (for a given accuracy) the wavelet transform of a function 'automat-
142
G. Beylkin and J. Keiser
ically' places significant coefficients in a neighborhood of large gradients present in the function. We simply remove coefficients below a given accuracy threshold. This combination of basis expansion and adaptive thresholding is the foundation for our adaptive pseudo-wavelet approach. In order to take advantage of this 'adaptive transform' and compute solutions of (1.1) in wavelet bases using O(Ns) operations, we have developed two algorithms: the adaptive application of operators to functions, and the adaptive pointwise product of functions. These algorithms are necessary ingredients of any fast, adaptive numerical scheme for computing solutions of partial differential equations. The algorithm for adaptively multiplying operators and functions is based on a 'vanishing-moment property' associated with the B-blocks of the so-called Non-Standard Form representation of a class of operators (which includes differential operators and Hilbert transforms). The algorithm for adaptively computing ](u), e.g., the pointwise product, is analogous to the method for evaluating nonlinear contributions in pseudo-spectral schemes. The spectral expansion of u is projected onto a 'physical' subspace, the function f(u) is evaluated, and the result is projected into the spectral domain. In our algorithm, contributions to ](u) are adaptively computed in 'pieces' on individual subspaces. Each of our adaptive algorithms uses O(Ns) operations, where N8 is the number of significant coefficients of the wavelet representation of the solution of (1.1). The adaptivity of our algorithms and the analogy with pseudo-spectral methods prompt us to refer to our overall approach as an
adaptive pseudo-wavelet method. The outline of this paper is as follows. In Section 2 we use the semigroup approach to replace the nonlinear differential equation (1.1) by an integral equation and describe a procedure for approximating the integral to any order of accuracy. We provide a brief review of wavelet "tools" relevant to our discussion in Section 3. In Section 4 we are concerned with the construction of and calculations with the operators appearing in the quadrature formulas derived in Section 2. Specifically, we describe a method for constructing the wavelet representation, derive the vanishing-moment property, and describe a fast, adaptive algorithm for applying these operators to functions expanded in a wavelet basis. In Section 5 we introduce a new adaptive algorithm for computing the pointwise product of functions expanded in a wavelet basis, and discuss the calculation of general nonlinear functions. In Sections 4 and 5 we give simple numerical examples illustrating the algorithms. In Section 6 we illustrate the use of these algorithms by providing the results of a number of numerical experiments. Finally, in Section 7 we draw a number of conclusions based on our results and indicate directions of further investigation.
143
Pseudo-Wavelet Algorithms for Nonlinear P D E s
w
The semigroup approach and quadratures
We use the semigroup approach to write the partial differential equation (1.1) as a nonlinear integral equation in time. We then approximate the integrals to arbitrary orders of accuracy by quadratures with operatorvalued coefficients. These operators have wavelet representations with a number of desirable properties described in Sections 4.1 and 4.2. The semigroup approach is a well-known analytical tool that is used to express partial differential equations in terms of nonlinear integral equations and to obtain estimates associated with the behavior of their solutions (see e.g. [37]). The solution of the initial value problem (1.1) is given by u(x, t) - e(t-t~
+
f;
(2.1)
e(t-')z~A/'f(u(x, 7))dT,
where the differential operator Af is assumed to be independent of t and the function f ( u ) is nonlinear. For example, in the case of Burgers equation, the operator Af - o and f ( u ) - ~1U 2 , s o that A l l ( u ) - uux appears as products of u and its derivative. Equation (2.1) is useful for proving the existence and uniqueness of solutions of (1.1) and computing estimates of their magnitude, verifying dependence on initial and boundary data, as well as performing asymptotic analysis of the solution, see e.g. [37]. In this paper we use equation (2.1) as a starting point for an efficient numerical algorithm for solving (1.1). A significant difficulty in designing numerical algorithms based directly on (2.1) is that the matrices representing these operators are dense in the ordinary representation. As far as we know, it is for this reason that the semigroup approach has had limited use in numerical calculations. We show in Sections 4.1 and 4.2 that in the wavelet system of coordinates these operators are sparse (for a fixed but arbitrary accuracy) and have properties that allow us to develop fast, adaptive numerical algorithms. Discrete evolution schemes for (2.1) were used in [11] and further investigated in [12]. The starting point for our discrete evolution scheme is (2.1) where we consider the function u(x, t) at the discrete moments of time tn - to + n A t , where At is the time step. Let us denote un - u(x, tn) and Nn H ( f ( u ( x , tn))). Discretizing (2.1)yields UnZr. 1 -- e q l A t U n + l _ l
-3[- A t
(
.1
7Nn+l + Z
~mNn-m
)
,
(2.2)
m--0
where M + 1 is the number of time levels involved in the discretization, and 1 _< M. The expression in parenthesis in (2.2) may be viewed as the quadrature approximation of the integral in (2.1). To simplify notation, we suppress the dependence of the coefficients 7 and/~m o n l.
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G. Beylkin and J. Keiser
The discrete scheme in (2.2) is explicit if 7 - 0, otherwise it is implicit. For a given M, the order of accuracy is M for an explicit scheme and M + 1 for an implicit scheme due to one more degree of freedom, 7. This family of schemes is investigated in [12] and is referred to as exact linear part (ELP) schemes. Applying this procedure to Burgers equation (1.5), we approximate I(t) =
(2.3)
e(t--r)s to
and list the results for m = 1 and m = 2. For m = 1, equation (2.3) can be approximated by I(t) - ~10s or
I(t) -
(u(to)ux(to) + u ( t l ) u x ( t l ) ) + O((At) 2),
~I0f.,i(u(to)u~(tl)
+ u(tl)ux(to)) + O((At) 2),
where 01:,m -
(e marl" -
I)s -I,
(2.4) (2.5)
(2.6)
I is the identity operator, u(ti) - ui and v(ti) - vi. Note that (2.4) is equivalent to the standard trapezoidal rule. For m - 2 our procedure yields an analogue of Simpson's rule 2
ci,iu(ti)u~(ti) + O((At)3),
I(t) - E
(2.7)
i=0
where c0,0
-
gx0s
- 1s
Ci,1
--
~0s
C2,2
--
610E,2_~_1~.
(2.8)
(2.9) (2.10)
For the derivation of higher order schemes (m > 2) and the stability analysis of these schemes we refer to [12], since our goals in this paper are limited to explaining how to make effective use of such schemes in adaptive algorithms. {}3
Preliminaries
a n d c o n v e n t i o n s of w a v e l e t a n a l y s i s
In this section we review the relevant material associated with wavelet basis expansions of functions and operators. In Section 3.1 we set a system of notation associated with multiresolution analysis. In Section 3.2 we describe the representation of functions expanded in wavelet bases, and
Pseudo-Wavelet Algorithms for Nonlinear PDEs
145
in Section 3.3 we describe the representation of operators in the standard and nonstandard forms. In Section 3.4 we discuss the construction of the nonstandard form of differential operators, following [5]. Much of this material has previously appeared in a number of publications, and we refer the reader to e.g. [22, 16, 36] for more details. M u l t i r e s o l u t i o n analysis and wavelet bases
3.1
We consider a multiresolution analysis (MRA) of L2(N) as 9"" C V2 C V l C V0 C V - 1 C V - 2 C ' ' ' ,
(3.1.1)
see e.g. [21, 22], such that 1. ~ j e z Vj = {0} and Ujez l/) is dense in L2(R), 2. For any f E Le(R) and any j E Z, f(x) E Vj if and only if f(2x) E Yj-l~
3. For any f E L2(R) and any k E Z, f(x) E V0 if and only if f ( x - k ) V0, and
E
4. There exists a scaling function ~o E Vo such that {~o(x- k)}ke z is a Riesz basis of Vo. In our work, we only use orthonormal bases and will require the basis of condition 4 to be an orthonormal rather than just a Riesz basis. 4'. There exists a scaling function ~o E Vo such that {~o(x- k)}ke z is an orthonormal basis of V0. As usual, we define an associated sequence of subspaces W j as the orthogonal complements of Vj in Vj_I, Vj-1 = Vj ~
Wj.
(3.1.2)
Repeated use of (3.1.2) shows that subspace Vj can be written as the direct sum
vj -
wj,.
(3.1.3)
j'>j
We denote by ~o(.) the scaling function and r the wavelet. The family of - 2-J/2~(2-Jx- k)}ke z forms an orthonormal basis of functions {r - 2-J/2r z forms an orthonormal Vj and the family {r basis of W j .
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G. Beylkin and J. Keiser
An immediate consequence of conditions 1, 2, 3, and 4 ~ is that the function ~a may be expressed as a linear combination of the basis functions of V - - l , L! -1
~(x) -- ~
E
hkqo(2x- k).
(3.1.4)
g k ~ ( 2 x - k).
(3.1.5)
k=0
Similarly, we have L1-1
r
= vf2 E k--0
L!
The coefficients g - {hk}L=11 and G - {gk}k=l are the quadrature mirror filters (QMFs) of length Lf. In general, the sums (3.1.4) and (3.1.5) do not have to be finite and, by choosing Lf < oc, we are selecting compactly supported wavelets, see e.g. [22]. The function r has M vanishing moments, i.e., 0,
~r
0 _< m 1,
and v ( z ) + v(1 - z) = 1.
(3.3.9)
In this case the interaction between scales for differential operators is restricted to nearest neighbors where ]j-j~] _ 1. On the other hand, Meyer's wavelets are not compactly supported in the time domain which means the finger bands will be much wider than in the case of compactly supported wavelets. The control of the interaction between scales is more efficient in the nonstandard representation of operators, which we will discuss later. Another property of the S-form which has an impact on numerical applications is due to the fact that the wavelet decomposition is not shift invariant. Even if the operator T is a convolution, the B~ and F~ blocks of the S-form are not convolutions. Thus, the S-form of a convolution operator is not an efficient representation, especially in multiple dimensions. An alternative to forming two-dimensional wavelet basis functions using the tensor product (which led us to the S-form representation of operators) 9!
.!
Pseudo-Wavelet Algorithms for Nonlinear PDEs
A I
2 1
B1
4 S B 1B 1
A 2
3 B2
4 S B21 B 2
As
Bs Bs
B
3
2
F 1
r'2 4
4
F, S F1
153
4
F2
Fs
s F 2,
s
r~
4
$
s
A4 B 4 5
F 4 T4
Figure 3. Organization of the standard form of a matrix.
is to consider basis functions which are combinations of the wavelet, r and the scaling function, ~(.). We note that such an approach to forming basis elements in higher dimensions is specific to wavelet bases (tensor products as considered above can be used with any basis, e.g., the Fourier basis). We will consider representations of operators in the nonstandard form (N S-form), following [8] and [5]. Recall that the wavelet representation of an operator in the N S-form is arrived at using bases formed by combinations of wavelet and scaling functions, for example, in L2(N 2)
Cj,k (x) r
(y),
Cj,k (x) ~j,k, (y),
~j,k (~) Cj,k, (y),
(3.3.10)
where j, k, k' E Z. The NS-form of an operator T is obtained by expanding T in the 'telescopic' series
T - ~ (QjTQj + QjTPj + PjTQj), jEz
(3.3.11)
where Pj and Qj are projectors on subspaces Vj and W j , respectively. We observe that in (3.3.11) the scales are decoupled. The expansion of T into the NS-form is, thus, represented by the set of operators
T = { & , B ~ , r j } j e z,
(3.3.12)
G. Beylkin and J. Keiser
154
L
Figure 4. Schematic illustration of the finger structure of the standard form.
where the operators Aj, Bj, and F j act on subspaces V j and W j,
Aj Bj Fj
--
QjTQj QjTPj PjTQj
9 Wj--+Wj, "
Vj---}Wj,
(3.3.13)
9 Wj-+Vj,
see e.g. [8]. I f J _ n is the finite number of scales, as in (3.1.12), then (3.3.11) is truncated to J
To - ~ ( Q j T Q j
+ QjTPj + PjTQj) + P j T P j ,
(3.3.14)
j=l
and the set of operators (3.3.12) is restricted to
T o - { { A j , B j , F j } ~ - J1 , T j } ,
(3.3.15)
where To is the projection of the operator on Vo and T j is a coarse scale projection of the operator T,
Tj - P j T P j " V j --+ V j ,
(3.3.16)
using (in L2(]R2)) the basis functions
J,k
J,k, (y),
(3.3.17)
Pseudo-Wavelet Algorithms for Nonlinear PDEs
155
~ii}i~i~iii!iii~iiil!iiiiii!i!!ii!~
1 IW F i g u r e 5. Organization of the non-standard form of a matrix. Aj, Bj, and Fj, j = 1, 2, 3, and 7"3 are the only non-zero blocks.
for k, k ~ E Z. Figure 5 shows the NS-form of a matrix for J - 3. The price of uncoupling the scale interactions in (3.3.11) is the need for an additional projection into the wavelet basis of the product of the N S - f o r m and a vector. The term "nonstandard form" comes from the fact that the vector to which the NS-form is applied is not a representation of the original vector in any basis. Referring to Figure 6, we see that the NSform is applied to both averages and differences of the wavelet expansion of a function. In this case we can view the multiplication of the N S-form and a vector as an embedding of matrix-vector multiplication into a space of dimension M2 n - J ( 2 g + l - 1), (3.3.18) where n is the number of scales in the wavelet expansion and J _< n is the depth of the expansion. The result of multiplying the N S - f o r m and a vector must then be projected back into the original space of dimension N - 2 n. We note t h a t N < M < 2 N a n d , forJ-n, wehaveM-2N-1. It follows from (3.3.11) that after applying the NS-form to a vector we arrive at the representation J
Z Z
j = l kEIF2,.,_ j
J
g3kqOj,k(X). j = l kElF2n_ i
(3.3.19)
G. Beylkin and J. Keiser
156
\ d'
d1
\ \ S
l
2
d
S2
d3 S3
Figure 6. Illustration of the application of the non-standard form to a vector.
The representation (3.3.19) consists of both averages and differences on all scales which can either be projected into the wavelet basis or reconstructed to space V0. In order to project (3.3.19) into the wavelet basis we form the representation, J
sk~g,k(x ), j--1 kE]F2n_ j
(3.3.20)
kEF2n_ J
using the decomposition algorithm described by (3.2.7) and (3.2.8) as follows. Given the coefficients {~J }j=l J and (dJ}g=l, we decompose {~1} into {~2} and {~2} and form the sums {s 2} - {~2 + ~2} and (d 2} - {(~2 § (~2}. Then on each scale j = 2, 3 , . . . , J - 1, we decompose {s j} = {~J + ~J} into {~j+l} and {~j+l} and form the sums {s j+l} - {~j+l + ~j+l} and {dJ+l} _ {~j+l + dj+l}. The sets {s J} and {dJ}J_l are the coefficients of the wavelet expansion of (Tofo)(x), i.e., the coefficients appearing in (3.3.20). This procedure is illustrated in Figure 7. An alternative to projecting the representation (3.3.19) into the wavelet basis is to reconstruct (3.3.19) to space V0, i.e., form the representation
(3.2.1)
(Pof)(x) = E S~176
(3.3.21)
kE Z
using the reconstruction algorithm described in Section 3 as follows. Given
157
Pseudo-Wavelet Algorithms for Nonlinear PDEs
{,~0}
~
{~1 _[._,~1} __ {81 }
._.),
"'"
-'+
{ ~1 + ~1} _ { d 1}
{SJ "Jr"8J}-- {8 J} {dg + i1g ) - {d J }
Figure 7. Projection of the product of the NS-form and a function into a wavelet basis.
{~0)
t-"
{S 1} -- {~1 + ~1)
"'"
t--" {8 J - l } --" {~J-1 .~_ ~J-1 }
{d l} _ {dx ..[_~1)
...
{d J - l ) _ {~J-1 ..[..6~J-1 }
/....
{8 J} {d J}
Figure 8. Reconstruction of the product of the NS-form and a function to space Vo. the coefficients {~J } ] 1 and {dJ }j=l, g we reconstruct {dJ} and {sg} into {g J - l } and form the sum {s J-1 } = {~g-1 + gg-i }. Then on each scale j - J - 1, J - 2 , . . . , 1 we reconstruct {~J} and {dJ} into {gj-1} and form the sum {s j - l } = {~j-1 + gj-1}. The final reconstruction (of {d 1} and {s 1}) forms the coefficients {s ~ appearing in (3.3.21). This procedure is illustrated in Figure 8. 3.4
The nonstandard
f o r m of d i f f e r e n t i a l o p e r a t o r s
Following [5], in this section we recall the wavelet representation of differential operators 0 p in the N S-form. The rows of the N S-form of differential operators may be viewed as finite-difference approximations on the subspace V0 of order 2M - 1, where M is the number of vanishing moments of the wavelet r The N S-form of the operator 0 p consists of matrices A j, B j, F j, for j - 0, 1 , . . . , J and a 'coarse scale' approximation T g. We denote the J ]~2i,1, and 0,i,t, 3 for j - 0, 1 , " - , J, and elements of these matrices by hi,l, s i,l J " Since the operator 0 p is homogeneous of degree p, it is sufficient to compute the coefficients on scale j = 0 and use
4
-
-
2- Jz ~
(3.4.1)
We note that if we were to use any other finite-difference representation as coefficients on V0, the coefficients on Vj would not be related by scaling
158
G. Beylkin and J. Keiser
and would require individual calculations for each j. Using the two-scale difference equations (3.1.4) and (3.1.5), we are led to 9 ~-~LI-I ~-~LI-1 j-1 a~ - 2 z..~k=0 z..,k,=O gkgk' S2i+k_ k, , t3Jl ~
-
~-.~LI-I ~-~LI-1
j-1
~-~LI-1
j-i
(3.4.2)
2 z_.,k=O ~k'=O gkhk, S2i+k_k, , LI-1
2 z.,k=O ~-~k'=O hkgk' s2i+k_k,.
-
Therefore, the representation of 0 p is completely determined by s~ in (3.3.6), or in other words, by the representation of 0 p projected on the subspace V0. To compute the coefficients s o corresponding to the projection of 0 p on V0, it is sufficient to solve the system of linear algebraic equations
8 0 -- 2 p
801 "b ~
a2k-1(801_2k+l
"-[- 821+2k_1)
,
(3.4.3)
k=l
for - L f + 2 _ 1 _< Lf - 2 and L! - 2
E l=-Lf+2
1p s o - (-1)Pp! ,
(3.4.4)
where a 2 k - 1 are the autocorrelation coefficients of H defined by L I-l-n
an=2
~
hi hi+n,
n=l,...,Lf-1.
(3.4.5)
i--0
We note that the autocorrelation coefficients an with even indices are zero, a2k = O,
k - 1 , . . . , L I / 2 - 1,
(3.4.6)
and a0 - x/~. The resulting coefficients s~ corresponding to the projection of the operator 0 p on V0 may be viewed as finite-difference approximations of order 2 M - 1. Further details are found in [5]. We are interested in developing adaptive algorithms, i.e., algorithms such that the number of operations performed is proportional to the number of significant coefficients in the wavelet expansion of solutions of partial differential equations. The S-form has 'built-in' adaptivity, i.e., applying the S-form of an operator to the wavelet expansion of a function, (3.2.3), is a matter of multiplying a sparse vector by a sparse matrix. On the other hand, as we have mentioned before, the S-form is not a very efficient
Pseudo-Wavelet Algorithms for Nonlinear PDEs
159
representation (see, e.g., our discussion of convolution operators in Section 3.3). In the following sections we address the issue of adaptively multiplying the NS-form and a vector. Since the NS-form of a convolution operator remains a convolution, the AJ, B j, and FJ blocks may be thought of as being represented by short filters. For example, the NS-form of a differential operator in any dimension requires O(C) coefficients as it would for any finite difference scheme. We can exploit the efficient representation afforded us by the N S-form and use the vanishing-moment property of the B j and F j blocks of the N S-form of differential operators and the Hilbert transform to develop an adaptive algorithm. In Section 4.1 we describe two methods for constructing the N S-form representation of operator functions. In Section 4.2 we establish the vanishing-moment property which we later use to develop an adaptive algorithm for multiplying operators and functions expanded in a wavelet basis. Finally, in Section 4.3 we present an algorithm for adaptively multiplying the NS-form representation of an operator and a function expanded in the wavelet system of coordinates. w
N o n s t a n d a r d form representation of operator functions
In this section we are concerned with the construction of and calculations with the nonstandard form (NS-form) of operator functions (see, e.g. (2.2)). We show how to compute the NS-form of the operator functions and establish the vanishing-moment property of the wavelet representation of these operators. Finally, we describe a fast, adaptive algorithm for applying operators to functions in the wavelet system of coordinates. 4.1
T h e nonstandard form of operator functions
In this section we construct the NS-forms of functions of the differential operator 0x. We introduce two approaches for approximating the NS-forms of operator functions: (i) compute the projection of the operator function on V0, Po f (Oz)Po, (4.1.1) or, (ii) compute the function of the projection of the operator,
f (PoO~Po ).
(4.1.2)
The difference between these two approaches depends on how well i~(~)l 2 acts as a cutoff function, where ~(x) is the scaling function associated with a wavelet basis. It might be convenient to use either (4.1.1) or (4.1.2) in applications. The operator functions we are interested in are those appearing in solutions of the partial differential Equation (1.1). For example, using (2.1)
160
G. Beylkin and J. Keiser
with (2.5), solutions of Burgers equation can be approximated to order (At) 2 by
u(x, t + ~xt) - eA'Lu(x, t)
--1-OE, 1 [U(X, t)OxU(X, t + At) + U(X, t + At)Oxu(x, 2
t)]
(4.1.3)
where s = vO 2 and Os is given by (2.6). Therefore, we are interested in constructing the NS-forms of the operator functions e ~ts
and
(e~'~ -- I)~-~
OE,1 =
(4.1.4)
(4.1.5)
,
for example. In the following we assume that the function f is analytic. In computing solutions of (1.1) (via, e.g., (4.1.3)) we can precompute the N S-forms of the operator functions and apply them as necessary. We note that if the operator function f is homogeneous of degree m (e.g., m = 1 and 2 for the first and second derivative operators), then the coefficients appearing in the NS-form are simply related, see e.g. (3.4.1). On the other hand, if the operator function f is not homogeneous then J , ~ ,k', and k,~ via (3.3.6) and compute the coefficients ak,k, we compute Sk,
7g,k, via equations (3.4.2) for each scale j - 1, 2 , . . . , J _< n. We note that if f is a convolution operator then the formulas for sk_ 0 k, are considerably simplified (see [5]). We first describe computing the N S-form of an operator function by projecting the operator function into the wavelet basis via (4.1.1). To compute the coefficients J = 2-J 8k,M
/_~oo oo ~ ( 2 - J x -
let us consider f(Ox)~(2-Jx-
k)f(Ox)~(2-Jx-
If?
k') - x / ~
k')dx,
oo f(-i~2-J)~(~)e-iek'ei2-~Xed~,
(4.1.6)
(4.1.7)
where ~(~) is the Fourier transform of ~(x),
~(~)- ~
1 / _ ~~176
~ ~o(x)e~dx.
(4.1.8)
Substituting (4.1.7) into (4.1.6) and noting that 8k,kl j " , we arrive = S~_k, at
s~ -
f (-i(2-~)l~(()12e~d(.
(4.1.9)
Pseudo-Wavelet Algorithms for Nonlinear PDEs
161
We evaluate (4.1.9) by setting
~' -
f(-i2-J(~ + 2~k))l~(~ + 2~k)l~,
f0 27r ~
(4.1.10)
kEZ or
s~ where
g(~) - ~
g(~)ei~Id(,
f(-/2-~(~ + 2~k))i~(~ + 2~k)l ~.
(4.1.11)
(4.1.12)
kEZ
We now observe that for a given accuracy e the function 195(~)12 acts as a cutoff function in the Fourier domain, i.e., 195(~)12 < e for I~! > 77 for some > 0. Therefore, equation (4.1.10) is approximated to within e by K
(7(~) -
Z
f ( - i 2 - J ( ~ + 27rk))lq5(~ + 27rk)12'
(4.1.13)
k=-K
for some K. Using (4.1.13) (in place of g(()) in (4.1.11) we obtain an approximation to the coefficients s~, N-1
~ - -~ Z ~ ( ~ ) ~ ' .
(4.1.14)
n--0
The coefficients s~ are computed by applying the F F T to the sequence {g(~n)} computed via (4.1.13). In order to compute the NS-form of an operator function via (4.1.2), we use the DFT to diagonalize the differential operator 0x, and apply the spectral theorem to compute the operator functions. Starting with the wavelet representation of 0x on Vo (see Section 3.4 or [5]) of the discretization of 0x, we write the eigenvalues explicitly as L
k, + s-l e-27rz~.k, ), Ak -- So + ~ j ( sl e21r, ~-
(4.1.15)
/=1
where the wavelet coefficients of the derivative, sl - s ~ are defined by (3.3.6). Since f ( A ) - ~ ' f ( A ) ~ --1, (4.1.16) where A is a diagonal matrix and ~- is the Fourier transform (see [37]), we compute f (Ak) and apply the inverse Fourier transform to the sequence
l(~k),
N
~0 _ Z I ( ~ ) ~ ' ~ (~-~)~'-~ , k=l
(4.1. it)
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G. Beylkin and J. Keiser
to arrive at the projection of the operator functions ](0~) on the subspace V0, i.e., the wavelet coefficients sT. The remaining elements of the N S form are then recursively computed using equations (3.4.2). 4.2
V a n i s h i n g m o m e n t s of t h e B-blocks
We now establish the vanishing-moment property of the B-blocks of the N S-form representation of functions of a differential operator described in Section 4.1 and the Hilbert transform. We note that a similar result also holds for the B-blocks of some classes of pseudo-differential operators, see e.g. [31]. Additionally, we note that these results do not require compactly supported wavelets, and we prove the results for the general case. In Section 4.3 we use the vanishing-moment property to design an adaptive algorithm for multiplying the N S-form of an operator and the wavelet expansion of a function. Proposition 1. If the wavelet basis has M vanishing moments, then the B-blocks of the N S - f o r m of the analytic operator function f (Ox), described in Section 4.1, satisfy q-c~
1 ~l - O,
(4.2.1)
l-~--oo
for m - O, 1, 2, . . . , M - 1 and j - 1 , 2 , . . . J.
Proof:
Using definition (3.3.6), we obtain lm~l = l----oo
F
r
k)f(Ox)Pm(x)dx.
(4.2.2)
oo
We have used the fact that -~- CX3
Z
Im~(x - l ) - Pm(x),
(4.2.3)
l'--oo
where Pm (x) is a polynomial of degree m, for 0 _ m ___M - 1, see [30]. Since the function f(.) is an analytic function of 0x, we can expand f in terms of its Taylor series. Therefore, the series for f(Ox)Pm(x) is finite and yields a polynomial of degree less than or equal to m, f (Ox)Pm (x) - Pm' (X),
where m' _ m. Due to the M > m vanishing moments of r integrals (4.2.2) are zero, and (4.2.1) is verified.
(4.2.4) the
Pseudo-Wavelet Algorithms for Nonlinear PDEs
163
Proposition 2. Under the conditions of Proposition 1, the B-blocks of the N S - f o r m of the Hilbert transform
1 /? -f(s) - - ds,
(7-lf)(x)- -p.v. 7r
oo
8 --
(4.2.5)
X
(where p.v. indicates the principle value), satisfy" nt-oo
E
lm/~i' - O,
(4.2.6)
l=-c~
for0 e}
kE]F2n-
whereas for the error we have
[[(Pof)~(z) -(Pof)(x)[[2 --
(4.3.6)
j
/
Z Id~ [2 j=l {k:ld~l_<e}
<eglr/2,
(4.3.7)
where Nr is the number of coefficients below the threshold. The number of significant wavelet coefficients is defined as Ns = N - Nr, where N is the dimension of the space V0. We define the e-accurate subspace for f, denoted D) C V0, as the subspace spanned by only those basis functions present in (4.3.6), D) - V j U {span {r
" 14[ > e},
(4.3.8)
for 1 p. In any event we are led to evaluate the double sum in (5.2.3), which can be done using the adaptive pseudo-wavelet algorithm described in Section 5.1.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
175
If the function f is not analytic, e.g., f ( u ) - iul, then the primary concern is how to quantify an appropriate value of jo, i.e., how much refinement (or 'oversampling') is needed to take advantage of the interpolating property (5.5). On the other hand, determining jo may become a significant problem even if f is analytic. For example if the Taylor series expansion of f ( u ) does not converge rapidly, e.g., f ( u ) - e u, we may be led to consider the following alternatives. In the first approach, we begin as above by expanding e u in the 'telescopic' series J
eP~ -- egJlt -- E
egJ-lU -- egju'
(5.2.4)
j--1
and using P j - 1 - Q j + Pj to decouple scale interactions. We then arrive at J e P~ - e PJu + ~ eP~U(e Qju - 1). j=l
(5.2.5)
Since the wavelet coefficients Q j u are sparse, the multiplicand e Q~u - 1 is significant only where Q j u is significant. Therefore, we can evaluate (5.2.5) using the adaptive pseudo-wavelet algorithm described in Section 5.1, where in this case the mask is determined by the significant values of e Q j u - 1. The applicability of such an approach depends on the relative size (or dynamic range) of the variable u. For example, if u ( x ) - a sin(2~x) on 0 0, and for each a threshold c such that
s
there exists
117:'J(u) - IL (u)IlL= ~ ~, The fact that the collection {wJ'k}(n,j,k)eCjx z is a Hilbert basis (complete orthogonal system) in Vj means that there exists a finite linear combination of elements of this collection such that
Proof:
IIVJ(u)
c~kw~'kllL2 r (n,j,k) E I • Z } ,
(2.2.4)
- for the Shannon entropy :
HI(u)--
~
14'kl 2 lnl kl.
(2.2.5)
(n,j,k)EI•
For a given entropy, the best basis of u will be defined as the one that has the smallest entropy. Of course, the best basis depends on the chosen entropy. Our interest in this discussion lies in the possibility of minimizing the number of degrees of freedom to approximate a function to given accuracy. If the cardinal entropy is chosen, the number of degrees of freedom (given by the "significant" coefficients that are larger than ~ in absolute value) will be the entropy. If the Shannon entropy is chosen, the size of the coefficients comes into play, and the number of degrees of freedom will be given by the number of significant coefficients in the best (Shannon) basis representation. Thus, for a given ~, the definition of He~ leads to the smallest number of degrees of freedom. This is important for the numerical simulation as it directly influences the size of discretization matrices we have to work with. Another reason for preferring the cardinal entropy is that it is much cheaper to evaluate. In Subsection 3.2 we illustrate numerically the comparison between the two entropies for a less obvious criterion: the accuracy with respect to the number of degrees of freedom (see Figure 2 in Subsection 3.2).
P. Joly et al.
208 2.3
A d a p t a t i o n to t h e p e r i o d i c case
Our interest is to introduce a discrete method based on the wavelet packets to approximate the solution of some initial boundary value problem governed by partial differential equations. The boundary conditions can be either of Dirichlet, Neumann or periodic type. The main feature of the method is to be intrinsically adaptive inside the domain, hence the problem of the correct treatment of the boundary conditions is a separate issue. This allows us to present the method in the case of periodic boundary conditions. The corresponding bases will be derived from the ones on IR described previously by periodization of the generator functions win'k as follows:
f ( x + r).
V f E L2(IR), f(x) - Z
(2.3.1)
rEZ
It is then straightforward to check that VjEIN,
Vk" 0_k 0, x e [0, 1],
(5.1.1)
with initial condition
u(0,
= u~
1 . The approximation involves a discrete and viscosity parameter v - 400~ parameter in time, At, that is a positive real number, and a discrete parameter in space, J. The solution u ( m A t , .) is approximated by an element u m - u ~ of V J defined by the following algorithm.
ALGORITHM: Initial step: Let A(u~ be the indices of the best basis associated to u~ (see equations (4.3.2-4.3.3)), the notation ~o and A(u~) is as explained in Section 4.3. Set A ~ - fi.(u ~g). (m + 1) th step: Let u TM be given by:
Z
(j,n,k)EA m
where A TM C A ( u ~ 9 i) Define first ~m and ~m as explained in (4.3.1). 9 ii) Define A m+l as follows: A m+l - { ( j , n , k ) e A(u~
that are neighbors of ( j ' , n ' , k ' ) e rim},
(for the definition of neighbors see Subsection 4.3). 9 iii) Compute u m+l written as" U m + l --"
~
Ol.~k(~'~+ 1) Win'k ,
(j,n,k)EA m+l
such that it satisfies the discrete problem: V ( j , n , k ) E A m+l,
P. Joly et al.
224 Urn+l_
fo
At
(tm
w~'k dx
= js
dftm* dwJn,k 1 f01 d(fi,~) d x - -~ dx 2 w j'k dx, dx dx
-u whereu,m = ~3 U m - ~ 1 U m -
1.
The time discretization that is presented here is the simplest one and, of course, can be easily improved, but this is not the point of this paper. Remark. The previous algorithm will certainly tend to increase the dimension of A m+l . Thus, in order to keep the size of the basis for the representation of u minimal, we have from time to time to compute again the best basis of u m and use the set of indexes A(u m) in place of A(u~ for further time steps (see the next section).
5.2
Details on the implementation
In order to solve (5.1), we need to construct the stiffness matrix related to this adaptive problem. Since the linear set spanned by the selected best basis functions (with indices in A TM ) which has to be used to perform the Galerkin procedure changes at each time step, and since from time to time the definition of the best basis may also evolve, it is worthwhile to have some computation and storage done in a preliminary stage. For this purpose, we consider the stiffness matrix with respect to the entire basis (~J'k)o 0. Here H = H~ (fl) or H H1 (Vt), respectively, where H ~ ( f l ) is the closure of C ~ (Ft) with respect to the [!" IIg m-n~ Of course, when A is the biharmonic operator, one has H - H02(~t) or similar spaces incorporating mixed boundary conditions. The second example represents a different type of problems which still fits into the present framework. In order to solve an exterior boundary value problem AU-0
in IR3 \ f l ,
U-f
on 0~t,
(2.8)
it is tempting to transform this problem into a boundary integral equation. For instance, the indirect method leads to the equation on the boundary
Au - f,
A - I + 2K
(2.9)
where K involves the double layer potential operator
Ku(x) "-[1/2-
Oa(x)]u(x) +
( x - y) iy - xl (y)dy,
(2.10)
and 0~(x) denotes the interior angle at x E 0 ~ located on an edge of 0~. In this case it is known that H - L2(F), F := 0gt. Solving (2.9) for u leads to the solution of (2.8) by evaluating a singular integral.
Nonlinear Approximation and Adaptive Techniques
241
Alternatively, the direct method yields an integral equation of the first kind
Au - Vu-
1
(-~ + K ) f,
where V is the single layer potential operator
47rl x _ y] dy.
Vu(x) 012
It is known that this fits into the above framework with H - H-1/2(F), H* - HI/2(F). Of course the latter context requires a proper definition of Sobolev spaces on surfaces or manifolds. This is usually done by "lifting" Sobolev spaces from domains in IRd with the aid of an atlas and a partition of unity (see e.g. [9, 19]). We shall comment on this issue later in more detail. Thus in what follows, H typically stands for a Sobolev space H t = Ht(f~) (or some subspace determined by boundary conditions) while for t _> 0, H -t is to be understood as the dual of H t. We conclude by pointing out one more property of the class of operators under consideration. Note that in the above examples A has support of measure zero or has a Schwarz kernel with certain asymptotic properties. More precisely, we will assume that
(Av)(x) - f K(x, y)v(y)dy, f~
where K;(x, y) is smooth off the diagonal x - y and where we require that whenever d + p + Ipl + [r'l > 0 v tOx/.t CgyK~(x, Y)I r
(3.1.18)
Then ~ (f) E H a+s (IR d) if and only if f E Ha(IRd), s + a E (-~, 7) and
(3.1.19)
IlZs/IIH,+o(R,') • II/IIHo( ").
The Besov spaces Bq(Lp(IRd)), a > O, 0 < p,q < ~ , are smoothness spaces in Lp(lRd). The index a is the primary index and gives the order of smoothness (analogous to the number of derivatives). The second parameter q gives a finer scaling. For example, the space B~(L2(IRd)) = Ha(IR d) and B~(Lp(IRd)) is the Lipschitz space of order a in Lp(IRd), provided a > 0 is not an integer. Besov spaces are usually defined by Fourier transforms or moduli of smoothness (see e.g. [25]). There is, however, an equivalent definition in terms of wavelet decompositions which we shall employ. In fact, for the above range of a one has ~" ~ IIfl] B;(/,(nd)) q "~ IET) r
III-q(a/d+l/U-1/P)l(f,r
E
< co,
(3.1.20)
o
whenever .f E Bq(Lp(IRd)). We shall mainly be concerned with a particular scale of these spaces which will replace the role of the Sobolev spaces when treating nonlinear approximation. If a > O, we let
7 "-(aid + 1/2) -1
(3.1.21)
so that 7 < 2. Then the Besov space Ba(IR a) "- B~(L.,.(IRd)) is the set of all functions in L2(IR a) which have a wavelet decomposition (3.1.14) and the wavelet coefficients of f satisfy
IISII o(, ) •
I(f,r
< cr
(3.1.22)
S. Dahlke et al.
246
Then, (3.1.22) gives an equivalent quasi-norm for B a. Note that B~ d) = L2(IR d) with equivalent norms. As a gets larger, the spaces Ba(IR d) get smaller" Ba(IR d) C BZ(IRd), a _ ~. Wavelets of the above type are still of limited use for the numerical treatment of operator equations. Below we will briefly indicate how to obtain wavelet bases with the above properties (3.1.16), (3.1.17), (3.1.22) in several other cases of practical relevance. 3.2
W a v e l e t s on t h e i n t e r v a l
As the simplest example of a bounded domain let us consider first gt = [0, 1]. This case deserves particular attention because it will also serve as a core ingredient in constructions for more complex situations. The common starting point (see e.g. [1, 8, 12, 17]) is to construct collections Ok = {r : m E Ak} C L2([0,1]) such that the spaces Sk := span Ok are nested and contain all polynomials up to a certain desired degree. Taking some dual pair ~,~5 as in (3.1.3) and fixing such that for k > ko we have supp~(2 k. -m),supp~5(2 k. - m ) C (0, 1), m = t~,..., 2 k - e, the collections Ok are comprised of these interior translates 2k/2~(2 k. --m) together with certain boundary functions which are needed to preserve the desired degree of polynomial exactness. If ~ has exactness order N, these boundary functions are simply obtained by truncating the expansions (3.1.12). For instance, for the left end of the interval one adds the N functions L
~k,t-~+ ~(x) "-
E
k0 we can still identify the indices (k,m) with dyadic cubes I = 2 - k ( m + [0, 1]). Defining 7) + "- Uk>koZ)k and 79 "- De U Z)+, we obtain essentially the same format as above:
f -- E (f' ~I)r -Jr Icv~
E
(r
•
~)I)~)I,
(3.2.2)
111-2~l(f, ~,)12,
(3.2.3)
(f,
as well as
Ilfll~([o,~]) ~ ~
I(f,,~,)l 2 +
IC79c
~ (r
•
or
I]fl[~([o,1]) x ~ [(f, r I~v~
~+
~ (~,i)c.o •
I(f, r
T,
(3.2.4)
where 7 := (c~ + 1/2) -1 . Of course, in this case one has # ~ ~ = 1, but in anticipation of the tensor product case below this redundance is useful. Also one should note that for notational simplicity, we have suppressed the fact that, due to boundary modifications, ~~ actually depends on I. Again the range for which (3.2.3) is valid is ( - ~ , 7), where "7"-sup{a'~eH
a(IR)},
~.-sup{a.qSeH
a(]R)}
(see [17]). The case c~ - 0, of course, recovers the Riesz basis property. Since by construction the spaces Sk "-- span ~k are exact of order/V, one has
/
1
xrr
- O,
I e D +,
r - O,...,N -1.
(3.2.5)
0
3.3
T h e i s o p a r a m e t r i c case
Taking tensor products of wavelets on [0, 1] immediately yields biorthogonal wavelet bases on the unit d-cube [::1 "- [0, 1] d with analogues of (3.2.3), (3.2.4), (3.2.5). One can push this line a little further in the following direction. Suppose that for some d ~ _> d, ~ is a regular mapping from IRd into IRd', i.e., ~ is smooth and its J acobian is bounded away from 0. Let
S. Dahlke et al.
248
f~ . - n(r-1). Sobolev spaces or Besov spaces on Ft can be defined by lifting corresponding spaces from [2 with the aid of n. In fact,
(f , g) .-- / f(~(x))g(~ (x)) Idet ~' (~ -1 (x))ldx []
is a natural inner product which can be used to define Sobolev norms. On the other hand, / .
(f, g) "- / f(a(x))g(a(x))dx
(3.3.1)
. I
[]
induces equivalent norms whenever a is sufficiently regular. Taking tensor products of the above mentioned wavelets on the interval readily yields biortho~gonal wavelet bases 9 - { r 9 I E Dc} U {r " r E ~o, I E D+}, - {r " r E ~} on D. Here we have used the convention r "- r for I E De. Of course, in this case one has # ~ o _ 2 d - 1 and the structure of the sets De, D + is clear from the tensor product construction. Then the collections .-
.-
o
e
(3.3.2) t~f) .-- {~/~ .-- ~ I o / ~ - 1 . ~ I E i~},
are obviously biorthogonal relative to the inner product (.,.) in (3.3.1), which again satisfy (3.2.3) and (3.2.4). The moment conditions take the form (P,r 0, ( r E ~o • D+, (3.3.3) whenever P o n-1 is a polynomial of coordinate degree less than .N. Here and in the following we reserve the notation De for those dyadic cubes associated to the scaling functions on the coarsest level. The importance of this case will become clearer below. 3.4
W a v e l e t s on m a n i f o l d s
When d = d' = 2 the above construction yields, for instance, wavelet bases on various planar domains. However, the case d' > d is important too. In fact, the examples in Section 2 show that one needs wavelets defined on manifolds which are embedded in some higher dimensional Euclidean space. The simplest case is the d-torus. Functions defined on the d-torus correspond in a one-to-one way to l-periodic functions f(x + rn) = f(x), m E Z d. Clearly every compactly supported function r/in L2(IR d) is easily periodized by [r/](x) "- ~ ~(x + k). (3.4.1) kEZ d
Nonlinear Approximation and Adaptive Techniques
249
Moreover this is easily seen to preserve orthogonality relations,
f g(x)f(x)dx
-
0
]Rd
~
f[f](x)[g](x)dx
i.e.,
=0.
El
Thus, given wavelets r ~I on IRd, the functions [r [~I] form corresponding wavelet bases on the d-torus. The ease of this construction is exploited in many papers. Again, the case d = 1, the circle, deserves special attention. Suppose C is any smooth closed curve (without self-intersection) in IR2. Then C can be written as a parametric image C = ~([0, 1]) of a smooth 1-periodic mapping ~. Thus combining periodization with the isoparametric approach from Section 3.3 immediately provides wavelet bases on C giving rise to analogous norm equivalences and moment conditions. These wavelets can be used to discretize, for instance, boundary integral equations of the type mentioned in Section 2, arising from exterior boundary value problems for planar domains with smooth boundaries. When the curve is not smooth but has corners, it may have to be subdivided into smooth sections and wavelet bases can be obtained by piecing together parametric images of wavelets on the interval. This gives stable bases for L2. However, for the characterization of smoothness spaces, this is not sufficient. Here the transition between adjacent segments requires special care. We will briefly indicate a systematic approach to this problem below in the context of a more general situation. Note that example (2.10) requires wavelets defined on two-dimensional closed surfaces in IR3. In such a case periodization does not help. Instead one can use the tools developed in computer aided geometric design where such surfaces are modeled as a union of parametric patches. Thus assume that F is a piecewise smooth d-dimensional manifold of the form M
F-
[..Jf'i,
FiNFt-0,
i~l,
(3.4.2)
i--1
where Fi = tci(O) and tci are regular sufficiently smooth parametrizations. Again one can consider function spaces 9V(F) where 9V(F) = H~(F) or F ( F ) - B~(Lp(F)) and the range of c~ depends on the global regularity of F. For instance, when F is at least Lipschitz it makes sense to consider Sobolev spaces with index c~ < 1. For practical purposes and for the sake of constructing wavelets on F the characterization of 9v via an atlas and a partition of unity is rather useless. An interesting alternative was offered in [9] where a characterization of $'(F) is directly based on a decomposition of F into patches Fi. The following brief indication of the basic ideas is taken from [19] where an attempt is made to make the existence statements
S. Dahlke et al.
250
from [9] constructive and where the details of the following comments are given. First one orders the patches Fi in a certain fashion. If Fi n Fl "-- ei,l is a common face and i < l, then ei,l is called an outflow (inflow) face for Fi (F1). 0F~, 0F~ are called the outflow and inflow boundaries of the patch Fi. Let F~ denote an extension of Fi in F which contains the outflow boundary OFt in its relative interior, and whose boundary contains the inflow boundary 0F~ of Fi. Now suppose that Ei is an extension operator from the domain Fi to F~ such that m
IIE~flI~(F,) ~< IlfllJ=(F,), II(E~'f)l"ll:=(r,), f~r mEAk
Nonlinear Approzirnation and Adaptive Techniques
253
take L2 (ft) into L2 (IRd) and its adjoint
(Pf~)* f "- E
(f ' Cek,m)~pk'm
rnEAk
takes L2(IR d) into L2(f~). Moreover, the mapping OO
E f " - p eko s "Jr Z
e - P ke) f (P/~+I
(3.5 . 5)
k=ko is an extension satisfying (3.5.1) for any r < 7. Due to biorthogonality, evaluating E*(bi requires computing the inner products (~I, q5ek,m)rte for levels k larger than the level of I. In numerical implementations, by using decay properties, this can in turn be restricted to finitely many levels depending on the required accuracy. In view of the above comments, we shall assume in the following that we always have a pair of biorthogonal bases 9 - {@I "~) C ff~, I C De U D + } and ~ - { ~ I " ~ E ~ , I C Dc tO D +} where Dc corresponds to functions on the coarsest level, while for I E 79+ the ~t, ~ I play the role of wavelets. Again the sets ~} will generally depend on I but will always contain at most finitely many functions. Setting as above D := 79c tO79+, on one hand moment conditions of the form (3.2.5) or (3.3.3) hold, while on the other hand relations like
IlSll r (r,(a))
• ~ ~ III-q(~/e+l/2-~/V)l<S,~I>l q I ED ~bEq~
(3.5.6)
and
Ilfll (n)
IIl- /al(f,6,)t
c~ C (-5', 7),
(3.5.7)
are valid when ft is a domain or manifold of dimension d as discussed above. As in all the above examples, it will be convenient to identify always the indices I with dyadic cubes of volume III. So far, we have outlined several principles to construct wavelets on various types of domains and manifolds. If one wants to employ such wavelet bases for solving an operator equation, the issue of boundary conditions is, of course, important. When dealing with boundary integral equations on a closed manifold this problem does not arise. It may also not be severe in connection with natural boundary conditions for elliptic problems on bounded domains. Appending essential boundary conditions is, in principle, a possibility to avoid incorporating boundary conditions in the trial spaces and to preserve possibly
254
S. Dahlke et al.
many favorable properties of wavelets defined on simple domains [41]. For domains which can be represented as a union of parametric images of a cube, the approach outlined above also facilitates incorporating essential boundary conditions in the wavelet spaces. We dispense here with elaborating further on this issue and refer to [19, 36] for details of corresponding recent progress in this problem.
3.6
Wavelet discretization of operator equations
We return now to the operator equation (2.1) where in the following H = H t and H* = H - t , where either H t = H t (gt) when f~ is a closed surface, or when natural boundary conditions are assumed, or H t is a closed subspace of H t (Ft) determined by boundary conditions so that A is injective on H t. In fact, we will assume that (2.2) holds with H = H t. The standard Galerkin method for approximating the solution u of (2.1) begins with a finite dimensional space S c H t and finds the function u s E S such that ( A u s , s) = ( f , s), s e S. (3.6.1) By choosing a basis {sk } for S, (3.6.1) becomes a system of linear equations (a(si, s j ) ) i , j c = f,
(3.6.2)
with f := (fi) and fi := (f, si), c the vector of coefficients of u s with respect to this basis and the matrix (a(si, sj))i,j the stiffness matrix. In the sections that follow, we shall be interested in the efficiency, in which u s approximates the exact solution u of (2.1). The typical choices for S in the standard finite element theory are spaces of piecewise polynomials on some partition associated to f~. An analogous choice in the context of wavelets are spaces S = S j : - span{r : r C ~ , III < 2-Jd}, or more generally S = SA := span {r : (r I) C A}, where A is some finite subset of V := { ( r r E ~ , I E :De U :D+}. The efficiency of Galerkin methods depends on: (i) the approximation power of the spaces S, (ii) properties of the stiffness matrix (condition number and sparsity). We shall see in the following sections how the accuracy of the approximation of u s to u depends on the regularity of u. The properties of the stiffness matrix, including its amenability to preconditioning, is a central theme in finite element methods amply reported on e.g. in [16, 18, 48, 49]. Wavelet discretizations offer the following advantages with regard to (ii). To describe this, for A C V as above, let PAY "-
(Y, (r
(3.6.3)
Nonlinear Approximation and Adaptive Techniques
255
Note that under the assumption (2.2), which we will quantify as
c'~llAvll.-, < Ilvll., 0 and choose an initial admissible tree J0. Set Bs = { J E J:(Jo) : R ( J ) > e}, Js = J0. If B6 = 0 stop. Otherwise, for I E B~ do: 9 replace J~ by J~ UC(I). 9 replace Bs by (B~ \ {I})U {J e C(I) : R ( J ) > e}. Since [ ] P j ~ f - fl[L2(n) ~ 0, n --+ oc, Jn -- {I E 7;)'11 [ < 2 -nd} the above algorithm terminates for every r :> 0 after finitely many refinement steps, i.e., eventually one obtains BE = q), and the resulting tree J~ has the property that R ( J ) 0 and T := ( a i d + 1/2) -1 be as in Theorem 2. If g e B ~ ( L ~ ( ~ ) ) for any/~ > a and # > 7, then
IIg- PJ.flIL ( )
1, the right-hand side f is contained in L2(fl). Therefore, On the other hand, Theorem 8 implies that u E H3/2(f~) - ~ -~3/2(L2(fl)). 2 we know from Theorem 12 that u E B~(L,(a)),
By interpolation and embeddings of Besov spaces, we can conclude that u is in a family of Besov spaces B~(Lq(f~)) for a certain range of parameters q and s, i.e., u E Bq(Lq(f~)) whenever ( 1 / q , s ) i s in the interior of the quadrilateral with vertices (1/2, 0), (1/2, 3/2), (s*/d + 1/2, 0), (s*/d + 1/2, s*). Therefore, to compute the range of parameters s for which u is contained in B~(L~(f~)), T = (s -- 1)/d + 1/2, we have to determine the intersection of the lines 1 1 2s* and
1 q
1 2
s-1 d
which is the point ( s * / ( 3 d ) + 1/2, s * / 3 + 1). An application of Proposition 1 with t 1 yields the result. To illustrate this result, we consider the example where d - 2. If c~ >_ 2, then s* - 3. Hence, in this case, the nonlinear method gives an H 1approximation to u of order up to n -I/d, whereas a linear method using n terms could only give n - 1 / 2 d in the worst case. In general, a priori knowledge about the Besov regularity of the solution u to (2.1) would give lower bounds for the errors produced by any adaptive method. Conversely, if we knew that a particular adaptive scheme
271
N o n l i n e a r A p p r o x i m a t i o n and Adaptive Techniques
is asymptotically as efficient as best n-term approximation in I[" IIH', its performance would allow us to infer the regularity of u. Of course, since the wavelet coefficients of the solution u are not known a priori, one cannot apply Remark 3 directly. There are several possible ways of dealing with this problem. Let d~(g) denote the sequence of wavelet coefficients of g relative to 9 , i.e., d~,~(g) - (g,r A C V, and analogously d,i,(g ). By (3.6.12) the solution u of (2.1) is determined by dq~(Z_tu) = A - l d c , ( Z t f ) .
(7.1.6)
Recall from ( 7 . 1 . 2 ) t h a t the best n-term approximation of u in I1" IIH' corresponds to the best n-term approximation of Z _ t u in the L2-norm I1" IIL~ which, by Remark 2, corresponds to selecting the n largest terms Id,~,;~(:T_._tu)l = IAI-tld,~,)~(u)l. It is known that in certain cases the decay properties of (3.6.13) of the infinite matrix ,4 imply similar decay properties for A -1, perhaps with different parameters, see e.g. [51]. In such a case the largest coefficients of Z - t u are expected to appear in a 'neighborhood' of the (accessible) largest coefficients f~ - IAItd,i,, ~ (f). The effect of the smearing caused by the application of A -1 can in principle be estimated by the same methods as used in connection with matrix compression [18]. However, this assumes that the singular behavior of u is completely governed by the righthand side f. Next we shall describe a somewhat different approach from that of [14]. To motivate this let us briefly recall first a basic strategy employed by many adaptive finite element schemes. A key observation is the equivalence between the validity of two-sided error estimates and the so-called saturation property. This issue is discussed in [5] in the context of finite element methods. The basic reasoning can be sketched as follows. Suppose that S C V C H t are two trial spaces with respective Galerkin solutions u s , u v . By orthogonality one has
l i l y - ~sll _ I1~- ~sll, where I1" II denotes again the energy norm. Moreover, one easily checks that one has II ~ - uvll < ~11'-'- usll
(7.1.7)
for some/3 < 1 if and only if
(1 - 9~)'/~11~ - ~sll < lily - ~sll.
(7.1.8)
Thus, if the refined solution u v captures a sufficiently large portion of the remainder (7.1.8), the global energy error is guaranteed to decrease by a factor/3 when passing to the refined solution u v and one has the bounds
It~v - ~ s l l _
I1~- ~ s l l _ (1 - / ~ ) - ' / ~ l l ~ v
- ~sll,
(7.1.9)
272
S. Dahlke et al.
which are computable. In practice, one controls the local behavior of u v us and refines the mesh at places where (an estimate for) this difference is largest. This results in balancing the error bounds. Although this has been observed to work well in many cases, the principal problem remains that something like (7.1.8) has to be assumed to prove convergence of the overall adaptive algorithm. It is perhaps worth stressing that wavelet analysis allows us to remedy this conceptual deficiency and derive much stronger information about remainders. In fact, we shall see below that the assumption (7.1.8) about the unknown solution u can be replaced (quite in the spirit of the previous comments) by some (rather weak) information on the accessible data f. To this end, let us first relate the type of estimates (7.1.9) to n-term approximation. Instead of minimizing the error for a given allowance of n terms one can minimize the number of terms needed to meet a given error tolerance. Specifically, given any strictly decreasing sequence {~'i}ie~, we can look for a sequence {A(Ti)}ielN of index sets A(Ti) C X7 such that a#i(r,),t(U) • Ti,
i e iN.
(7.1.10)
The following observation is an immediate consequence of Remark 3. R e m a r k 4. One has I 1 ~ - PA(~,)~II • ~ ,
i e IN,
(7.1.11)
and the sets A(Ti) can be chosen to be nested, i.e., A(Ti) C A(Ti+I),
i e IN.
(7.1.12)
Let, with a slight abuse of notation, UA denote the solution of Galerkin problem (3.6.1) with S := SA := span {r
: A E A}.
If A C/~ we have II~ - ~x II~ - I I ~ - ~All ~ - I I ~ A - ~xll ~,
since the Galerkin approximation is an orthogonal projection relative to the energy inner product. Therefore, we obtain R e m a r k 5. Consider the following sequence {Ai}ie~: (i) Fix some A 1 C V and a < 1. Define T1 := IlU- UA1 II.
Nonlinear Approximation
and Adaptive Techniques
273
(ii) Given A i, choose A i+1 c V, A i C A i+1 such that II~A'
-
ZtA'+l
II _> ~llu - UA, It,
(7.1.13)
while for any A C V with A i g A and # ( A \ A ~) < # ( A i+1 \ A i) one has (7.1.14) [lUm, -- UAII < ~ l l u UA, II, Set ~ + 1 -
(iii)
(1 - ~2)1/211u - UA, II,
Replace i + 1 by i and go to (ii).
Then one has I l u - ~A~ II • ~ # A o + c , , ( ~ ) ,
n ~ IN,
(7.1.15)
where c is some constant. In practice, it will generally not be possible to realize the above strategy of capturing a significant portion of the remainder by a possibly small set of additional indices, since the exact estimation required in (7.1.13) and (7.1.14) is generally not possible. However, it will be possible to bound quantities of the form IlUA --us for A C A, from below and above by computable local quantities times constants which are independent of the sets A, A but different from one. 7.2
A posteriori error estimates
Suppose that for some A C V, SA is the current trial space and that we have computed the solution Uh of (3.6.1) (within some appropriate tolerance). According to Remark 5, the next step is to estimate the error I l u - Uhll in the energy norm in a way that indicates how to select next a bigger set A C V, A C A, of wavelet indices so that on one hand, A stays still possibly small while on the other hand, the error Ilu - u AII is guaranteed to decrease by a certain amount. As mentioned before, selecting the index sets A implicitly corresponds to creating possibly nonuniform meshes. In fact, the spaces S n - - span{r : 1~1 _< 2-n} correspond to uniformly refined meshes, and taking only subsets of the complement bases {r : I~1 = e} corresponds to a nonuniform refinement. To this end, we exploit the commonly used fact that the error in the energy norm can be estimated by the residual in a dual norm which, at least in principle, can be evaluated. In fact, since rA "-- AUA -- f -- A ( u A -- u),
by (3.6.4) and (2.5), one has C1
IIrAIrH-, ~
Ilu - ~All ~ c~ IIrAIIH-, 9
(7.2.1)
274
S. Dahlke et al.
Expanding the residual rA by the dual basis ~ and taking the Galerkin conditions P2 AUA -- P i f (7.2.2) into account, yields
rA-xev
xeV\A
Bearing (3.6.6) in mind, and quantifying the constants in (3.5.7), ensures the existence of finite positive constants c3, c4 such that
C3( E ]'~12tI(rA'@X)I2) 89-~ IIrAl[H-t -~ C4( E IAI2t I(rA, %Dx)12)1 XCV\A XCV\A (7.2.3) Thus, in principle, the nonnegative quantities -
.-
:
* I 0 positive numbers C1, C2 such that C2/~+2t -~- 2
5
(7.2.5)
~2 < C.
For each A E V, we define the influence sets V~,c
9-
{~' e v
9 Illog ~1 - [log ~'11 ~< log 2 ~2 x and m i n { l $ 1 - 1 , I~'1-1} d i s t ( ~ ,
~x,)
< c11 },
where f ~ denotes the support of ~ . The sets V~,~ describe that portion of the sum E~,i - Z (Ar r A'EA
appearing in the residual weights ~ (7.2.4), which is significant. In fact, the remainder e,x "E (Af,x,, r ~'EA\V~,~
can be estimated as follows [14]. Proposition 3. For e x and ~ , ~ as above there exists a constant c5 independent of f and A such that
( ~
~V\A
I~Xl=~I~,1=)-~ ~ c5~ II~AII.
('7.2.6)
Note that, again by (3.5.7),
II lr II ll.,
(Z
)~EA
so that the right-hand side in (7.2.6) can be evaluated by means of the wavelet coefficients of the current solution UA. Moreover, one can even give an a priori bound. In fact, the stability of the Galerkin scheme assured by (3.6.5) says, on account of the uniform boundedness of the P~ in H -t, that
IluAII ~< IIP~flIH-, ~< IlfllH-*.
(7.2.7)
276
S. Dahlke et al.
As for b) above, by construction, the significant neighborhood of A in V\A NA,~ "-- {A E V \ A 9A N Vx,~ r 0} (7.2.8) is finite: #NA,~ < co. Outside NA,~ the quantities 6~ in (7.2.4) are essentially influenced by wavelet coefficients of f. But this portion is essentially a remainder of f. In fact, by (3.5.7), 1
XeV\(huN^,~) _< c6
inf
VE,.~AO NA, ~:
IIf-
VlIH-, ~ c6
in f I I f -
VESA
vllH-,,
for some c6 < oo. This suggests defining I
d),(A, e ) " - ]Altl I
~
I
(Ar
,~,)u~,, I'
A E V \ A.
I
A'EAAVx,~
Note that, in view of (7.2.8), dx(A,e)-0,
AEV\A,
(7.2.9)
ACNx,~.
The main result can now be formulated as follows [14]. Theorem 13. Under the above assumptions, one has 1
vllH-,)
+ c;~ [IfllH-, + c6 in_f IIf vESa
)~ENA,e
as well as, ( Z ~ENA,e
d~'(A'e)2) 89
1 Ilu - uAll +
C1C3
c'se [IfllH-~
+ c6 in_f Ilf - VIIH-~" vESA
Moreover, for any A C V, A C A, one has
( E
1
C1C3
[luA
-
UAll+c~ c IlfllH-~+c6 in f Ill yES^
-
VllH-,,
277
Nonlinear Approximation and Adaptive Techniques
This result provides, up to the controllable tolerance
r(A,
~) " -
c~c II/lln-* + c6 in_f II/-- villi-,, vESA
computable lower and upper bounds for the error I l u - uill. Usually under more specialized assumptions, results of a similar nature have been obtained also in the finite element context (see e.g. [27]). Furthermore, one expects that nonlinear problems can be handled by combining such estimates with known abstract results. 7.3
C o n v e r g e n c e of an a d a p t i v e r e f i n e m e n t s c h e m e
In the present setting, it can be shown with the aid of Theorem 13 that under mild assumptions on the right-hand side f a suitable adaptive choice of A enforces the validity of the saturation property (7.1.8). We continue with the notation of Subsection 7.2. The following theorem was proved in [14]. Theorem 14. Let tol > 0 be a given tolerance and fix 0 E (0, 1). Define 9-
--+
ClC3
,
2C2C4
(7.3.1)
choose iz > 0 such that 1-0
#C* _ ~1 zer
Fully Discrete Multiscale BEM
295
the boundary integral operator in (2.12) is strongly elliptic if the direction b(x) is not tangential anywhere on F. Example D: Exterior Stokes flow. Here we are interested, for in determining the velocity field and the pressure distribution Newtonian, incompressible viscous flow exterior to a smooth and surface F in IRa. For illustration we consider the exterior Dirichlet Other cases can be handled similarly, see, e.g., [17]. The governing equations are -uAU+gradp
U
= -
example, (U,p) of bounded problem.
0, d i v U = 0 i n f ~ C , f on F
(2.14)
where f is a prescribed velocity field on the surface of the body satisfying fr f" n ds - 0 and u > 0 denotes the viscosity of the fluid. We require in addition that the fluid is at rest at infinity, i.e. -1
I U ( x ) l - o(1),
I g r a d U ( x ) l - o(Ix I -)2) I p ( x ) l - o(IxL-~), I g r a d p ( x ) l - o(Ixl
for Ixl ~ ce.
The fundamental velocity tensor, the so-called Stokeslet, is given by
{
1
G(z-v)-~
I~-vl
-1
z+
-
(x - y)(x 13 y)7-
I~-v
}"
We represent (U,p) as double layer potentials of an unknown density u : F ~ IR3 as follows: 3
E
U~(x)
j,k=l
/,
/.. uj(y)Tijk(y -- x)nk(y)ds v,
(2.15)
,]1
3
p x,
uj (y)Iljk (y -- X)nk (y)ds v
where
T~k(~) --
(2.16)
3 ~ci~cj~Ck
4~ I:~15
and
Iljk(~)--~
--~+31~15
,
Letting in (2.15) the point x tend to F, we obtain the boundary integral equation (2.5) with the hydrodystatic double layer potential 3
(Ku)i (~)
E
j,k=l
/,
[_ uj(v)T,j~(v Jl"
- ~)nk(v)a~,
9 e
F.
T. yon Petersdorff and Ch. Schwab
296
We observe that due to the classical estimate I( 9 - Y)" n(Y)l < c ( r ) I x - yl ~
x, y e r 3
valid for smooth, bounded surfaces F, the double layer kernel Y~k=l Tijk (X-y)nk(y) admits the estimate 3
Z Tijk(x - y)nk(y) _ max{a2-L2a(L-L)2 a(L-r), 2 -t, 2 -~' }
with some number a > 1.
(3.3.10)
305
Fully Discrete Multiscale B E M Then, for u 6 HS(F), fi 6 HS(F) there holds
I iiAvLiir162 >- c:' iivL IIo - C~-~(d§ ') IIvL llo "
307
Fully Discrete Multiscale B E M This gives, for sufficiently large a, V v L E V L.
(3.3.18)
2. We have
Its- ~!1o 0 depending only on the global shape of F and on the domains of analyticity of the charts {~j } and of the functions s~ in (3.1.8) such that for every u' E blo, u' ~ u, Kjj, (u, v) admits, as a function of v, an analytic continuation (for convenience again denoted by
gjj,(u, ~)) ~o~
~ B(u', r)'- {~ e r
whore0 0 depends only on the domains of analyticity of the chart ~-1. The assertion of the lemma follows from the definition (3.1.8) of the kernel after possibly reducing the value of 7. Since Kjj, (u, v) is a composition of real analytic functions with ranges included in the proper domains of analyticity, it admits a complex analytic extension (e.g.via the classic power-series argument) as a function of v E B(u ~, r). Case ii) j ~ j' and dist(Fj,Fj,) >_ ~ > 0. Here the assertion of the theorem is true for all u, u ~ E L/~ since the kernel is nonsingular and real
314
T. yon Petersdorff and Ch. Schwab
analytic in u ~ E/go for every u E U ~ and vice versa. Notice that the value of r in (4.2.2) must possibly be reduced depending on ~. Case iii) j ~ j' and dist(Fj, Fj,) - 0. This is the case when two surface pieces are adjacent, i.e.Fj n Fj, is either a line or a vertex. Fix u, u ~ E/go such that
-
I> o m
Then it follows as in case i) that for fixed u E/go the function Kjj, (u, v) is analytic in v E B(u~,r) where r is as in (4.2.2) and vice versa. The numerical quadrature rules Q j j , are constructed on the reference domain lg ~ i.e. the unit square S - ( - 1 , 1) 2 or the unit triangle T - {(Xl,X2) 9 - 1 < xl < 1 , - 1 < x2 < Xl}. To this end, the kernel K j j , ( u , u ' ) is mapped from/g~ x/g~', to the reference domain/go via the affine transformations ~
T~'U~ ----+U ~ Denote by
K j j, (U, U')"-- g j j , ((T 1 ) - l ( u ) , (T/:)--1 (Ut))
(4.2.3)
the transported kernel and let
djg, "- dist (r(j), r ( j ' ) ) ,
r ( j ) . - (~ o 7-/) -1 (u 0)
(4.2.4)
denote the Euclidean distance of the images F ( J ) , F ( J ~) of/go under the respective coordinate transformations. We consider first the case where U ~ - S, i.e.we have quadrilateral elements. ~
L e m m a 5. Assume that djj, )> 0 and that/go = ( - 1 , 1) 2. Then there exists a constant ~/ > 0 which depends only on the kernel, the boundary F and its parametrization such that (i) t'or every ul, u2, u~ E [-1, 1], Kgg,(u, u') admits an analytic extension to U~l E s with it ~ p' - 1 + 72 d gj, (4.2.5)
and max max K j j , (u, u') < M~ Ul.U2.Ul2E[--1.1] UllEep,
j,
(4.2.6)
Analogously, for every Ul, U2, U~ E [--1, 1], K j j, (~, U') admits an analytic extension to u~2 E E;, with p' as in (4.2.5) and
I-
I
max max K j j, (u, u') < M / ~I.U2.U~E[--1.11 u~EC.,
j,
(4.2.7)
Fully Discrete Multiscale B E M
315
(ii) Conversely, for every U'l, u'2, u2 E [-1, 1], Kgg,(u, u') admits an analytic extension to Ul E s with
pand max
1 + 721djj,
!
max K g g, (u,
Ull,Ul2,U2e[--1,1 ] Ul eE.
!
) < M~
(4.2.8)
j,
(4.2.9)
Analogously, for every u~, u~, ul E [-1, 1], K j j , ( u , u') admits an analytic extension to u2 E s with p as in (4.2.8) and max
max
u'1,u~,ul E[-1,1] u2es
!-g gg, (U, U') I< M~
j,
.
(4.2.10)
Here the constant M is independent of J and J'. Proof: Assume first that 1 - l' - 0, i.e.T~ is affine, but does not change the area. Then the assertion (i) follows from Lemma 4, since for every point u E U ~ and every u' E U ~ there exists B(u', r) such that K j j , ( u , v) is an analytic function for v E B(u',r). Therefore we can select C2, such that U --6 CC s CC U B(u',r(u,u')). u ' E/d ~
The analyticity of K j j , (u, v) and its homogeneity implies also the bound (4.2.6) for l' - O. The proof of (ii) is analogous. The case l, l' > 0 is then obtained by a scaling argument. We reduce the case that U ~ - T of triangular elements to the case where U ~ - S = ( - 1 , 1) 2 via the degenerate mapping (sometimes also called the "Duffytransformation") u - 9 (~), u' - 9 (~'), with ~ given by
O(~) --
~1 ) --1 -~-(~1 -[- 1)(~2 + 1)/2 "
(4.2 11)
We define in this case the transformed kernel by
~,,. (~. ~'/- ~,. ( ( ~ / l o ~(~/. (~:/-~ o ~(~'/)
(4 ~ 1~/
Note that an application of n x n tensor product Gaussian quadrature to K j j , ( ~ , ~l) in the unit square corresponds to using a conical product rule for K j j , in the unit triangle. To estimate the quadrature error on triangles, we need an analog of Lemma 5 for the transformed kernel (4.2.12).
316
T. von Petersdorff and Ch. Schwab
L e m m a 6. For the kernel Kjj,(~, ~') in (4.2.12) statements (i) and (ii) of Lemma 5 remain true/'or Kjj,(~, ~'), with possibly different, but absolute (i.e.independent of J, J') constants 7 and M in (4.2.5)-(4.2.10). Proof: As in the proof of Lemma 5, we first consider l - l' = 0. As before, we obtain from Lemma 4 the analyticity of the kernel Kgg,(u, u I) defined in (4.2.3) on T x T. It therefore remains to show the following: if f(u) is analytic in T then f(r is an analytic function of ~ E S. When f is analytic in T, there exists Ro > 0 such that
f(u) -
~_, ~1 f(a) (~0)(u- Uo)" V u 0 E T , V [ U - U o [ < R o . c~EIN~
(4.2.13)
We show that (f o ~)(~) can be expanded into a convergent power series about any point ~0 E T for which uo - r To this end, elementary estimates show that I= - uol ~ = I ~ ( ~ ) - ~ ( { o ) 1 2
< 3 I{ - {ol ~
for ~,~o E T. This shows that inserting u - &(~) into (4.2.13) yields a power series for f(r which converges for sufficiently small [ ~ - ~olNow the assertion follows for l = l' = 0 as in the proof of Lemma 5. For l, l' > 0, we use once again a scaling argument. Lemmas 5 and 6 will be used when the surface is subdivided either exclusively into quadrilaterals or exclusively into triangles. The arguments in the proof can, however, also be used in the case of mixed partitions of F consisting of both quadrilaterals and triangles. Corollary 1. Statements (i) and (ii) of Lemma 5 remain valid for the ker-
nels and
o
4.3
Q u a d r a t u r e for t h e n o n s i n g u l a r i n t e g r a l s
We analyze the quadrature for the nonsingular integrals, i.e.for those entries
A~j, - fr fr K(x, y)r162
(y)dszdsy
(4.3.1)
of the stiffness matrix for which dist (suppCj, suppCg,) > 0
(4.3.2)
317
Fully Discrete Multiscale B E M
(note that (4.3.2) implies (cOg, Cg, > - 0). According to (3.2.4), every Cg(x) is a scaled and transported copy of some r a piecewise polynomial basis function of 12r 1. As indicated in Remark 7, we will determine the number of Gauss points in such a way that the consistency estimates (3.3.6) and (3.3.7) still hold for each block. This and corresponding estimates for the singular integrals in the next section imply, via Theorems 1 and 2, optimal (up to logarithmic terms) convergence rates of the computed solution (i.e., with compression and quadrature) in the boundary energy norms as well as at interior points of the domain. Finally, we estimate the complexity of evaluating the nonsingular integrals. The numerical evaluation of the singular integrals is the topic of the following section. Basic error estimate. On the reference element, the multiwavelets Cg(x) are piecewise polynomial functions. This allows to derive quadrature error estimates from corresponding results for polynomial density functions. Lemma 7. Let Lt~ - ( - 1 , 1 ) 2. Let A,A' E IN0 and J = ( j , A , k , u ) ~_ fix, J' - (j', A', k', v') E fix' be such that dgg, > O. Let further
E
0_<max{ C~l,oc2} 1. Since #(u') is, for fixed u~ 6 [-1,1], a polynomial in u~ of degree _< d, it is sufficient to refer to Lemma 5 to show the analyticity of fjg,(u, u') in (4.3.3) as a function of u~. The analyticity of f j j, in the remaining three variables is seen in the same fashion. This completes the proof for the case A = A' = 0. The case that A > 0 and/or A' > 0 is deduced from the previous one by a scaling argument, keeping in mind the growth of 7r and of # for large complex arguments and the bounds for M1 and for M2 derived in Lemma 5. The following lemma presents a corresponding estimate for triangles. L e m m a 8 . L e t U ~ = T = { ( X l , X 2 ) ' - I < Xl < 1 , - 1 < x2 < Xl}. With the notations as in Lemma 7, det~ne
Lg,(v, v ' ) : = fgj,(O(v), r
+ 1)(v[ + 1).
(4.3.5)
Let
~(u)-
Z
o l' be such that djg, _ "),-12-1' (4.3.9) where dgj,^ is as in (4.2.4), and let fjg, (u, u') be as in (4.3.3). Let approximations A j j , to all nonzero entries A j j , of All, for which (4.3.9) holds be computed by ^ A j j, - Q,u' Qn2gjj, (u, u') (4.3.10) where
g j j, (u, u') =
/jj,(u,u') if r(j),r(j') quadrilateral, /jj,(r if r(J) triangular and r(j') quadrilateral, / j j , (u, r if r(J) quadrilateral and r(J') triangular, r if r(J),r(J') triangular, /jj,(r
and the product Gaussian quadratures are applied to each of the four pieces Cg and (bj, are polynomials in local coordinates. on which the multiwavelets ^ We denote by Au, the block of elements obtained from numerical quadrature. Assume further that the nonzero entries of Au, for which (4.3.9) is violated are computed exactly. Then there hold the block consistency estimates Ilfikl,l' -- ~kl,l'][ ~__C2-(d+l)(2L-l'-l)21'-I (4.3.11) oo
[[fikl,l, - fikl,l, ][1 ~_ C2-(d-J-l)(2L-l'-l)21-1',
(4.3.12)
provided we select the number of Gauss points in (4.3.10) according to nl >__9(d, J, J'),
n2 >_ 9(4, J', J),
(4.3.13)
where V~(d, J, J') - ~
+
2 log 2 "7 + l - l '
+ (d + 2)(2L - l - l ' )
2 log2 (1 + .),2t+lt~jj,)
Proof: We apply Lemmas 7 and 8 with ,k = l, ,V = l', 7r - C j , and # - C j, to each of the four pieces on which Cj and C j, are, in local coordinates, polynomials. Due to the way the Cg are normalized, we have
Ii~]!- 0(2*3,
I!~11--0(2*'3,
Hence we have, for J, J' such that (4.3.9) holds, the error estimate
A j j,
-- 2~jj,
]
{(1
~ )--2n2-~-(d)}
+ (1 + .),21'dj j,
(4.3.14)
Fully Discrete Multiscale B E M
321
Consider first (4.3.11). We have here a trial wavelet ~/)J, and a test wavelet Cg of levels I and l', respectively, which satisfy, according to (3.3.5), dgg, ~ 5ll' .'--_....L__
2
1+
2 log 2 7 + l - l' + (d + 2)(2L - l - l') 2 1 o g 2 ( l + 7 2 1 d j j ,)
This implies the asserted bound for nl. The bound for n2 is obtained completely analogously, with 1 and l' interchanged. Lemma 9 addresses only the case that (4.3.9) holds. As explained above, however, many elements in off-diagonal blocks as e.g.ALo are such that (4.3.9) does not hold. This will be overcome by binary subdivision of the larger of the two panels F(J) and F(J') until (4.3.9) is satisfied for all subelements. We assume w.l.o.g that 1 > l' and that diam(F(J')) >_ diam(F(J)). The following lemma shows how to construct the required subdivision of F(J').
322
T. yon Petersdorff and Ch. Schwab
L e m m a 10. Assume diam(F(J')) _> diam(F(J)), (4.3.2)and that (4.3.9) does not hold, i.e., that 0 < djg, < ~'-12 -t'.
(4.3.15)
Then there exists a binary partition {r(j) 9J E A(J, J')} of r(J') such that d j j >__~-12-~ for all J e A(J, J'). Moreover, the number of sube/ements F(J) C r(j') is bounded by Ih( J, J')l _ d + 2 ( / + 1) I l + 1' (4.4.18) respectively n2 _ d +
(d + 1 ) ( 2 L - l - l ' ) 2 log 2 5
l' { log 2 5'
(4.4.19)
nodes where 5 E (1, 4.62) depends only on the regularity of the chart aj (for regular charts it is uniformly bounded away from 1; a precise value was obtained for genera/surfaces in [34]). Then the block consistency estimates (4.3.11) and (4.3.12) are preserved and the total quadrature work for the singular integrals (4.4.9) is bounded by CNL (log NL)2 kernel evaluations. Proof: The function H({,i5) + H({ + i5,-i5) is analytic for i5 ~ 0. It remains therefore to investigate the situation at i 5 - 0. We write H({,iS) + H({ + i5,-i5) - H({,iS) + H(O,-i5) + i5" (01H)(r where r
~),-i5),
~) is analytic. By Lemma 12, this yields
H({,/5) + H(Ct + ~,-~) - Rjj,(?t,)) + Rjj,(?I,-~) + ~. (01H)(~(i5, ~),-i5). Thus the integrand is weakly singular at i5 - 0. The analyticity of the integrand f with respect to { follows from that of the kernel Rjj, of Lemma 12 and the charts, whereas the analyticity with respect to ~ can be shown as in [34], Theorem 1. Next we estimate the quadrature error using a tensor product argument as in the proof of Proposition 7. We use, however, for the nl-point quadrature error the estimate
oh g(x)Zrp(x/h)dx - G nl gZrp( . /h) _ 2d +
(d + 1 ) ( 2 L - l - l ' ) 2 log 2 ~
l' ~ log2 ~'
(4.4.24)
respectively
nodes where ~ is as in Theorem 6. Then the block consistency estimates (4.3.11) and (4.3.12) are preserved and the total quadrature work for the singular integrals (4.4.21) is bounded by CNL (log NL) 3 kernel evaluations.
Fully Discrete Multiscale BEM
335
Proof: From the definition of H in (4.4.9) we see that [dsu[, [dsu,[ are, in the (r ~, 77,O) coordinates, (piecewise) analytic functions with domains of analyticity independent of I and l ~. The product r162 is, in triangular coordinates, a polynomial of the form E C0~0~1~2r~~ cti_~4d
We claim that the integrand is (piecewise) analytic in the triangular coordinates and independent of the level 1. The independence of 1 follows directly from its definition. The analyticity as a function of (77,0) for ~ ~ 0 follows from H(u; v) being analytic in u and also in v ~ 0, since ~ ( 1 - 0) and 1 + ~0 (resp. 1 - ~ and 1 + 77- 0) do not vanish simultaneously for any (77, 0) E [0, 1]2. This also shows the analyticity with respect to ~ provided ~ 0. The analyticity at ~ = 0 (and thus the uniform analyticity in ~) follows from H(u; v) being pseudo-homogeneous of degree - 2 in v. Using the pseudo-homogeneous expansion of H(u; v) with respect to local polar coordinates in v (see, e.g., [33]), we see that the factor ~2 introduced by the triangular coordinates cancels the singularity of H. This shows that the integrand is regular at ~ = 0. Analyticity at ~ = 0 is then shown as in [34]. To estimate the quadrature error, we use a tensor product argument. For the double integration in (~,~) E (0, h) 2, we use the error estimate (4.4.6) and for the integration in (77,0) E (0, 1) 2, we use Proposition 5. This is analogous to what was done in the proof of Theorem 6. We get the error estimate
S - [Sj~ - G~le~le~2e~2 fl ~ C2 l-~l' (2 -(/-[-1)(2n1-[-1-4d)
-~-(~-2(n2-2d)) .
Matching this error bound with the block consistency estimates yields the lower bounds for quadrature orders n l and n2. The work estimate is obtained as in the proof of Theorem 6, bearing in mind that we now have to sum up 22/(nl)2(rt2) 2 over all blocks (thus the power of 1 + 1 in the denominator is reduced by one resulting in one additional power of L).
r(J) and F(J') share a vertex. The final case where supports contain a common vertex is also treated using a special coordinate transformation similar to [10, 30]. With F(u, u') = H(u, u ' - u ) , the integral takes the form 21-[-l'
_
1=0
21+r
2--0
i "--0
ltl F ( - u l , u2; ul, u~2)du2duldu~2du~1 ~=0
fU h ~sl ~uh ~s"1 F(-ul,UlS;Ul,t ~tiS)UldSdul UldS t ' du '1, 1--0
--0
~----0
~--0
T. von Petersdorff and Ch. Schwab
336
using u2 = UlS, u'2 - u2s'. We move the integrals over Ul, Ui outside and split the integral over (ul, u~) E [0, h] 2 into two triangular parts uh
1=0
fu h
~=0
A(ul,u2)du~dul
A(ul, Ul t)ul dt dul
-
+
1=0
0
~=0
0
d(U'lt, Ul)U'ldtdu'1
using u~ = u l t and Ul - u'lt, respectively. The first term in the sum is
2l+l' ~ h
/~
Ul --0
the
second
0
term
~sl ~sl F(-Ul, Ul s; ult, ul ts)u 3 t ds'ds dt dul, --0 ' --0 is
analogous.
Now
note
that
the
function
u ~ F ( - u l , UlS; ult, ults) is analytic, hence standard Gaussian quadrature can be used. We can now estimate the error as in Theorems 6, 7 using the tensor product argument of Proposition 6. This will yield again lower bounds for n l, the number of nodes for the Ul integration, and for n2, the number of nodes for the u2 integration of the form (4.4.23), (4.4.24), with 2d replaced by 3d. The quadrature work is estimated as follows:
L
l
/=0
/'=0
1' (see Remark 9). The quadrature orders are given by Lemma 11 and we assume that ni (k) = ni (J, J', k) are equal to the lower bounds in (4.3.18), (4.3.19). Since by assumption 3' = 1 and 1 > 1~, we estimate
_ 1. The strong Cauchy inequality
l a(vl, wm)[ _< rain{l, 7 m / v / a } [] vl [[a [[ wm [[a
(2.2.4)
holds true for a11 Vl E Vt and for ali Wm E Wm. Further, let j ~ l. Then, ]a(wj,wl)] 1 and a fixed noise level e > 0 the convergence rate Pa tends to zero. Moreover, p,,
-
p,,(O
-
as l
oo,
(3.3.8)
where the constant in the above O-expression does not depend on 1 and L or on a. 2. In view of our general assumption that the noise level ~ > 0 is fixed, the additive iteration admits a convergence rate Pa which is bounded smaller than 1 uniformly in 1 provided that Z]lmin < V/-~/(2C~7 (Cr/-[- 2)) or ~lmin ( ( ~/a0(r The quantity a0(e) was introduced in (2.1.4). In this case/min depends on r and it increases when r decreases. 3. If/min is fixed independently of ~ and 1 then the additive iteration may even diverge for r too small. Still, we can use the additive iteration as a preconditioner for the conjugate gradient method applied to (1.2). For more details on preconditioning the conjugate gradient method we refer, e.g., to Deuflhard and Hohmann [11] and Hackbusch [17]. R e m a r k . Readers familiar with multilevel preconditioners in the context of elliptic PDEs may wonder why we do not replace A -1 as well as B~-1 /min /min l. are scaling function spaces of order N (see Lemma 7 below). The edge functions satisfy, as the interior functions, a kind of refinement equation. The corresponding coefficients are tabulated in [6]. The Daubechies scaling function and wavelet of order 2 are shown in Figure 2 as well as the necessary modifications for the left boundary. R e m a r k . Since the wavelet basis in Wld is an orthonormal basis, the corresponding Gramian matrix Bt coincides with the identity matrix, which simplifies the matrix representations of both iterations (3.1.4) and (3.1.6) (see (3.2.9) for the additive iteration). L e m m a 7". Let Vld be the spaces (4.2.1) with underlying scaling functions ~lOk ~91l,k, and ~z,k of order N Further, assume that f 6 H r(O, 1) for some r 6 { 1 , . . . , N } . Then,
IIf - PlflIL 2 _ 0.001 is numerically sufficient (see Figure 5). In both examples above the convergence rates with respect to the linear splines are clearly smaller than those with respect to the Daubechies wavelet. The main reason for this observation is the higher regularity of the linear B-spline: The Daubechies wavelet of order 2 is Hhlder continuous with exponent 0.55 (see, e.g., Daubechies [9]) whereas the linear B-spline is Lipschitz continuous. The following example illustrates the performance of the iteration (3.2.8) by an approximate solution of (4.1.1) with kernel kl and exact righthand side g(x) x 2 ( x 2 - 4x + 9)/12 + (1 - cos(31rx))/lr2/18. Thus, f*(x) - (1 - x) 2 + cos2(37rx/2). In our computations we supposed that g~ is known only at the discrete points j 51 with g~(j tit) = g(j 5l)+ ej where the random errors {ej} are distributed uniformly in [-e,s]. The integrals in the (/31)j's are evaluated by the trapezoidal rule. Table 1 contains the number s of iteration steps to yield an Euclidean norm of the residue smaller than 10 - 4 . a . ]lz~ll. Here, {z~}~ denotes the sequence of iterates generated by (3.2.8) with starting guess either z~ - 0 or -
Z0 -- U/min'~ "-- qf~/,/mint A-1/min~'~l'lmin~l (3.2.11) (see Lemma 6). Our stopping criterion guarantees a relative accuracy IIz~ - ~tll/liz~ll bounded by 10 -4. The approximate solutions with respect to the discretization levels I -- 6 and 1 - 11 together with the minimum norm solution f* are displayed in Figure 6. We considered an absolute error e - 0.04 and got the optimal regularization parameter by trial and error. It took 14 (1 - 6) and 1 (1 - 11) iteration steps (L = 4) to bound the relative accuracy by 0.01. In the remainder of this section we compare the additive and the multiplicative iterations numerically. All computations presented are based on the approximation spaces spanned by the linear B-spline.
375
Wavelet Multilevel Solvers for Ill-Posed Problems 0.6 0.4 ~
0.2 0.0
1, usually require a sort of boundary modification in the system {r t for each j >_ 0, those of the r with support intersecting with a certain neighborhood of the boundary 0fl have to be replaced by their boundary-adapted counterparts r while functions are simply dropped from the system if supp r Cl fl = 0. See [7, 33] for examples in the case d > 1 where local support of the functions and local polynomial reproduction in the resulting subspaces are taken as the important features to be preserved. In other papers (such as [24, 25]), the boundary adaption is incorporated via a Schmidt orthogonalization process and may lead to functions with global support. In this note, we considerably simplify the task by concentrating on the following class of domains. Let Q = (0,1) d denote the open d-dimensional unit cube, and introduce the notation Qj,~ = 2 - J ( Q - a) for the dyadic cubes of level j >_0 (c~ e zd). We say that a bounded open domain fl C ]R d is aligned with the cube structure if for some j0 _> 0 there is an index set Y_~C Z d such that fl = ~.J Q~o,~. aEIo In addition, we exclude the possibility of slits and cuts (what is actually needed is the extension property for HS(fl) (s > 0) to hold). The last assumption can be removed if the following construction is slightly changed, compare the discussion in [32] for the case of finite element multilevel schemes. After scaling, we may assume that j0 = 0, which we will do throughout the paper. For domains that are aligned with the cube structure, a boundary modification is not necessary, simply set (I)j,a -- {r
" supp r
N fl # q}},
Then, Cj,~ is an algebraic basis in Vj,a - (span{r which we call nodal basis in Vj,~.
,
j k 0,
j k O.
417
Scaling Function and Wavelet Preconditioners
1. Suppose that {r satisfies (A1)-(A4), and that f~ is aligned with the cube structure (jo = 0). Then {Vj,~ } is a multiresolution analysis with the following properties:
Theorem
(i) For any gj - E(l,~) cl,.r
c Vj,n,
ilgi[12ix(a) ~" .., Z
(l,~)
2 -jd [cl,,~I2 9
(2.1)
(ii) For 0 < s < min(t, 2), we have CX)
ilfll2H.(~) v
i?f
~.
oo
~
g~eyj,~: =~j=o gJ ~=0
22J~l]gj[125=(~)
(2.2)
for all f 6 H s (~). In other words, the system (X)
~, - (.J %,,
(2.3)
j=O
is a frame in Hs(Q) after normalization, i.e.,
Iifll~.. II ,d
~
(f, Cj,~)Hs(~) I 2
II _t'j[
j=o (l,~)
V f C HS(~).
(2.4)
L2(12)
In the above statements, • stands for a two-sided estimate, with positive constants that may only depend on ~, s, d, and on {r The summations ~(l,a) are with respect to index pairs representing the basis functions in ~j,a .
We sketch the proof of Theorem 1 for the sake of completeness only. By assumptions (A1)-(A2), and using the usual dilation arguments, we observe there is a function r with support that for each cube Qj,z c supp r in Qj,z, such that ][r l _ O. With the usual tensor product generalize to d > 1.
aS defined above satisfy In particular, Ct E Ht(lR) form an orthogonal basis construction, the results
The refinement equations for (A4) which are important for the evaluation of the values of functions from Vj at intermediate points and for the implementation of the multilevel algorithms (see the definition of the matrices Ij entering (2.17)) can be found in [16, Section III]. Figure 1 depicts the classical linear, Lagrangian finite element (a), the finite element created from AFIF scaling functions (b), and the classical quadratic Lagrangian finite element (c). (More precisely, for (b) the graphs of the nontrivial AFIF basis functions r r r on [0, 1] are shown, analogously for (a) and (c).) The linear finite element and AFIF element have similar approximation properties, both contain piecewise linear (but not quadratic) functions within their span. On the other hand, the AFIF element and quadratic finite element have similar cardinality. That is, both have three basis functions that intersect a single element in one dimension. This fact should be kept in mind when we compare the performance of algorithms based on the different choices in terms of numbers of iterations for a fixed error reduction or in terms of condition numbers of the multilevel preconditioned systems (see the next section). Though the AFIF functions do not have explicit expressions and are only implicitly defined via recursions involving the refinement equation, the accurate calculation of terms that typically arise, such as v'j,~'j,~, dx
or
Vr
vet',a, dx
[10 0]
can be performed efficiently (see [23] or, in general, [11]). For example, the elemental mass and stiffness matrices for the above AFIF case are given by [Me]=
0 0
25 0
Scaling Function and Wavelet Preconditioners
.........................................................
1.0
425
i
!.................................... :.:.;.,
0.8
................... ~
.................................... ~............... :.:.._..... :.: ..............
0.6
...................................... - . - ~ : . : ....... :.:...,.:.: .......................................
0.4
..................................... :....:.-.:.: ........... : . ~ < , .......................................
o 9149
..-. jo preserve the L2-orthogonality property for such domains in any dimension d k 1. Even though we did not explicitly deal with the case of essential boundary conditions in Section 2, zero Dirichlet boundary conditions can be incorporated in a similar way (by keeping only those r ~ with support in gt in the basis). This has been utilized in the study of multigrid methods in [27]. We finish this section with a short description of multiwavelets {r r associated with the AFIF multiresolution analysis. As derived in [16], the AFIF wavelet functions are constructed to be interpolants at the quarter integer points and orthonormal to each other and the AFIF scaling functions. The function space W0 generated by the two wavelet functions is the orthogonal complement of V0. Several choices are discussed in [16]. The one which we present here as most convenient for our purposes consists of a symmetric r - r and an antisymmetric r - r Both functions are supported on [0, 2], and are depicted in Figures 2 and 3, respectively. This particular choice of wavelet functions makes the analysis on a finite domain much easier. The expressions for the wavelet mask coefficients which enter the matrices/~j from (2.20) can be found in [16, section IV] for the 1D case (one needs to be careful with the normalization factors introduced in [16]). Wavelet systems restricted to the interval [0, 1] have been described in [16, Theorem 4.4]. For our case, the choice of a basis for the complement spaces Wj,[0,1] as defined by (2.18) consists of all nontrivial restrictions of the symmetric ~,~1[0,~] and only those antisymmetric Cj~,~ with support completely contained in [0, 1]. It should be emphasized that the resulting restricted function spaces Vj,[0,1] and Wj,[0,1] are still orthogonal complements. This fact generalizes immediately to d > I and rectangular domains via tensor product arguments. As in [15, Section 10.1], the multivariate wavelet spaces are defined as spans of tensor products of univariate scaling functions from Vj-1 and univariate wavelets from Wj (except for pure
J. Ko et al.
428
C-products). For a 2D-rectangle, this means to take the span of all
Cj-I (Xl)~j(X2), Cj(Xl)r (X2), ~j(Xl)~)j(X2), Cj--1, Cj denote the generic basis functions from the above defined
where Vj-1 and Wj on the respective 1D-intervals. This construction automatically ensures L2 orthogonality of the multivariate wavelet spaces for different j for rectangular domains. Domains f~ with re-entrant corners such as L-shaped domains need more care (compared with the above construction for the rectangle it is enough to modify a few functions in the vincinity of the re-entrant corner to preserve full orthogonality as required in Theorem 3; we leave this as an exercise to the reader). In any case, elementary rules guarantee the applicability of Theorem 3 for the AFIF elements, and justify the optimality of the corresponding C-algorithm as described at the end of Section 2. Since the underlying system r is an orthogonal basis in L2(f~), this C-algorithm should perform extremely well for problems with dominating L2-elliptic part. For the corresponding numerical experiments, see Section 4. w
Numerical examples
An important goal of this paper is to assess the numerical performance of the class of multilevel r and C-preconditioning methods introduced above. In particular, we include standard multilevel finite element solvers (linear and quadratic elements) into the comparison. A motivation is that until now relatively few empirical results on wavelet preconditioning methods are documented, and fewer still make a serious attempt to calibrate performance to standard finite element formulations. Careful studies of the numerical performance of these algorithms are critical to establish viable, worthwhile directions for future research. Our numerical studies are still preliminary, and concern 1D and 2D Neumann boundary value problems for the Poisson equation. Most studies of the numerical performance of wavelet and scaling function preconditioners for Galerkin formulations have utilized a model equation in one dimension [3, 1, 9, 19]. While studies of one-dimensional boundary value problems are seldom of interest in applications per se, there remain important conclusions that can be drawn from this class of problems. In particular, the numerical examples in one dimension set precedents in optimal complexity that are realized in some more general problems over classes of domains in higher dimensions. This is the reason we include them here. The two-dimensional tests concentrate on simple domains like the unit square and L-shaped domains. Finally, robustness with respect to singular perturbations caused by a zero-order Helmholtz term is investigated. Some related experiments comparing Cand C-algorithms for linear finite elements on square domains can be found
429
Scaling Function and Wavelet Preconditioners
in [21]. Our numerical experiments show that (i) the multilevel preconditioning techniques presented in this paper are amenable to both scaling function and wavelet constructions, (ii) all selections (AFIF scaling and wavelet functions, Daubechies scaling functions, linear FEM and quadratic FEM) achieve asymptotic optimal complexity without tailoring the underlying function systems to the domain, if the latter is well-aligned with the cube structure. Also, we see that for the standard elliptic problems, multilevel finite element preconditioning of BPX type yields, as a rule, a better performance than wavelet-based methods. This supports our opinion that the ongoing development of wavelet-like solution methods for PDEs should include a thorough testing and comparison with conventional methods for partial differential equations. 4.1
1D t e s t s
We consider the weak formulation in Hi(0, 1) of the two-point boundary value problem d2 u
dx 2 ~-u
- 2 x 3 + 3x 2 + 1 2 x - 6,
x e (0, 1),
u'(0)- u'(1)-0, which has the exact solution u(x) = 3x 2 - 2x a. For the discretizations and solvers, we employ (i) linear finite elements, (ii) quadratic finite elements, (iii) Daubechies scaling functions, and (iv) AFIF scaling functions as defined in Section 3. We exclusively use the C-algorithm (i.e., the pcg method with the multilevel preconditioner based on the scaling functions) discussed in Section 2. As scaling factors (see (2.15)) we choose
which corresponds to multilevel diagonal (or Jacobi) scaling, and seems to be the most reliable choice on the average. An analogous choice is made for the scaling factors dj,i in the C-algorithms below. The results of the numerical study are summarized in Table 1 and depicted graphically in Figure 4. A first interesting conclusion in considering the results of Table 1 is that the multilevel r for the AFIF and the Daubechies scaling functions yield nearly identical results. Both methods require just under 30 iterations to reduce the residual to a value of 10 -6 of its starting value. This fact is quite counter-intuitive in light of the fact that many authors have noted the extremely poor conditioning of the Gramian matrix associated with truncating the Daubechies wavelets to the interval. Because the performance of these two formulations are so
430
J. Ko et al.
10 3
o 10 2
10~ i : : .........................
10 0
0
100
200
300
l o..e $ -multilevel,DaubechiesFEM B--e $ -multilevel, AFIF FEM A..A $ -multilevel, Linear FEM v - - v $ -multilevel, Quadratic FEM
: i
' i
i i
i i
i i
i i
400
500 Dim.
600
700
800
900
1000
Figure 4. Number of Iterations (Error Reduction by 10-6), 1D, log- Plot.
Table 1. Number of Iterations to Decrease Residual by 10-6, 1D. [!
I/ Daubechies l[
AFIF
!1 .....L'inear .
[i J [i 3 4 5 6 7 8 9 10
b
!] Quadratic ]1
n j l i t e r I[ ....n j... l i t e r [[ n j ] iter [I n j ..... i 12 12 17 14 9 5 17 20 20 33 19 17 8 33' 36 21 65 22 33 11 65 68 24 129 23 65 12 129 132 24 257 25 129 12 257 ' 260 26 "513 25 257 12 513' ....516 29 10'25 24 513 13 1025 1028 29 2049 26 1025 13 2049 . . . . . . . . . .
iter li 12 15 16 16 17 17 18 17
similar, in the remaining numerical examples, we will only consider scaling functions and wavelets associated with the AFIF basis. In comparison, the classical linear finite element basis and classical quadratic elements require 13 and 17 iterations, respectively, to achieve the same decrease in the residual. Finally, as noted earlier, these numerical experiments should be evaluated keeping in mind the cardinality of the masks representing the elemental mass and stiffness operators. In Section 3 we showed that the AFIF
Scaling Function and Wavelet Preconditioners
431
10 3
~ 0
10 2
'5
10~ 0
5000
10000
Dim.
15000
Figure 5. Number of Iterations (Error Reduction by 10-6), 2D, Square Domain, log-Plot. elemental mass and stiffness matrices have 3 x 3 -- 9 entries. Of course, it is well-known that for the classical linear and quadratic finite elements we have 2 x 2 = 4 and 3 x 3 = 9 entries, respectively. In contrast, it is shown in [26] that in the order 3 Daubechies case these elemental matrices have 5 x 5 -- 25 entries. In higher space dimensions, the differences are even more significant. 4.2
2D
tests
In this section, we report on similar performance results for the model problem -Au+u-f
Ou On
= 0
in
~,
on
0~.
Table 2 summarizes the number of iterations required to reduce the initial residual by a factor of 10 -6 using the C-algorithm in the case ~t - [0, 1]2. For this particular problem, the number of iterations approaches a value of approximately 10, 20, and 30, for the linear, quadratic, and AFIF elements, respectively. The performance of the r in comparison to the Jacobi preconditioner (i.e., when diagonal scaling of the discretization matrix A j is used as preconditioner) is depicted in Figure 5. Similar results are summarized in Table 3 and Figure 6 for an L-shaped domain. Again, all calculations are carried out using the C-multilevel algorithm, and
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J. K o et al.
Table 2. Number of Iterations to Decrease Residual by 10-6, 2D, Square Domain. AFIF nj
3 4 5 6 7
289 1089 4225 16641 66049
Jacobi,
10 3
--9
m--m
Linear iter 81 9 289 11 1089 12 4225 13 16641 12
iter 28 29 30 31 31
AFIF
nj
Quadratic nj iter 289 19 1089 20 4225 20 16641 20 66049 21
FEM
Jacobi, Linear
FEM
.................................................................................
................... i
r e.9 L..
'~
J
10 2
!
d
Z
.
o ,/~ . . . . . . . . . . . . .
.,~ . . . .
.O
l
Furthermore, it can easily be seen [18, 27, 4, 8] that the components an(t) are uncorrelated to the second order:
< an (t)am (t) > = ~n,m)~n.
(1.5)
One of the major difficulties raised by the above procedure is due to the fact that, for domains which are translationally invariant (homogeneous) in some directions (e.g., Ri,j(Xl,X2,X3,X'l,X'2,x'3) - Ri,j(Xl x~, x2, x~, x3, x~) if invariant along the Xl direction), the proper orthogonal decomposition is identical to the Fourier decomposition in these specific directions. In the boundary turbulence problem, for instance, the spanwise (xl) and streamwise (x3) directions are homogeneous [3, 8]. Accordingly, the proper eigenfunctions take the form: e 2 ~ ( k ~ + k ~ ) ~ ~i,kl,k3 (~) (x2) The ~(n) ~i,kl,k3 (x2) functions are often rather well localized for flows dominated by coherent structures. On the contrary, the harmonic part of the decomposition is not localized at all, and we are led to a paradox: we want to construct a low dimensional system (ODE) that will reproduce the behavior of localized coherent structures. Among all possible decompositions of the velocity field, we choose the one which is optimally adapted (in the L 2 average sense) to the given velocity field. However, in the presence of translation symmetry, this results in nonlocal modes, and consequently nonlocal models. A possible solution to this is simply to choose an appropriate length scale containing 'enough' structures to afford a reasonable interaction, and to impose periodic boundary conditions in the homogeneous direction(s). Then we obtain the dynamical model by Galerkin projection onto the most energetic modes coming from the proper orthogonal decomposition. This is the strategy adopted in the paper by N. Aubry et al. (we refer the reader to [3, 7, 18] for further details concerning in particular the modeling of the energy transfer to small scales as well as the interaction with the mean flow). Basically, what is postulated when using such an approach is the fact that interactions are primary local and that the influence of missing neighboring structures at the boundary of the modelled subdomain can be supplied by the coupling with the 'opposite end' afforded
444
J. Elezgaray et al.
by periodization. This approach is in fact widely used in computational fluid dynamics in order to reduce problem size (see [19] for a study of the periodization procedure in the context of channel flows). Due to their good spatial and scale/wavenumber localization properties, it seems natural in this context to use wavelet bases [29] to replace Fourier bases in homogeneous directions. In order to investigate this conjecture, we chose the one space dimension Kuramoto-Sivashinsky (KS) equation as a suitable model problem. With periodic boundary conditions on sufficiently long intervals (length L), this PDE exhibits rich spatio-temporal behavior [24, 9], in which zeros of the scalar variable u(x, t) are created and destroyed in irregular yet structured events (see Figure la) closely reminiscent of the burst/sweep cycles of the turbulent boundary layer. The rest of the paper is organized as follows: in Section 2, we discuss the fact that, while wavelet bases are not optimal in the L 2 sense, representations of the KS solutions in terms of the finitely many scale generations of the Perrier-Basdevant [33] periodic wavelets do almost as well as the optimal Fourier mode representations of comparable dimension in capturing the average energy (L 2 norm)[6]. Thus, it seems reasonable to exploit the spatial localization property of wavelet projections in an attempt to extract local models which capture the relevant dynamics in a limited part of a large spatial domain. Such local models can be extracted by Galerkin projection onto subspaces spanned by "close" wavelets, but, while short term transients display power spectra comparable to that of the full system [14], typical solutions eventually decay to fixed points. Exploratory analyses indicate that the drastic loss of symmetry (typically, only a 292 symmetry survives from the initial rotational 0(2) invariance in phase space) is responsible for the absence of long term recurrent dynamics in such local models. In Section 3, we show [32] that provided that periodic boundary conditions are used on the short subdomain, the loss of symmetry induced by wavelet projections is sufficiently mild that key global features of the dynamics are preserved. Section 4 explores the ability of local models of the KS equation, obtained with a small set of Fourier modes supported on a short subinterval, to reproduce coherent events typical of solutions of the same equation on a much longer interval [11]. Finally, Section 5 explains how wavelet decompositions afford a natural connection between large scale properties of the KS equation in a large domain (extended system limit), and its behavior in a small domain [13]. 1
1We assume the reader is familiar with orthogonal wavelet bases [33, 29]. In all the numerical simulations below (except in the simulations reported in Table 1), we used periodic fifth order spline wavelets, as described in [33].
Kuramoto-Sivashinsky Equation
445
100
8O t 60
40
20
0
50
100
150
200
250
300
350
400
100 10-1 10-2 (b)
I
10 -1
k
10~
F i g u r e 1. (a) A numerical simulation of Equation (2.1) with periodic boundary conditions on the interval of length L - 400. (b) The spatial power spectral density of the solution obtained by time averaging. (Note" logarithmic scales on both axes).
J. Elezgaray et al.
446
w Near optimality~ of wavelets and symmetry breaking local models The Kuramoto-Sivashinsky equation was originally proposed as a model for instabilities on flame fronts and 'phase turbulence' in chemical reactions. In this paper, we will use the formulation"
Ut + L-2uzz + L-4uxzxz + 2L-luu~ - 0, 0 _< x _ 1,
(2.1)
with periodic boundary conditions. The Fourier transform of the linear part of (2.1) with respect to x, --
(k,t)
(2.2)
shows that the uzz term in (2.1) is destabilizing and acts as an energy source, while u ~ is a stabilizing or energy sink term whose effect increases with (spatial) wavenumber. More precisely, (2.2) shows the existence of a range of linearly unstable wavenumbers ]27rk[ < L, with a peak at 27rlk[ = L/~f2, corresponding to the wavenumber of the most rapidly growing linear mode. The form (2.1) has the advantage over the form considered in, say [2] or [23] that it preserves the spatial average
ft(t) =
/o 1u(x, t)dx,
(2.3)
which is therefore customarily set to zero. When the KS equation is integrated on a sufficiently long domain a characteristic pattern emerges, which we illustrate in Figure la for the specific case L = 400. Here the value of u(x, t) is indicated by a gray scale (black = minima, white = maxima) over the (x, t) plane. A typical spatial wavenumber emerges, as indicated also by the clear peak in the power spectrum of Figure lb, which occurs near the maximum linear growth rate 27rlk[ - L/v~. This average wavenumber is maintained largely by interactions among a range of active wavenumbers up to and above the linear stability cut off 27r]kl - L, causing the continual appearance and disappearance of peaks and troughs, or equivalently zeroes, in u(x, t). These coalescence and creation events, and the dominant wavenumber 27r]k[ - L/x/~, are the coherent structures of our model problem. A physical explanation for the appearance of the chaotic state depicted in Figure la is as follows: the linear stability analysis above shows that the system tends to create 'cells' of a preferred wavelength. However, the creation of these cells takes place randomly in space, so that the cells are either compressed or stretched by neighboring cells [9]. This results in the creation or annihilation events clearly visible in the space-time representations of the solutions of the KS equation.
Kuramoto-Sivashinsky Equation 2.1
447
Comparing Fourier and wavelet decompositions in the L 2 a v e r a g e sense
As stated in the introduction, one clear advantage of wavelet bases is their spatial localization. Unfortunately, when the equation is translationally invariant, wavelet basis are not optimal in the L 2 average sense. Let us be more precise. If
u(x, t)
-
E
qEZ
uq(t)eq(X), eq(x)
-
e 2iTrq=
(2.1.1)
denotes the Fourier decomposition of any solution of the KS equation, and
u(x, t) -
E
aj,k(t)~j,k(X),
(2.1.2)
j>_0,ke[0,2J-1]
a wavelet decomposition, then it can be proved [8, 4] that, for any n >_ 1, n
n
(l~(~)(t)i 2) >_ ~ ,
i=1
(2.1.3)
i=1
where we have sorted the wavelength q indices (resp. the (j, k) wavelet indices) in decreasing average energy order:
(]ttq(i) (t)! 2) _> (lUq(i+l)(t)!
(2.1.4)
2)
and
(a2(i),k(i) (t)) >-- (a2(i+l),k(i+l) (t)).
(2.1.5)
Now, as shown in [6], it is possible to obtain an estimate of the amount of energy lost (in the average) in going from the Fourier {uq(t)} decomposition to the wavelet {aj,k(t)} decomposition. Namely, if for a given e, we need N(e) Fourier modes in order to satisfy
N(~)
~(TUq(~)(t)i 2) - ~ (luq(~)(t)i 2) N(c) slightly bigger than N(e). We refer the reader to [6] for a detailed discussion of this estimate, and here merely give some numerical results which confirm the above estimate. Let us first note that, due
J. Elezgaray et al.
448 Table
Number of modes 64 (j = 6) 96(j = 6, 5) 127(0 0,
(5.1)
where ~(x, t) is a stochastic forcing. However, the introduction of unknown parameters in the description of the small scales makes the effective corn-
J. Elezgaray et al.
460
U2
U3
1 0 -1 2O
1 0 -1 20 0
0
o~,
0
o~,
Figure 5. Local model with varying length: (a) The periodized window of the full solution u2(x, t); (b) Solution of a local model supplied with Ls(t) and l(t). putation of u unclear. A completely different approach is that of Zaleski [38], later pursued by Hayot, Jayaprakash, and coworkers [16] in one and two dimensions. These authors use a coarse-graining procedure to integrate out short wavelength degrees of freedom k > A, for a suitable cutoff A. If u(k, t) denotes the Fourier transform of u(x, t), then equation (2.1) may be rewritten as: k2
~(k, t) - - ~ 4 . ~ ~ ( k , where
t) + g(k, t) + / ( k , t),
(5.2)
g(k, t) - -2i 27rk
Z u(q,t)u(k-q,t) (5.3) Iql_A or Ik-ql>_h
involving the coupling between at least one small scale mode and any other mode, is uncorrelated with the large scale modes for time differences longer than the (shortest) characteristic time scale of the large scales Tt: T -1
:f(k, t)u(-k, t - r)dt ~ 0 as T
-~ ~ ,
7- > rl, Ikl < i .
(5.5)
Kuramoto-Sivashinsky Equation
461
In other words, the assumption that only the linear term is renormalized yields a closed expression for u in terms of three point correlation functions mixing short and long wavelength modes. It then remains to check that the above definition of u is independent of the parameters A, k and the time delay T, and that the 'stochastic forcing' f(k, t) has the expected correlation properties. Although this method works well, it is not constructive, in the sense that one effectively needs to integrate the full KS equation in order to compute these three point correlations, and eventually u. In order to show how wavelet projections can yield an effective equation for the large scales, we use an approach reminiscent of that in the previous section, although here the strategy is rather to start from the 'full' equations, without any approximation, and successively drop coupling terms until an already known equation is reached. Hereafter, we will use alternatively the notations u(x, t) = ~ aj,k(t)ff~j,k(X ) and u(x, t) = ~ a s ~ s ( x ) for the wavelet decompositions (a = (j,k)), and split the wavelet coefficients into small ({a > }) and large ({a < }) scales, corresponding respectively to the values of j > jo and j - (t) - n s~7 >< s~ a~ is a slowly varying additive forcing term. Let us first consider the small scales equation (5.7) without the forcing terms 51s>~ and f>s, i.e., in isolation from the large scales. The key observation is that the equations governing the statistical properties of the a > variables satisfying these unforced equations (5.7) are identical to those of the KS equation with parameter f , - L2 -(j~ up to rescaling and small corrections. Namely, if we define the "boxes" Bk, k = 0 , . . . 2j~ - 1, specified by the sets of indices (j, k') with j _> j0 + 1 and Ik2-(Jo+l) - k'2-J I equations into 2j~ independent sets of equations, each one including only variables a > belonging to the same Bk box, in a manner that preserves the statistics but not necessarily the dynamics. To prove this claim, we rely primarily on the fact that the evolution equations of the moments < as(t)a~(t + T) > (< 9 > is again a time average) can be periodized, and on the scaling relations between the
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462
coupling coefficients la~ and na~7 for equation (2.1) with parameter L and L = L2-(J~ which are:
Iaz(L) n~z~(L)
-la~z:(L)(l+O(exp-2~"'"/s)) -
2-(J~
(L) (1 + O(exp -2j'~''/S)),
where a = (j, k), c~1 = (jl, kl), jl = j - (jo + 1), kl = 2 -(j~
(5.8) etc., and
jmin - min(j, j',j"). See [13] for details. Assembling these observations, approxione can easily see that the rescaled variables aal = 2(J~ mately satisfy the moment equations arising from equation (2.1) for length parameter L. Consequently, one gets the relation claimed above, between the equation governing the statistical properties of solutions of (2.1) with parameter L and those of (5.7) with no forcing. Let us now investigate how the approximations performed on the unforced small scale equations (5.7) survive, when one includes the slow forcing terms coming from the interaction with the large scales. Our intuition is that these approximations can hold provided the amplitude of the forcing is small and its evolution is well separated in time from the small scales (a similar idea constitutes the main ansatz of the work by C. C. Chow et al. [9] ). If jM corresponds to the scale of the most energetic Fourier mode k = L/27rv~, i.e., 2 -(jM+I) < 27rv~/L < 2 -jM, we expect that the above scaling relations between statistical quantities obtained for two different values of L will hold even in the presence of the slow forcing terms 8/>> and f > , provided that jo < jM. In order to illustrate this point, we present in Figure 6 the second moments and the probability distribution functions of the wavelet coefficients aj,k for several scales j and for the two parameter values L = 50 and L = 400 (that is, jM ~ 5 and jo = 2), with the appropriate rescaling. The most energetic scale j = 6 as well as the second moments of these distributions are in a rather good agreement. This means that (i) the scaling relations obtained for the unforced equations do imply approximate scaling relations for the statistical properties of the solutions for length values L and L, and (ii) the forcing is weak enough to preserve the relations obtained for the unforced equations. We note however that the same comparison with j0 -- 3 would yield very poor agreement. The reason is that the asymptotic dynamics for the KS equation with L = 25 is a fixed point (see [24]): this system is simply too "short" to exhibit sufficiently rich dynamics in isolation (but see below). We conclude that the choice of the cutoff j0 is actually dictated by at least two conditions: (i) the order of magnitude of the forcing terms 8>> and ]>, and (ii) the (asymptotic) dynamics of the KS equation for the parameter value L. Let us now focus on the large scale equations of motion: 9
< -'*>'Y and f< (t) - ~ > a.>r + /-> '~ > af~ a~ ~ " Following our initial picture of the small scales as a set of 2 j ~ independent boxes slowly driven by the large scale variables, we wish to check its validity with regard to the statistics of the forcing terms 61,~(t)a~+ 2-(i~
+ +
(o>>
,yl~(S))a~ (5.10)
a 7< 9
(Here we momentarily drop the implicit summation convention, and make use of the approximate scaling relation (5.8)). The comparison between simulations 1 and 2 will substantiate an important point of this section, namely, that the statistics of couplings between small and large scales in equation (2.1) can be computed from the interaction with a collection of independent low-dimensional systems. Simulation 3 no longer takes the large scale evolution from Simulation 1, but instead uses
l.
+
E
, o,~~.y
a~ a~
la~ (L)a~
a.y> (5.11)
as governing equations for the large scales. The ensemble formed by equations (5.10) for k = 0 , 1 , . . . , 2j~ - 1 and (5.11) constitutes a completely autonomous model for the statistics of the KS equation (neither fitting parameters other than the cutoff scale j0 nor external forcing terms are needed). We considered the statistics of the two ( a ~ ( t ) a ~ ( t - T)) and three point correlations ( a ~ ( t ) a ~ ( t ) a ~ ( t - T)). These enter, for instance, in the energy budget equation (T = 0)
da j=0
|
l
J
!
|
!
!
4 10 -5
-1.
10 -4
0
IO-
T>>j=I
Figure 7. a) Probability distribution function of the transfer T> > term for j - 0, corresponding to the integration of the full KS equation (solid line), simulation 2 (long-dashed line) and simulation 3 (short-dashed line), for j0 = 2. b) Same as in a) for the j - 1 transfer T> >. 9< < <
+Ta
>>
,
(5.12)
and have been studied in the RNG approach to the Navier-Stokes equations [39, 12]. The time average of the transfer T >> is shown [39, 12] to correspond to a negative linear correction to the l~a < a 2s > term. As can be seen in Figure 7, the average < T >> > for a = (0, 0) is negative, although very frequently the "instantaneous" transfer is actually positive. Simulations 1 and 2 seem to agree fairly well for any j _< j0. The same remark is also valid for the other values of a, as well as for the "cross" transfer term T . The good agreement between the two statistics stresses the fact that all the information needed to compute the interaction between small and large scales (up to a good approximation) is actually encoded in the dynamics of the individual Bk boxes. However, the comparison of the undelayed three-point correlation functions obtained in Simulation 3 with those of simulations 1 and 2 shows clearly that, as one could expect, the agreement deteriorates as j approaches j0. The pdf's of the transfer term are close for j = 0, deviate for strong (but rare) values of the transfer for j = 1, and differ by a factor -~ 4 for j = 2 (not shown in Figure 7). Such disagreement should be expected. In fact, the periodization approximation involved in our model neglects nonlinear couplings of the type n , where the modes ~Z and ~ belong to different boxes, and a corresponds to some large scale mode. Simulation 2 shows that this approximation is correct as soon as the small scales are forced with the right statistics. On the other hand, Simulation 3 shows that these missing interactions can significantly change the statistics of the large scales close to the cutoff scale j0.
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The conclusion of these numerical simulations is clear: forcing a set of 2j~ independent subsystems with the large scales computed from the full simulation of equation (2.1) adequately reproduces energy transfer between large and small scales, provided the phase space of each of the uncoupled boxes Bk is large enough. However, when the large scales are generated autonomously by a "closed" large scale model coupled to the independent subsystems, the agreement is satisfactory only for scales well separated from the cutoff scale j0. w
Conclusion
In this paper, we have reviewed several recent approaches to the problem of modeling the local dynamics of extended turbulent systems in which most of the energy is contained in coherent structures. We first showed that, in an L 2 average sense, wavelets can do almost as well as Fourier modes in systems with translation invariance, in which the latter are optimal. However, a naive extraction of model equations from the wavelet projections of the original equation can be unsuccessful if the symmetries of the problem are not preserved (at least approximately). The study of wavelet projections of the KS equation on short intervals revealed that, from the point of view of capturing the correct dynamics, wavelets are comparable to Fourier modes, although the resulting equations are more massively coupled than the corresponding Fourier projections, which in addition preserve exactly the 0(2) symmetry of the KS equation. Moreover, in the analysis of the resulting models in phase space we are driven to consider combinations of wavelets which represent 'global' functions almost indistinguishable from Fourier modes. This led us to consider local models obtained by projecting onto a small set of Fourier modes supported on a short subinterval. Somewhat surprisingly, short systems having periodic boundary conditions on both fixed and varying lengths subdomains can reproduce the characteristic local events of much longer systems. Furthermore, we find strong evidence for a resonant effect, in which particular length ranges give superior tracking and long term asymptotic behavior. Finally, we considered a coarse-graining procedure to obtain an effective equation for the large scales of the KS equation. The method is very natural when written in wavelet bases, and no fitting parameters are needed in order to model the behavior of the small scales. It should be evident from all the above results that much remains to be done in order to understand why such local models behave so well. A first hint could be given by the short term tracking estimates suggested by A. Mielke [30], which shows that for parabolic equations such as KS, finite disturbances only make their presence known within a nonlinear cone of influence in the (x - t) plane (see [11]). Thus, we expect the effect of
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467
neighboring 'boxes' to be important for local models only after a finite time has elapsed. One can also give a probabilistic estimate on short term tracking in phase space [11], and estimate the average (over initial conditions) divergence between the solution of the full equation and our local models. However, despite such open questions, and from a physical point of view, our initial conjecture that short systems possessing the appropriate dynamical 'components' can reproduce the coherent structure interactions of extended systems, appears to be validated.
Acknowledgments This work was partially supported by DoE Grant DE-FG02-95ER25238 (PH) and NATO 92-0184. HD also thanks the Swedish Fullbright Commission and the Department of Mechanics at the Royal Institute of Technology in Stockholm, Sweden, for their support.
References [1] Armbruster, D., J. Guckenheimer, and P. Holmes, Heteroclinic cycles and modulated traveling waves in systems with 0(2) symmetry, Physica D 29 (1988), 257-282. [2] Armbruster, D., J. Guckenheimer, and P. Holmes, KuramotoSivashinsky dynamics on the center unstable manifold, SIAM J. Appl. Math. 49 (1989), 676-691. [3] Aubry, N. , P. Holmes, J. L. Lumley, and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid Mech. 192 (1988), 115-173. [4] G. Berkooz, Turbulence, coherent structures and low dimensional models, P h . D . thesis, Cornell University, 1991. [5] Berkooz, G., in Studies in Turbulence, T. Gatski et al. (eds.), Springer, New York, 1992, p. 229. [6] Berkooz, G., J. Elezgaray, and P. Holmes, Coherent structures in random media and wavelets, Physica D 61 (1992), 47-58. [7] Berkooz, G.,. P. Holmes, and J. J. Lumley, Intermittent dynamics in simple models of the wall layer, J. Fluid Mech. 230 (1991), 75-95. [8] Berkooz, G., P. Holmes, and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid Mech.
25 (1993), 539-575.
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[9] Chow, C. C. and T. Hwa, Defect-mediated stability: an effective hydrodynamic theory of spatiotemporal chaos, Physica D 84 (1995), 494. [10] Dankowicz, H., Chaos in low- and high-dimensional systems, Ph.D. thesis, Cornell University, 1995. [11] Dankowicz, H., P. Holmes, J. Elezgaray, and G. Berkooz, Local models of spatio-temporally complex fields, Physica D 90 (1996), 387-407. [12] Domaradzki, J. A., R. W. Metcalfe, R. S. Rogallo, and J. Riley, Analysis of subgrid-scale eddy viscosity with use of results from direct numerical simulations, Phys. Rev. Lett. 58 (1987), 547. [13] Elezgaray, J., G. Berkooz, and P. Holmes, Large-scale statistics of the Kuramoto-Sivashinsky equation: a wavelet based approach, Physical Review E 54 (1996), 224. [14] Elezgaray, J., G. Berkooz, and P. Holmes, Wavelet analysis of the motion of the coeherent structures, in Progress Wavelet Analysis and Applications, Y. Meyer and S. Roques (eds.), Ed. Fronti~res., 1993, p. 471. [15] Glauser, M. N., W. K. George, and D. B. Taulbee, Evaluation of an alternative orthogonal decomposition, 38th. Annual A.P.S., Tucson, 1985. [16] Hayot, F., C. Jayaprakash, and Ch. Josserand, Long-wavelength properties of the Kuramoto-Sivashinsky equation, Phys. Rev. E 47 (1993), 911. [17] Herzog, S., The large scale structure in the near-wall region of turbulent pipe flow, Ph.D. thesis, Cornell University, 1986. [18] Holmes, P., J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996. [19] Jimenez, J. and P. Moin, The minimal flow unit in near-wall turbulence, J. Fluid Mech. 225 (1991), 213-240. [20] Jolly, M. S., I. G. Kevrekidis, and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D 44 (1990), 38-60. [21] Kardar, M., G. Parisi, and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889.
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[22] Kelley, A., The stable, center stable, center, center unstable and unstable manifolds, J. Diff. Eqns. 3 (1967), 546-570. [23] Kekrevidis, I. G., B. Nicolaenko, and J. C. Scovel, Back in the saddle again: a computer assisted study of the Kuramoto- Sivashinsky equation, SIAM J. Appl. Math. 50 (1979), 760-790. [24] Kuramoto, Y., Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys. 64 (1978), 346; Sivashinsky, G. I., Nonlinear analysis of hydrodynamic instability in laminar flames, Part I: Derivation of the basic equations, Acta Astronautica 4 (1977), 1176-1206; Hyman, J. M. , B. Nicolaenko, and S. Zaleski, Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces, Physica D 23 (1986), 265-292. [25] Leibovich, S., Structural genesis in wall bounded turbulent flows, in Studies in Turbulence, T. Gatski et al. (eds.), Springer, New York, 1992, p. 387. [26] Lemari4, P. G., Une nouvelle base d'ondelettes de L2(R'~), J. Math. Put. Appl. 67 (1988), 227-236.
[27]
Lumley, J. L., Stochastic Tools in Turbulence, Academic Press, New York, 1971.
[2s]
L'vov, V. S. and I. Procaccia, Comparison of the scale invariant solutions of the Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations in d dimensions, Phys. Rev. Lett. 69 (1992), 3543; V.S. L'vov and V.V. Lebedev, Interaction locality and scaling solution in d + l KPZ and KS models, Europhys. Lett. 22 (1993), 419; V. L'vov, V. Lebedev, M. Paton and I. Procaccia, Proof of scale invariant solutions of the Kardar-Parisi-Zhang and Kuramoto-Sivashinsky equations in 1+ 1 dimensions: analytical and numerical results, Nonlinearity 6 (1993), 25-47.
[29] Y. Meyer, Ondelettes et Operateurs, Hermann, Paris, 1990. [30] Mielke, A., personal communication, Montr4al, September 1993.
[31]
Moin, P., Probing Turbulence via large eddy simulations, AIAA 22nd Aerospace Sciences Meeting.
[32]
Myers, M., P. Holmes, J. Elezgaray, and G. Berkooz, Wavelet projections of the Kuramoto-Sivashinsky equation. I. Heteroclinic cycles and modulated traveling waves for short systems, Physica D 86 (1995), 396-427.
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[33] Perrier, V. and C. Basdevant, Periodical wavelet analysis, a tool for inhomogeneous field investigation- Theory and algorithms, La Rech. Aeros. 3 (1989), 53-67. [34] Robinson, S. K., Coherent motions in the turbulent boundary layer, Ann. Rev. Fluid Mech. 23 (1991), 601.
[35]
Sneppen, K., J. Krug, M. H. Jensen, C. Jayaprakash, and T. Bohr, Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation, Phys. Rev. A 46 (1992), 7351.
[36]
Yakhot, V., Large-scale properties of unstable systems governed by the Kuramoto- Sivashinsky equation, Phys. Rev. A 24 (1981), 642. See also H. Fhjisaka and T. Yamada, Theoretical study of a chemical turbulence, Prog. Theor. Phys. 57, (1977), 734 for related work.
[37]
Yakhot, V. and S. A. Orszag, Renormalization-group analysis of turbulence, Phys. Rev. Lett. 57 (1986), 1722; Renormalization-group analysis of turbulence, I.: theory, J. Sci. Comput. 1 (1986), 3-51.
[3s]
Zaleski, S., A stochastic model for the large scale dynamics of some fluctuating interfaces, Physica D 34 (1989), 427-438.
[39]
Zhou, Y. and G. Vahala, Reformulation of recursive-renormalizationgroup-based subgrid modeling of turbulence, Phys. Rev. E 47 (1993), 2503.
Juan Elezgaray CRPP-CNRS, Av. Schwietzer 33600 Pessac, France
[email protected] Gal Berkooz BEAM Technologies, Inc. 110 N. Cayuga St. Ithaca, NY 15850 gal~cam. cornell.edu
Kuramoto-Sivashinsky Equation Harry Dankowicz Department of Mechanics Royal Institute of Technology S-100 44 Stockholm Sweden
[email protected] Philip Holmes PACM, Fine Hall Princeton University Princeton, NJ 08544-1000
[email protected] Mark Myers
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T h e o r e t i c a l D i m e n s i o n a n d t h e C o m p l e x i t y of S i m u l a t e d Turbulence
Mladen Victor Wickerhauser, Marie Farge, and Eric Goirand
Abstract. A global quantity called "theoretical dimension" is defined which is roughly proportional to the number of coherent structures that expert observers count in simulated two-dimensional turbulent viscous flows. This paper reviews some previously published computations of this quantity for a few academic examples and for a small number of flows computed from random initial vorticity fields.
w
Introduction
Evolution equations describing complicated phenomena like turbulence and nonlinear wave propagation sometimes produce coherent features such as shock fronts and traveling vortices. These coherencies permit an approximate description of the evolving state by relatively few parameters, regardless of how many free parameters were initially used in the numerical resolution of the equation. The goal of this paper is to discuss automatic methods for extracting such low-rank approximations to complicated phenomena, and to present results of one such method applied to two simple examples: Burgers equation with dissipation, as previously computed in one spatial dimension [8], and the incompressible Navier-Stokes equation, previously analyzed in two spatial dimensions [7]. New data is contained in Figures 4 and 5, and Tables 2 and 3. The rank reduction method will be a kind of lossy compression; the solution at any instant in time will be written as a superposition of orthogonal phase atoms,, defined below, and then only those component atoms whose amplitudes exceed some threshold will be retained. Coherence will M u l t i s c a l e W a v e l e t M e t h o d s for P D E s W o l f g a n g D a h m e n , A n d r e w J. K u r d i l a , and P e t e r O s w a l d ( e d s . ) , pp. 4 7 3 - 4 9 2 . C o p y r i g h t (~)1997 by A c a d e m i c P r e s s , Inc. All r i g h t s of r e p r o d u c t i o n in a n y f o r m r e s e r v e d . ISBN 0-12-200675-5
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474
be detected by counting the number of retained components; when this count is low, the instantaneous state will be considered coherent. To count the relative importance of the retained components in such phase atom expansions, their amplitudes will be weighted using the entropy functional defined as follows. For any nonzero sequence a = {a(n) : n = O, 1, 2 , . . . } with I[all2 - ~ n la(n)l 2 < c~, put
H(a) de.._f Z 'a(n)'2 .
ilal12
log
('a(n)'2) ilali2
.
(1.1)
As usual, 0 log 0 is evaluated by continuous extension as 0. This is called the "entropy functional" because it is the entropy of the discrete probability distribution given by p(n) = la(n)12/llall2. In [19, p.278], and many other places it is shown that if M > 0 is the count of nonzero elements a(n), then 0 _ 1 in normalized units, where Ap is the uncertainty in position and Ak is the uncertainty in momentum. In the wavelet packet construction, Ap ~ 2" and Ak ~ 2 -8 in the same normalization, so that the product of the uncertainties is roughly as small as possible. Such functions, which cannot be significantly better localized in phase space, are evidently phase atoms. Fourier analysis with such waveforms or atoms consists of calculating the wavelet packet transform Wskp(f) -- (r f). Certain subsets of the indices (s, k, p) give orthonormal bases B, and for these subsets there is an inversion formula:
f-
E
(s,k,p)eB
(r
flr
(2.1.1)
Wavelet packets are rarely constructed explicitly. More often, one simply applies the fast discrete algorithm described in [4] to the sampled values of f, and thereby produces the coefficients W,kp(f). The underlying functions r can, however, be developed as follows. Introduce two (short) finite sequences {ha} and {g,~}, called conjugate quadrature filters, which satisfy
Theoretical Dimension of Turbulence
477
the relations: 1
Eh2n-Eh2,~+l= n
v/~,
gn-(-1)'~-lhl_n
VnEZ;
(2.1.2)
n
5:
E
hnhn+2m =
n
gngn+2m =
{,,0, ifotherwise; m- o,
(2.1.3)
n
E hngn+2m - O,
(2.1.4)
Vm E Z.
n
Next, define a family of functions recursively for integers k _> 0 by:
W2k(X) W2k+l (x)
= --
V~ E n hnWk(2X -- n), Vr2 ~ n gnWk(2x -- n).
(2.1.5)
Note that W0 satisfies a fixed-point equation. Conditions (2.1.2) through (2.1.4) ensure that a unique solution to this fixed-point problem exists, and that {Wk :k E Z} forms an orthonormal basis for L2(]R). The quadrature filter pair h, g can be chosen (see [6]) so that the solution has any prescribed degree of smoothness. Equations (2.1.2) through (2.1.5) all have periodic analogs as well, which can be used in the case of periodic boundary conditions. The experiments in this article used a periodic algorithm with the so-called "C 6" coefficients, given as hn and gn in Table 1. Table 1. "Coiflet 6" coefficients for orthogonal wavelet packets.
nl
< -2
-2 -1 0 1 2 3 >3
Low-pass filter coefficient hn 0 3.85807777478867490 x 10 -2 -1.26969125396205200 x 10 -2 -7.71615554957734980 x 10 -2 6.07491641385684120 x 10 -1 7.45687558934434280 x 10 -1 2.26584265197068560 x 10 -1 0 ..........
High-pass filter coefficient gn
0 -2.26584265197068560 7.45687558934434280 -6.07491641385684120 -7.71615554957734980 1.26969125396205200 -3.85807777478867490 0
x x x x x x
10 1 10 -1 10 1 10 -2 10 -2 10 -2
One-dimensional wavelet packets are defined from these Wk by the formula: ~)skp(X)
--
2-s/2Wk(2-Sx -- p).
As described in [4], taking those functions {r " (s, k,p) E Z} for which " (s,k,p) e Z} form a disjoint the half-open dyadic intervals { [ k , ~ ) cover of the unit interval gives an orthonormal basis subset Z.
M. Wickerhauser et al.
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The library of basis functions in two dimensions consists of all possible tensor products of the r functions with both factors sharing the same scale s. The definitions and formulas for this two-dimensional case may be found in [20]. Certain basis subsets can be described by disjoint tilings of the unit square, as follows. Let I be a half-open dyadic square [ ~ , 2,k|+1 )X [2, ,k-Y- kv2` +1 ) and put r174 (x, y) -- 2 - ' s W k . ( 2 - S x - - p x ) W k , ( 2 - S y - - p y ) . Then every basis in the library, for functions on the 2 s x 2 s grid, corresponds to a set of the form: {r
" I E Z, px E Z,py E Z, 0 _O. f
(2.2.1)
(2.2.2)
Theoretical Dimension of Turbulence
479
-0.5
I
l
I
I
I
I
I0
20
30
40
50
60
F i g u r e 1. Burgers' evolution from sin(2rx) at t -- 0 to t -- 1 in increments of 0.05. The constant u is the viscosity of the fluid and the function F0 = Fo(x) is the initial state at time t = 0. Consider one classical example: Fo(x) = sin(27rx). The graph in Figure 1 shows the evolution from this initial function at times 0, 0.08, 0.16, 0.32, 0.5, 0.75, and 1.00. The two arcs of the sine, one positive the other negative, are propagating in opposite directions to produce a steep slope at x = 1/2. The dissipation term A F produces the vanishing effect: the total energy in the solution tends to 0 as time increases. Without dissipation the slope at x = 1/2 would become infinite and a discontinuity would appear; the viscosity term controls how close the solution gets to singularity before dissipating. The apparition of a near-discontinuity means that the amplitudes of small-scale components in the solution are increasing, since they contribute the large derivatives. This phenomenon is better seen in Figure 2, which depicts the amplitudes of wavelet components of the signal arranged by scale. The evolution was computed with a Godunov scheme applied to the 1-periodic signal, using a space-step of 1/64 and a time step of 1/100. In [8], it was shown that the energy is decreasing in the biggest-scale wavelet components of the evolving function, while the energy in the smallest-scale ones is increasing. It was observed that one of the big-scale amplitudes already begins to decrease at time zero. The maxima of the smaller-scale amplitudes are reached later and later with decreasing scale. This last aspect can be better seen on Figure 2 which shows the absolute value of
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Abs[wavelet coef]
initial condition:sin(2*Pi*x)
60
50
40
30
20
10
0 .
.
.
.
.
.
2'0 .
.
.
.
40
,
.
.
.
.
.
a
60
,
,
,
o
|
................
80
i
I00 time/0.01
Figure 2. Amplitudes of the wavelet coefficients of Burgers' evolution from the initial function sin(21rx), shown in gray scale. the wavelet coefficients in gray scale: white is zero, black is the maximum. The graduations between 0 and 100 represent time; the others show the index of the wavelet coefficients. The first wavelet coefficient is the mean of the signal (actually 0), the second is the biggest-scale difference coefficient, the third and fourth are next largest difference coefficients, and so on. The last 32 are the smallest-scale difference coefficients, since there are a total of 64 samples of the signal. The analysis was done with "Coiflet 30" wavelets [6] because they have a large number of vanishing moments and are nearly symmetric. Ultimately, through dissipation, the function and thus all its wavelet coefficients decrease to 0. 2.3
Two-dimensional incompressible Navier-Stokes simulations on the torus
The classical simulation of two-dimensional decaying turbulent flows uses the incompressible Navier-Stokes equation with small viscosity. The Kraichnan-Batchelor theory in this situation [1, 9] postulates homogeneous mixing within the flow and supposes that the whole vorticity field is involved in the "cascade process" that transports enstrophy from large eddies to small ones, while energy is transferred from small to large scales. In contrast to this explanation, we believe that two-dimensional turbulent flows are generically inhomogeneous, and propose to model them
Theoretical Dimension of Turbulence
481
as a superposition of coherent rotational vortices embedded in a random quasi-irrotational flow. We have observed, in numerical simulations of twodimensional Navier-Stokes equations with random initial conditions, that isolated vortices result from the condensation of enstrophy into localized, well-separated structures. These structures are stable as long as they do not interact with one another, but during close encounters they experience strong deformations, which then excite some internal degrees of freedom. This gives rise to a local cascade or transfer of enstrophy toward small scales and to its concomitant dissipation. Consequently, only a limited active portion of the vorticity field, correlated to the coherent vortices, is responsible for the turbulent cascade. The remainder, or background portion of the field, is passively advected and plays a negligible dynamical role. The atomic view may be compared with the vortex methods of Winckelmans and Leonard [22], Marchioro and Pulvirenti [14], and Saffman [17]. It generalizes the simplest model used to approximate two-dimensional flows, that of superposed point vortices, by considering the flow to be a superposition of atoms that are chosen from among a library of smooth localized functions such as wavelet packets [4] or localized cosine functions [3, 13]. The additional parameters available to these atoms enable us to take into account the internal degrees of freedom of each vortex, which can be considered as a molecule. The goal will be to compute the number of "significant" atoms in a turbulent flow, i.e., those components whose amplitudes exceed a preset threshold. Those that correspond to the same locale can be interpreted as the principal components of a coherent structure. Their number evolves in time, with a generally decreasing trend due to the decay in enstrophy caused by dissipation, and gives a quantitative estimate of the number of coherent structures and the complexity of the turbulent flow. The analysis begins with a direct numerical solution of the NavierStokes equation describing the dynamics of a two-dimensional incompressible viscous flow. In the periodic plane S = (0, 27r) • (0, 27r) C ]a 2 and in the absence of external forcing, these take the following form: 0u cg----t-+ (u. V)u + V P -
uV2u = 0
in S • ]R+,
V.u=0
inS•
u(x, O) = Uo(X)
+,
in S.
Here u is the velocity field, P is the pressure field, and v is the kinematic viscosity. Periodic boundary conditions are also imposed. The equations are rewritten in terms of vorticity w and stream ]unction r defined by u
-
(ux u2
-
_
OXi
-
'
Oxl
Ox2
(2.3.1)
482
M. Wickerhauser et al.
The Navier-Stokes equations then become + J(r
Ot
- vV2w - O, -
V2r
(x, t) ~ S x IR+,
(x, t) ~ S x la +, xES.
~(x, 0) - ~0(x),
Again, periodic boundary conditions are imposed. The Jacobian operator in terms of these new variables is: J(r w)-
0r Ow
0r 0_~w
'
(~Xl 0X2
0X2 0Xl"
(2.3.2)
The functions w and r can be expanded in their Fourier series over the periodic domain S" w(x, t) - E &(k, t)e ik'x,
&(k, t) = ~1 fx es w(x, t)e -ik'x dx,
k
r
t) - E r
t)e ik'x,
r
t) = ~1 fx es r
t)e -ik'x dx.
k
A turbulent vorticity field such as the one depicted in Figure 3 develops from a random initial vorticity field w0(x) which is integrated for many time steps in the presence of time-periodic external forcing (at very low wavenumbers), until the vorticity field reaches a statistically steady state. Forcing is subsequently turned off and the same integration is continued in the decaying regime. A pseudo-spectral Galerkin method was used to integrate the NavierStokes equations; at each time step, all differentiation was performed in ~b, r coordinates and all multiplication in w, r coordinates. Both w and r are represented as finite Fourier series, or superpositions of the Fourier modes at wavenumbers 0 0
V x,~ e ira
A ( x ) ~ . ~ >_ 5 I~!2 .
(H2)
We also assume some weak regularity of the coefficients aa~, the weakest of which is being in VMO(IRn). What this precisely means will be explained further, but at the present time, we only vaguely describe it as a property of uniform continuity in the mean. These operators are naturally defined on appropriate Sobolev spaces" L is continuous from W I'p (= WI'p(IRn)) to W -I'p, and M is continuous from W 2,p to L p , 1 < p < c~. Our first task is to describe and to discuss the resolvents of L and M on these spaces. We will prove the following. Theorem A. Assume that the coefficients a ~ 1 0, such that if r ___Cp, then (M +r from L p to W 2'p.
is continuous
We will also give explicit constructions of the resolvents in question, with the help of wavelet bases. As an application, we will recover interior estimates for the solutions of inhomogeneous problems, due to Chiarenza, Frasca and Longo in the case of M ([5], see also [3]). These results form the content of Section 3. In Section 4, we consider again the above-mentioned resolvents when the regularity hypothesis on the coefficients is strengthened. In such a case it is natural to expect the invertibility of L + ~ and M + ~ to hold on a wider range of functional spaces, and it is our purpose to confirm and to describe this phenomenon. For simplicity, we only quote here a less precise version of the result to be proved: Theorem B. Assume that the coefficients are in the HSlder space C 8~, with O < so < l. Then, if l < p < oo, i) for every ~ > 0 , (L + ()-1 is continuous from W -l+s,p to W l+s,p when ii) if ~ >_ r (defined in Theorem A), ( M + r to W 2+s,p when 0 _ 3, the GMerkin scheme, applied to an equation of the form Lu + {u - f, where f E C~~ ~) and ~ > O, will converge in the W I'p topology for 1 < p < ce, if the coemcients are in V M O , and in the WI+s,P topology, 1 < p < cc , 0 jo, we use Theorem 4'. Indeed, we know by Corollary 1 that the functions (0a0~8~)j(~)>jo satisfy vaguelettes estimates with a size constant uniformly bounded with respect to jo. Hence, inequality (2.6.6) gives us
IlSc=~*II ~ _< C(p) N2-~o(A),
where
N~-,o (A) - ~
(3.1.13)
N~-~o (~,).
Summing up (3.1.12) and (3.1.13) we obtain the estimate IOSr
_ C(p)
We now choose jo = jo(r
(
IIAlloo Mo + r + N2-~o(A)
)
.
such that 1 4 -j~ g2-Jo (A)_> ~ ,
which is possible provided (: is large enough, in such a way that lim jo (r = c~. By construction we have
liSr
< c(p) (1 + liAlioo) Y~-~o(,)(A),
which proves Theorem 5.
(3.1.14)
513
Analysis of Elliptic Operators 3.2 T h e r e s o l v e n t of L
3.2.1 S o b o l e v e s t i m a t e s We begin by proving the invertibility results that we announced in Theorem A. They rely on a study of the resolvent of L very similar to the previous one. We again consider the functions 0h and the operator PC defined by (3.1.1) and (3.1.2), but we now view PC as a continuous operator from W -I'p to W I'p , 1 < p < co. This is possible thanks to Theorem 1, Lemma 2 and Corollary 1 once more. We then define the operator R; by the relation (L + ~)P~ = I d - R~ , where Ri is acting (and continuous) on W -I'p. The analog of Theorem 5 is the following. T h e o r e m 6. For every p E (1, co) we have lim [[Rt][w-l,p = 0 .
~--+~
(3.2.1)
Proof: We compute
For each a E { 1 , . . . , n}, let Ta be the operator defined by =
[a.z
-
(where we dropped the subscript r Then, (3.2.1) will be proved if we show that, for every integer j0 and for each a, ][Tal[Lp,W-I,p ~_ C(p)
4j ~
IIA[[~ 4Jo + [r + N2-~o (A)] ,
(3.2.2)
following the reasoning after (3.1.14). It is a consequence of Remark 1, in particular, of relation (2.4.13), that the operator Id + A is an isomorphism from L p to W -I'p , 1 < p < co. Thus, (3.2.2) is equivalent to 4j ~
[[T~(Id + A)[lp _< C(p) [[A[I~ 4Jo + [r + N2-J0 (A)] .
We have To,(Id + A) ~ , --[a~z - m x ( a ~ z ) ] (1 + 2j('x)) 0f~O~, .
(3.2.3)
514
J. Angeletti et al.
Since, by Lemma 2, the families ((1+2J(~))0/30~)~i, fl E ( 1 , . . . , n ) , satisfy the same vaguelettes estimates as the families (OaOf3Ox)Xel, with the same behaviour of the size constants, we can proceed to prove (3.2.3) exactly as we did for (3.1.14). The details are left to the reader, m As in the case of M, Theorem 6 implies that for every p E (1, c~), the operator (L + () is invertible from W -I'p to W I'p if ( is large enough. We can even prove a better result which is a slight improvement and an extension to the V M O case of a similar result due to Taylor [15]. T h e o r e m 7. For every ( > 0 and every p E (1, oc), the operator L + ( is invertible from W -1,v to W I'p. Proof: The case p - 2 follows from Lax-Milgram lemma. For the other values of p, we argue first with a weak version of the result to be proved. L e m m a 6. For every ( > 0, (L + ~)-1 extends to a continuous operator on L p 1 < p < oc This lemma is well known. It can be proved, for example, by writing (L + ~ ) - 1 _
e.-tL e - t ( dr,
~0~176
and noticing that the semigroup e -tL is uniformly bounded on L 1 and L ~ , by Aronson estimates [1]. Now let ( > 0 and p E (1, c~) be fixed, and let us choose ~ so large as to ensure the invertibility of (L + ~) from W -1,p to W 1,p. By the resolvent identity, we have (L + ~ ) - 1 _ (L + ( ) - 1 _ (( _ () (L + ()-1 (L + ( ) - 1 = (L + ~)-1 _ (r _ ~)(L + ~)-2 + (( _ ~)2 (L + ~ ) - I ( L + ()-1 (L + ~ ) - 1 . By Lemma 6 and the definition of ~, each term on the right maps W -1,p in W I'p. m At this point, we have proven Theorem A. We however would like to explicitly construct (L + ()-1 for every ( > 0, in the same spirit as we constructed (M + ()-1 for large (. Here also we have a formula analog to (3.1.3), that comes from (3.2.1) and Theorem 6, and reads (L + ~ ) - 1 _ P ; ( I d -
R;) -1
(3.2.4)
if ( is large enough. But we cannot reach the small values of ( with this method. We are thus led to a modification of our argument, and this is the reason why we now need some results about the Galerkin operators associated to (L + ().
515
Analysis of Elliptic Operators 3.2.2 Galerkin
operators
To begin with, we at first consider L + ~ as acting on W 1'2 9 If jo is an integer, we define (3.2.5) (L + ~)jo - 7rio ( i + ~) 7rio , the Galerkin operator associated to L + ff on Vjo. By the Lax-Milgram lemma, this operator is invertible on Vjo (with the L 2 topology), and we define Fjo,; = 7rio* [(L + ~)jo] -1 7t'jo. (3.2.6) This is the natural extension to all of L 2 of [(L + ~)jo] -1. L e m m a 7. r'jo,( is continuous from W -1'2 to W 1'2, and uniformly bounded with respect to jo, satisfying
Ilrjo,r
1
12 4j(x)s 1 < C 2 -j(x)(~+s).
(4.1.5)
It is known that if f is measurable,
Ill
IIw=,,
- a~
/.L ~
~
If(x + h) f(x)l 2 ihl~§ ~ dxdh.
(4.1.6)
We will use, because it simplifies the proof, an orthonormal basis of compactly supported wavelets of regularity at least equal to 1, denoted by (G~,):~eI. The associated scaling function is F. The reader need not be acquainted with the properties of F and the wavelets G x. It will suffice for the reader to keep in mind that supp F is compact, F is at least Lipschitz, F defines a partition of unity: V x E lRn
E
F(x-
k) - l,
kEZ n
the mean value of F is 1, and F is orthogonal to the wavelets at scales smaller than 1: V A E I, j(A) > 0, < F, Gx > = 0. (4.1.7)
J. Angeletti et al.
520
Finally, we note that condition (4.1.1) is equivalent to
flhl
_ 2j('x)s
Analysis of Elliptic Operators
525
But it is apparent that a E B M O 8 if and only if the scalars fRn T ~ , I, satisfy the so-called Carleson condition, which reads
)~ E
2
/R n T ~
_ C IQI,
(4.2.2)
Q~cQ
for every cube Q. Hence, T is continuous on L p when a E B M O ~, by virtue of the following lemma. L e m m a 11. Let T be an operator such that the family (T@x)~eI satisfies the estimates (2.2.1-3). Then the kernel of T is of Calderbn- Zygmund type, and T itself is Calderbn-Zygmund (i.e. bounded on L 2) if and only if condition (4.2.2) is fultilled. This lemma is classical in Calderbn-Zygmund theory. It is essentially equivalent to the David and Journ6 theorem [7]. The reader is referred to [16] for further references. We now know that when a E B M O ~, T is bounded on L 2. By Calderbn-Zygmund theory, it extends to L p spaces when 1 < p < c~, and by a simple argument to the operator U also. Indeed, we will prove that U = T*, (4.2.3) provided a is real-valued, which we can assume. Equation (4.2.3) is a consequence of the orthogonality relation between wavelets, and the diagonal form of A (in the case of the commutator [ ( - A ) ~/2, Ma], this part of the argument has to be slightly modified). We have
< U~,~
> =< (a-m~(a))~,AS~ =
2
-
m.
>
>,
thus proving (4.2.3) and the "if' part of Theorem 11. The converse part starts again with (4.2.1), which can be written as [As, Ma] : T* - T, always assuming a to be real-valued. By Lemma 11 we know that [A~, Ma] has a Calderbn-Zygmund kernel. We compute the image of 1 (see [7] for the precise meaning of this) which leads to
[AS,Ma](1) -- AS(a). If [A~, Ma] is continuous on L 2, it must map L ~ to B M O . then implies that a E B M O ~.
Corollary 6
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J. A ngeletti et al.
4.3 T h e r e s o l v e n t o f M
Let us now come back to our operators M and L, and see what happens. We use again the notation of Subsection 3.1. T h e o r e m 12. If the entries of A are in B M O ' , for some s E (0, 1), then Sr is continuous on W 8'p if 1 < p < oc, with lim
liS;liw,.p = O.
(~oo
Corollary 7. W 2+s''p
(4.3.1)
If ~ is large enough, M + ~ is invertible from W s''p to
0 < 81 < 8 < 1, 1 < p < oc
Let us prove Theorem 12. Since we already know that lim
IISr
~--~oo
= O,
it will be enough to prove that S; is continuous on the homogeneous space IU ~,p, with lim ][S~[[r162 = 0. ~-+oo
We have seen (Remark 1) that A8 is an isomorphism from IU s,p to L p. It is thus equivalent to prove that ASScA -s is bounded on L p, with lim
{--+oo
[[ A8 S~ A -s lip = 0.
We now compute AsS; A-" ~x, for each A E I. We recall that -
=
which gives As& A - ' ~
-- AS{[aaf~- m~(aaf~)] 2 -j()')` aa af~ e~} = [As, aa~] 2-J(x)saaazox +
-
(4.3.2)
^'
The first term in the right-hand side of (4.3.2) is treated as follows. By Corollary 1, the functions 2-Js0a0~0x, when j(A) - j, satisfy vaguelettes estimates with size constant Cj, where 2J(2-s) C j < _ C 22j + r < _ C r Hence, the operator T1, defined by Tl~.X = [As, aa~](2-J(~)sOaOZ~x), is continuous on L p, by Theorems 1 and 11, with [[TI[[p _ 1 and for every A,p (see for example [11] or [13] for the calculations leading to such an estimate). Since we have
< As2-J(Z)so~oBO~, ~
> -- 2 (j(z)-j(~))s mz,,,
we obtain
I< As2-J(Z)so~o~O~,~z >I O, [s'] O.
Theorem 13 and Corollary 7 together give Theorem B. However, an important step in proving our results on the operator L has not yet been explained, namely the study of the Galerkin operators and Theorem C. This is the subject of the last section. w
T h e G a l e r k i n a p p r o x i m a t i o n of t h e r e s o l v e n t of L
5.1 T h e c o n v e r g e n c e r e s u l t s
Let us now consider again the Galerkin operators (L + ~)j = 7rj(L + r
when ~ > 0. Recalling the notation of Subsection 3.2.2, we have Fj,r - 7r~{(L + ~)j}-lTrj, and, if f is some test function, u and uj are the solutions of the equations (L + r
(L + ()u = f ,
= 7rjf .
This is equivalent to u=(L+~)-lf
,
uj = rj,r f .
From the Lax-Milgram lemma, we know that lim [[u- uj[Iwx,2 = 0.
j--+oo
(5.1.1)
We address here the problem of describing the various topologies for which the analog of (5.1.1) holds. Keep in mind that Vjo will be equipped with various topologies in the sequel, and accordingly completed in some cases, though we will always keep the same notation. When we will need to specify the topology, we will denote by (Vj, E) the space Vj equipped with the topology induced by the space E. We will prove the following theorems, thus recovering Theorem C.
Analysis of Elliptic Operators
529
T h e o r e m 14. /f the coefficients aaz are in V M O , then lim
j--+oa
!1- - ujllw,,, = 0
(5.1.2)
for every p E (1, oo ) . T h e o r e m 15. If the coefficients a~z are in B M O 8, for some s E (0, 1), then lira [ [ u - ujllw~+o,, = 0 (5.1.3) j-+oo
for every p E (1, oe).
Both are consequences of the following theorem, which is a stronger version of Theorem 8. T h e o r e m 16. I f a ~ E V M O , the operators Fj,; are uniformly bounded from W - 1,p t o W I ' p , 1 < p < oo. The same is true from W - lq-s,p t o W I+*,p i f a ~ z E B M O ~, 0 < s < 1. As we pointed out before, the uniform boundedness from W -1,2 to W 1'2 is the only easy case. Before proving the others, let us show how we deduce Theorems 14 and 15. We begin with the V M O case, and a fixed exponent p in (1, oe). We claim that there exists a constant C such that
V j e IN" V v e Vj
Ilvllw,,,, _< c II~,(L + Ovllw-,,,,.
(5.1.4)
Indeed, we may write v - Fj,(w, w 6 Vj, because Fj,( is invertible on Vj. Since, by definition, we have
=j(L + r162
-w,
inequality (5.1.4) is equivalent to V j e IN V w e Vj
ttrj,r
_< c Ilwilw-,,,,
which is exactly the content of Theorem 16. Let us now prove (5.1.2). We have II- - - j l l ~ , . ,
< II~-ltw,.,
+ II~j- - u j l l w , . , .
Inequality (5.1.4) implies
I[lrju - uj[Iw~,~ 0 independent of j and r
To prove this, we fix
xo,yo E IRn, and consider O, a compactly supported function from the Lipschitz class such that
9
- O(yo) -I
and
o - yol
IlV ll - 1.
(5.4.4)
We then prove (5.4.2) for the perturbed Galerkin operator
eT'~,, j,r
-
-
(eTCFj,r
TM
where e +~r stands for the operator of pointwise multiplication by e •162 with 7 a positive constant to be chosen. Once (5.4.2) is proved we obtain
lr 2,rM (xo,yo)l
< C(r
e -~[~(~~162176
J. A ngeletti et al.
536
which gives (5.4.3) provided C(r and "), do not depend on xo and yo, and ~' "- ~/ovf~ for some absolute constant ~'o > 0. -~r is the inverse of the Galerkin operator Now, the operator e~r associated to e-~VL e ~r and to e-~r e ~ r By this, we mean that if we replace L by L, with
L f - - d i v A V f - ~/(AVf) 9V r - ~/div(fAVr
- ~/2(AV(I) 9V(I))f,
if we replace Vj by Vj to be described later, and if we set
(L + ()j
j(L + r
where
~j
e-'r r162
=
is a nonorthogonal projection onto Vj, then we have - -~j. [(L + r - 1-~j,
e~r162
as in formula (3.2.6). Thus we have to prove that (5.4.2) remains valid in the more general setting where L is replaced by L and Vj by Vj, for any (I) satisfying (5.4.4), and for "i, - ~/o v~. The replacement of L by L is easy to handle, since the higher order terms are not modified. It is the replacement of Vj by Vj which is the main source of technicalities. Recalling that ~ is the scaling function attached to (Vj)jez, the spaces Vj are generated by the functions
~ k ( X ) -- 2in/2 e -~[r162
~ ( 2 J x - k),
k E Z n.
By construction, if f E Vj, we have
kEZ" where Cjk = 2in/2 fR" f ( x ) e ~[r
~(2Jx -- k) dx.
The spaces (Vj)jez do not form a standard multiresolution analysis anymore. They, however, share enough properties for the generalization of (5.4.2) to hold" they satisfy the embedding condition
vj c Vj+l,
Analysis of Elliptic Operators
537
and if Wj is generated by the functions ~ (x) = e -~[r162
r
(x)
where j(s - j, then it is a nonorthogonal supplementary subspace to Vj in Vj+I, which can replace the usual wavelet subspace Wj. The main point here is that, although the functions ~x do not have vanishing moments, they have small enough moments. We stop here the description of step 3, and show why the estimate (5.4.3) implies the L p estimate for Fj,;, which is the last step. It is a consequence of Taylor's formula. Once more, we choose ~ large enough, and write ,-v
2M-2
~.j,~: + ( 2 M - 1)
(Z -- ~)2M-2 F2,M dz.
k=O
We already know that Fj,~ is continuous on L p, and the estimate (5.4.3) implies the same result for Fj,z2M. Hence we obtain the L p boundedness of F j,(:, uniformly in j. Theorem 16 is now completely proved, and so are Theorems 14 and 15. w
Conclusion
All the preceding results might be extended in several directions. They probably can be localized, in the sense that if the coefficients a~,z have better regularity on some open set, it seems reasonable to conjecture that the same property will be shared by the solution u of an inhomogeneous problem, and that the convergence of the Galerkin approximations uj will be better on this open set. Our results can also be extended to boundary value problems on regular open sets. Using a classical trick (see [15]), it is not hard to see, for example, that Theorem A generalizes to the case of Dirichlet condition, thus recovering results of Chiarenza, Frasca, and Longo [4]. We are currently developing this line of research, and hope to publish some new results soon. Finally another set of questions seem interesting to investigate, regarding the convergence of Galerkin schemes. The first question we think of is that of describing the properties of the approximation spaces Vj which are really needed for our results to hold. The second question is about the nonregular coefficients case: when the cofficients have no regularity at all, it is known that our W 1,p results are only true when p is close to 2. However, it is also known that u = (L + ~ ) - l f is HSlder continuous if f
538
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is in some reasonable space. This is the celebrated De Giorgi-Nash-Moser theory. We hence address the problem of proving the convergence of the Galerkin scheme in the appropriate HSlder space, for suitable spaces Vj. Though we are only dealing with linear operators, all the known proofs of De Giorgi-Nash-Moser estimates have a strong nonlinear structure. How to adapt this theory to the context of Galerkin approximations is the main difficulty to solve, a difficulty we find challenging.
Acknowledgments. It is a pleasure for us to acknowledge the influence of P. Auscher, who showed us the references [3] and [6], and of N. Lerner and G. M6tivier, especially regarding the results on nondivergence form operators. We also warmly thank M. Berg for her typing the manuscript, the referee for his useful comments, and the editors of this book for their patience and cooperation. References [1] Aronson, D., Bounds for fundamental solutions of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890-896. [2] Auscher, P., A. McIntosh, and P. Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal., to appear. [3] Chiarenza, F., LV-regularity for systems of PDEs with coefficients in V M O , in Nonlinear Analysis, Function Spaces and Applications, vol.5, Krbec, Opic, Rakosnik (eds.), Prometheus Pub. House, 1994. [4] Chiarenza, F., M. Frasca, and P. Longo, W2,p-solvability of the Dirichlet problem for non-divergence elliptic equations with V M O coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. [5] Chiarenza, F., M. Frasca, and P. Longo, Interior W 2,p estimates for non-divergence elliptic equations with discontinuous coefficients, Ricerche di Mat. 40 (1991), 149-168. [6] Coifman, R., R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611-635. [7] David, G., and J. L. Journ6, A boundedness criterion for generalized Calderbn-Zygmund operators, Ann. Math. 120 (1984), 371-387. [8] Escauriaza, L., Weak type (1, 1) estimates and regularity properties of adjoint and normalized adjoint solutions to linear non-divergence form operators with V M O coefficients, preprint. [9] Frazier, M. and B. Jawerth, A discrete transform and decomposition of function spaces, J. Funct. Anal. 93 (1990), 34-170. [lO] Frazier, M., B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS-AMS Regional Conf. Ser. in Math. 79 AMS, Providence, 1991.
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[11]
[12] [13] [14]
[15] [16] [17]
[ls] [19]
539
Lemari~, P. G., Alg~bres d'op~rateurs et semi-groupes de Poisson sur un espace de nature homog~ne, Th~se, Orsay, 1984. Lemari~, P. G., Ondelettes vecteurs ~ divergence nuUe, Rev. Mat. Iberoamericana 8 (1992), 91-107. Meyer, Y., Ondelettes et Opdrateurs, vol. 1-3, Hermann, Paris, 1990. Sarason, D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. Taylor, M., Pseudodifferential Operators and Nonlinear PDE, Birkhguser, Basel, 1991. Tchamitchian, P., Wavelets and differential operators, in Different Perspectives on Wavelets, I. Daubechies (ed.), Proc. Sympos. Appl. Math. 47, AMS, Providence, 1993, pp. 77-88. Tchamitchian, P., Wavelets, functions and operators, in Wavelets. Theory and Applications, G. Erlebacher et al. (eds.), Oxford University Press, Oxford, 1996, pp. 83-181. Tchamitchian, P., Inversion de certains op~rateurs elliptiques ~ coefficients variables, SIAM J. Math. Anal., to appear. Triebel, H., Theory of Function Spaces II, Birkh/iuser, Basel, 1992.
J.M. Angeletti Laboratoire de Math~matiques Fondamentales et Appliqu~es Facult~ des Sciences et Techniques de Saint-J~rSme 13397 Marseille Cedex 20, France et LATP, CNRS, URA 225
[email protected] S. Mazet Laboratoire de Math~matiques Fondamentales et Appliqu~es Facult~ des Sciences et Techniques de Saint-J~rSme 13397 Marseille Cedex 20, France et LATP, CNRS, URA 225
[email protected] P. Tchamitchian Laboratoire de Math~matiques Fondamentales et Appliqu~es Facult~ des Sciences et Techniques de Saint-J~rSme 13397 Marseille Cedex 20, France et LATP, CNRS, URA 225
[email protected] This Page Intentionally Left Blank
Some Directional Elliptic Regularity for D o m a i n s w i t h C u s p s
Matthias Holschneider A b s t r a c t . In this paper we discuss how the position-scale half-space of wavelet analysis may be cut into different regions. We discuss conditions under which they are independent in the sense that the Toeplitz operators associated with their characteristic functions commute modulo smoothing operators. This is used to define microlocal classes of distributions having a well-defined behavior along the lines in wavelet spaces, and it allows us to describe singular and regular directions in distributions. As an application, we discuss elliptic regularity for these microlocal classes for domains with cusp-like singularities. An extended version of the present paper may be found in [5].
w
Introduction
The classical definition of local singular directions of a distribution rl is given by the wavefront set (see, e.g., [6]). At a given point x it is, roughly speaking, the cone of all directions in which the Fourier transform of the localized distribution Crl does not decay rapidly, where r is any smooth function t h a t is supported in some neighborhood of x. More precisely, for fixed r E C ~ (IRa), x E supp r a direction ~ E IRa\{0} is regular if in some conic neighborhood ~/~ ~ we have k E 7 =~ I(r
rl)^(k)l -< Cg(1 + ]kl) - y , N - 0, 1, 2, . . . .
The complement of the regular directions is denoted by Ex,r directions at x are then defined as
The singular
F~z -- NCEz,r Multiscale
Wavelet Methods
Wolfgang Dahmen,
541
for P D E s
A n d r e w J. K u r d i l a ,
a n d P e t e r O s w a l d ( e d s . ) , p p . 541-5{}5.
Copyright (~1997 by Academic Press, Inc. All r i g h t s of r e p r o d u c t i o n in a n y f o r m r e s e r v e d . ISBN 0-12-200675-5
M. Holschneider
542
where the intersection is over all r E II7~~ n) with x E supp r However this concept of singular and regular directions does not always fit with what one would intuitively call a regular direction in a distribution. W h a t we mean is best illustrated by an example in 1lt2. Consider the set K - {(x,y) E lR 2 9x >_ O, lYi , Vm > O, ~(k) = o(k m)
(Ikl
-~ 0).
The Schwartz space of functions over the half-plane lI-In shall be denoted by S(IHn). It consists of those functions r for which the following norms are all finite
JJrlJ~,l,m,n'-
sup
(b,~)e~ ~
J(a + 1/a)k(1 +
Ibl)~O~'O2r(b,a)J
< oo.
Note that this means that r, together with all its derivatives, decays rapidly for large b and for large or small a. It can be shown that w:
So(~ ~)
x
s0(~ ~) -+ s ( ~ ) ,
(g, ~) ~ wg~
is continuous. The same holds for the wavelet synthesis defined through
r(b'a) l~-h n (X -a b
dbdaa
and we have that
./~4 : 8o ( IRn) x S ( lH '~) ~ So (lR ) ,
( h , r) ~ ~[ h r
is continuous, too. However, in this paper we will not discuss topologies on the microlocal classes that we will define below. This can be done in an obvious way and we want to streamline the discussion. We note here the following important fact. For admissible g E 80(IR n) and arbitrary h E $0(]Rn), the cross kernel (2.3) is a function in 3(]Hn). It is thus very well localized.
3.2
Wavelet analysis of S~(R n)
We denote the space of linear continuous functionals 77 : 8 0 ( ~ n) -+ C by S~(IRn). We consider it together with its natural weak-, topology.
Directional Elliptic Regularity
547
The space $~(IR n) can canonically be identified with S'(IR)/P(IRn), where P(IR n) is the space of polynomials in n variables. The wavelet transform of 7/E $~ (IRn) can now be defined pointwise as Wgrl(b, a) = rl(gb,a). This is a smooth function (see, e.g., [4]) that satisfies
IWgv(b, a)i 0. By duality we have that the mapping (this time for the fixed wavelet g e So (IRn))
is continuous. Here S' (]H~) is the dual of S(]H n) together with the weak-, topology. On the other hand, consider any r E $'(]Hn). Then we set for s E
~0(~ n)
=
Clearly, for h E 5:0 (lR~) Mh
:
r
Mhr
is again continuous. In the case of a locally integrable function r of at most polynomial growth we have
,~4hr(s) - - / ~ n r(b, a) W-Ks(b , a) dbda
3.3
M o r e g e n e r a l spaces
Many function spaces can be characterized in terms of wavelet coefficients. As a rule, the faster the wavelet coefficients decay, the more the analyzed function is regular. We come to the details. For this, consider a vector space B(IH n) of locally integrable functions
S(~t ~) C B(~t ~) C S'(~tn). Suppose in addition that B (IHn) is invariant under convolutions with highly localized kernels
r e S(]I-In), s e B(]H n)
==~ r , s, s , r ~_ B(]Hn).
The convolution of two functions over IHn is defined by (2.2). It then makes sense to pull back B(lH n) to a vector space of distributions over ]Rn. We shall denote this space of distributions by B(]Rn). It is defined through the following theorem.
M. Holschneider
548
T h e o r e m 1. For a distribution ~1 E S~(IRn), the following are equivalent. 9 There is a wavelet g E So(IR n) which is admissible in that it satisfies 12 da ,~ 1,
f0 ~176
for which we have
Wgrl e B(lHn).
9 For all h E So (IRn) we have
who
e
Proof: The passage from Wgr/to Whr/is given by the highly localized cross kernel (2.3). By definition, this operation leaves B(lI-I n) invariant.II Therefore it makes sense to speak of the space B(lFt n) associated with B (]I-In). It is precisely the space of distributions for which ~Ygrl E B (]Hn), where g is as given in Theorem 1. We still shall need an additional technical assumption on the spaces B(lI-In). Their multiplier algebra should contain the bounded functions
m e L~176
s e B(lH n) =~ m . s e B(]Hn).
This allows us to define the Toeplitz operators
.MhmWg :
B(IR n) --+ B(IRn).
For the rest of the paper we refer to the spaces B ( ~ n ) satisfying all stated properties and their pulled back counterparts over lFtn, B(I[:tn), as admissible local regularity spaces. We end this section with a remark concerning topology. Suppose that in addition B(]I-In) is a Banach lattice with norm
II
liB(w,)= llSIIB(W,).
Suppose further that for fixed I I E S(IH n) we have that
r~II,r is continuous. Then we can define a norm on BOR n) which makes it a Banach space by setting
IlalIB(R )- IlWgSIIB( ) 9
Directional Elliptic Regularity
549
This is reasonable, since for different wavelets satisfying the hypothesis of Theorem 1, we obtain equivalent norms. It is easy to verify the sufficient condition for B(IH n) to be stable under convolution with localized kernels in the case that B(lH n) is a Banach space. It is sufficient to find K and c such that for all s E B(IH ~) we can estimate
I]~(~" +~, ~')lls(.-I-) _ 1. m This immediately implies the following lemma which we shall use in the next section.
557
Directional Elliptic Regularity
Lemma 4. Let E D f~ be such that E c is well separated from fl. Then there is a set E, E D E D f~ such that E is well separated from E c and f~ is well separated from E c. Proof: Some F~(f~) with e small enough will do. m
4.4
C u t t i n g the half-space
Let us come back to our original goal of dividing the half-space into two sets of different regularity. As we already said, it is not possible to speak of regularity B(IH ~) inside a given set f~ C ]H~, since this notion is not independent under highly regular Calder6n-Zygmund operators, or to put it more simply, it might depend on the given wavelet used for the definition. However, if we require regularity B(]H n) in a region which is slightly larger than f~, it then follows that the same regularity holds true in f~ for any wavelet. By abuse of notation, let E (respectively, f~) denote the operator that restricts functions over IHn to the set E (respectively, to f~). That is, we have E : r ~-4 xEr, where ) ~ is the characteristic function of E. Theorem 2. Consider two sets E and f~ and suppose that ~ D f~ in such a way that Ec and ~ are well separated. Let g, g', h, h' E So(IR n) satisfy for some c > 1 (s = g, g', h, h'), c- 1
0 and some c > 0
(b, a) e ~'
~
It(b, a)l O,
Plugging this estimate into the previous expression, we have to estimate for (b, a) e F.,
A((b,a)) g f~
IP'(b',a')l db'da' a---7--,
(4.1)
I / a T _ b .'~j
with P'(b',a') = A ( ( b ' , a ' ) ) g P ( - b ' / a ' , l / a ' ) . Together with P, we have that P~ is highly localized. For A ___0, let us look at the following integral running over the complement of a non-Euclidian ball centered at (0, 1)"
/~ ((v,~,))>~
IP'(b',a')l
dbldal
a'"
Thanks to the high localization of P~, this function is decreasing faster than any power of 1/A as )~ ~ c~. Now, since E and F.~ are well separated we may use characterization (4.1) to conclude that the integral in (4.1) is estimated by F(A((b,a)) ~) for some e > 0. But this function is again rapidly decreasing as A((b, a)) ~ c~ and the proof is finished. II
Directional Elliptic Regularity
559
Note that the lemma we want to prove may be rephrased as follows: for all P E S(]Hn), we have that s ~ .=.' (P 9 (Es)) is infinitely smoothing in the sense that it maps functions of polynomial growth into rapidly decreasing functions over the half-space. P r o o f of T h e o r e m 2: The previous considerations imply the following: if we have P~l/Ygrl E B(]Hn), with g admissible, then for all g' E So(lR n) we have f~Wg, r/ E B(]I-In). To show this, note that the transition from l/Ygrl to Wg,~ is done by convoluting with a highly localized kernel P. Now, we may write aWg, rl = f l ( P 9 (EWgr/)) + fl(P 9 (ECWgrl)). Since by hypothesis B(]I-I '~) is invariant under multiplications with bounded functions and convolutions with P, the first term is again in B(IHn), whereas the second term is arbitrarily smooth. A slightly more complicated situation occurs in our theorem, since we can not conclude from fl4hr E B(]R "~) that r E B(]nn), since the wavelet synthesis is not injective. We can find a set E between E and f~,
such that E is well separated from the complement of E and f~ is well separated from the complement of .=.. This follows from Lemma 4. We may conclude that Z W , Mhr W,
= =(P1 *
e
where P1 = W f h for any admissible f E 80(lRn). In particular, we may choose f to be a reconstruction wavelet for g and thus it follows that P1 * l/Yg~ = Wg~. Now writing (as characteristic functions!) P~ = 1 - Nc, the last expression equals e(ei
9
- =(el
9
The set E is well separated from E c and thus the second term has rapid decay as A((b, a)) gets large. Let us call this function u. Then, since u is well localized we have r , u e 8(ln ), for all r E 8(]Hn). We therefore obtain, up to a function of rapid decay,
M. Holschneider
560
Now }4;g,~ = P 9 )/Yg~ for some P E $(]Hn). Therefore, since f~ C E is well separated from the complement of E we have at the beginning of the proof that fB/Yg,~ E B(IH ~) up to the well localized function f~u. But then clearly Mh, f~Wg, r/E B(]Rn). m Let ~ D f~ be open and let again f~ be well separated from the complement of E as before. Consider the two Toeplitz operators Tr~ = MhY, W~,
T, = Mh~Vg.
We then have proved the following
Corollary 1. We have that [
, T, ] =
T, - T,
is infinitely smoothing in the sense that it maps the tempered distributions in S~ (IRn) into smooth functions in So(IR~). 4.5
Microlocal classes
The theorems of the previous section may be used to define some very general microlocal classes. Suppose we are given two regularity spaces A(IR n) and B(IRn), and suppose in addition that B(]R n) C A(]Rn). Consider a set f~ C ]I-I'~. Since we are only interested in local properties we may suppose that f~ is bounded in the Euclidian norm. In order to avoid technicalities, we suppose that ~ is closed. The first type of local regularity classes corresponds to the idea that globally a distribution has a regularity described by A(IH n) whereas locally, in f~, we have some higher regularity of type B(]I-In). A dual idea would be to have the wavelet coefficients concentrated on the subset f~. That is, outside of f~, the wavelet coefficients are small, hence correspond to the higher regularity B(IH~), whereas inside f~, the coefficients are in A(]I-In). We want to make these statements more precise. Consider first the case of higher local regularity. Suppose that there is a sequence of closed sets {f~k), k = 1 , 2 , . . . with ~1 C . . . f ~ k C f~k+X"" C f~. We suppose that f~k converges to f~ in the sense that f~ -- Uf~k. k
Directional Elliptic Regularity
561
Suppose that ftk and ft~+ 1 are well separated for each k. Then clearly f~k and ~t~ are well separated for k < I. We then say that ~/E $~(lR n) belongs to the microlocal class A(f~; A,B) iff for some admissible g and all k we have E A(IRn) and Mgf~kV1]g~E B(lRn). By the results of the previous theorem it is clear that the definition does not depend on the specific wavelets nor on the family of approximating sets f~k. Indeed, by Lemma 3 we may take the family F1/k(ft) as universal family of approximating sets. Note however, that for arbitrary ft, the previous class might coincide with A(IRn). Indeed, in order to have an approximating sequence from the interior, of mutually well separated sets the "smoother" region can not be arbitrary thin. It must contain at least some non-Euclidian neighborhood of some set. Frequently one takes A = S~(]Rn) in which case one is only interested in the behavior of the wavelet coefficients around ft. In the next section we shall use this kind of classes to define directional regularity in distributions. Consider now the dual approach, where we want to formalize the idea of wavelet coefficients concentrated on ft. Suppose now that a sequence of open sets {~tk}, k = 1, 2 , . . . with
converges to gt in the sense that f~-- Nf~k. k
Again we require that (f~k)c and ft k+l are well separated for each k. We then say that r/E $~(IR n) belongs to the microlocal class E(~t; A, B) iff for some admissible g and all k we have
e A(IRn) and Mg(f~k)cwgT1 e B(]Rn). Note that in the case where B(~ n) = $o(lRn), this corresponds to the idea of having the wavelet coefficients concentrated on the set gt, where they satisfy the less restrictive regularity estimate given by A(]I-In). w
S o m e directional microlocal classes
We now propose to look at more specific examples of regularity classes, in particular, to those we mentioned in the beginning of the paper, that is, to classes related to the notion of singular or regular directions in distributions. Particular useful examples arise when we consider parabolic regions
M. Holschneider
562
or lines in wavelet space. As measure of regularity it is useful to consider the HSlder-Zygmund scale A s of spaces defined in wavelet space via tlslla -
sup
la-as(b,a)l.
Fix a vector ~ E IRn, I~I > 0 and consider the set E = E(~, 7) - { ( b - ~ , a -
) ~ ) " 1/2 > ~ > 0)
for some 7 > 1. We now say that r/ E Aa(IR ~) is locally of type ( a , ~ , 7 ) if it belongs to the microlocal class ~A,B with Ft - F~(E(~,7)) for some e > 0, A - S(IR ~) and B - A~(IR~). More explicitly, this means that the wavelet transform of s satisfies for some ~ > 0,
(b,a) E F~(S(~,-),))
=~ I W g u ( b , a ) l 0 at x E 0Ft in direction ~, ~ E lRn - {0}, if there is some c > 0 such that
{yly-
1
0 such that for (b, a) E re(E) there is an n-dimensional Euclidian ball of radius > A((b,a)) ~ around b that is contained in the complement of the influence region of ft. Denote by r aC ~ function which is identically 1 on the complement of the unit ball of radius 1 and which is supported on the complement of a slightly smaller ball. Then denote by Cb,a the family of translates and dilates of r Therefore if p is supported by fl, it follows that
(b, ~) e ~ ~ W~p(b, ~) = W~(r
~) =