Discretization Methods and Iterative Solvers Based on Domain Decomposition

Lecture Notes in Computational Science and Engineering Editors M. Griebel, Bonn D. E. Keyes, Norfolk R. M. Nieminen, Espoo D. Roose, Leuven T. Schlick, New York

17

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Barbara 1. Wohlmuth

Discretization Methods and Iterative Solvers Based on Domain Decomposition With 82 Figures and 25 Tables

Springer

Barbara I. Wohlmuth Institut fur Mathematik Universitat Augsburg Universitatsstrafse 14 86159 Augsburg, Germany e-mail: [email protected]

Cataloging-in-Publication Data applied for Die Deut sche Bibliothek - CIP-Einheitsaufnahme Wohlmuth, Barbara : Discreti zation methods and iterative solvers based on domain decomposition I Barbara I. Wohlmuth. - Berlin; Heidelberg ; New York; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer , 2001 (Lecture notes in computational science and engineering ; 17) ISBN3-540-41083-X

Mathematics Subject Class ification (1991): 65N30, 65N15,65FlO, 65N55, 65N50 ISSN 1439-7358 ISBN 3-540-41083-X Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the materia l is concern ed, specifically the rights of translation, reprintin g, reuse of illu str ations, recitat ion, bro adcasting , reproduction on microfilm or in any other way,and storage in data banks. Duplicatio n of this publication or parts thereof is permitted only und er the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obt ained from Springer-Verl ag. Violations are liable for prosecution und er the German Copyrig ht Law. Springer-Verlag Berlin Heidelberg New York a memb er of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2001 The use of general descriptive names, register ed names, trademarks, etc. in this public ation does not imply, even in the absence of a specific statement, that such name s are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: Friedh eIm Steinen-Broo, Estudio Calamar, Spain Cover production: design & production GmbH, Heidelberg Typeset by the author using a Springer TEXmacro package Printed on acid-free pap er SPIN 10725050 46/3142/LK - 5 43 21 0

Preface

DOMAINE: [domen] Ce domaine est encore ferrne aux savants DECOMPOSER : [dek3poze] Decomposer un probleme pour mieux le resoudre

• Micro Robert : Dictionnaire du francais primordial

The num erical approximation of partial differential equations, very often , is a challenging task. Many such problems of practical interest can only be solved by means of modern supercomputers. However , the efficiency of the simulation depends strongly on the use of special numerical algorithms. Domain decomposition methods provide powerful tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applicat ions in science and engineering. Although the first domain decomposition techniques were used successfully more than hundred years ago, these methods are relatively new for the numerical approximation of partial differential equations. The possibilities of high performance computations and the interest in large-scale problems have led to an increased research activity in the field of domain decomposition. However, the meaning of the term "domain decomposition" depends strongly on the context. It can refer to optimal discr etiz ation techniques for the underlying problems, or to efficient iterative solvers for the arising large systems of equations, or to parallelization techniques. In many modern simulation codes, different aspects of domain decomposition techniques come into play, and the overall efficiency depends on a smooth interaction between these different components . The coupling of different discretization schemes , the coupling of different physical models, and many efficient preconditioners for the algebraic systems can be an alyzed within an abstract framework . At first glance these aspects seem to be rather independent. However , all have one central idea in common: The decomposition of the underlying global problem into suitable subproblems of smaller complexity. In general, a complete decoupling of the global problem into many independent subproblems, which are easy to solve, is not possible. Since, the subproblems are very often

VI

Preface

coupled, there has t o be communication between t he different subproblems. Althou gh t he t erm optimal dep ends on t he context, t he pr oper handling of t he inform at ion t ransfer across the int erfaces between t he sub pro blems is of majo r impor t an ce for t he desig n of opt imal methods. In the case of discretization techniques, a priori estimates for t he discreti zati on errors have to be considered. They very mu ch depend on t he appro pr iate couplings across the interfaces which are often realized by mat ching condit ions. The jump across the int erfaces which measur es the non conformity of t he method has to be bounded in a suitable way. In t he case of it er ativ e solvers , t he convergence rat e and t he computational effort for one it er ation ste p measure t he qu ality of a method. To ob t ain scalable it er ation schemes, very ofte n, one has to includ e a suitable global prob lem of small compl exity. In t his work, both discretiz ation t echniques and it erativ e solvers are addr essed . A brief overvi ew of different approaches is given and new t echniques and ideas are proposed . An abst ract framework for dom ain decomposition methods is pr esented and an analysis is carried out for new t echniques of special inter est . Optimal est imates for the methods considered ar e est ablished and numerical results confirm the theoreti cal pr edicti ons. Chapter 1 concerns special discretiz ation methods based on domain decomposit ion t echniques. In particular , t he decomp osition of geomet rical complex structures int o sub domain s of simple shape is of special int erest . Another example is t he decomposition int o subst ructures on which different physical mo dels are relevan t . Then , for each of t hese subp rob lems , an optimal appro ximation scheme involving t he choice of the triangulation as well as t he discretiz ation can be chose n. However t o obt ain optimal discretiz ations for the global problem , t he discret e subpro blems have been glued together appropriately. Here, we focus on mort ar finite element methods. To start, we review t he standard mor t ar setting for t he coupling of Lagrangian conforming finite elements in Sect . 1.1. Both standard mortar formul ations - t he non conforming positiv e definit e probl em and the saddle poin t problem based on t he un constrain ed product space - are given. In Sect . 1.2, we introduce and analyze alternative Lagran ge mul tiplier spaces. We derive ab stract condit ions on t he Lagran ge mul tiplier spaces such t hat t he non conforming discretiz ation schemes obtain ed yield optimal a priori results. Lagrange multiplier spaces based on a du al basis ar e of special int er est . In such a case, a biorthogonality relation between the nodal basis fun cti ons of these spaces an d th e finit e element t race spaces holds . A main advantage of these new Lagran ge multiplier space s is t hat t he locality of the support of t he nod al basis functions of t he constrained space can b e pr eserved. Wi th this observation in mind , we introduce a new equivalent mortar formulation defined on t he un const rain ed pr odu ct space in Sect . 1.3. We show t hat t he non-symmetric formulation can be analyzed as a Dirichlet Neumann coupling. Based on the elimination of t he Lagran ge mul tiplier , we deri ve a symmetric positi ve definit e formul at ion on t he un constrain ed

Pr eface

VII

product sp ace, and t he equivalence to th e positive definite problem on th e constrained space is shown. Two formulations, a variational as well as an algebraic one, are presented and discussed . A st andard nod al basis for th e unconstrain ed product space can be used in the impl ement at ion . The stiffness mat rix associate d with our new variational form can be obtain ed from t he standard one on the un constrain ed space by local operations. Section 1.4 concerns two examples of non-st andard mortar sit uations. Eac h of them reflects an int erest ing feature of t he abstract genera l fram ework , and illustrates the flexibility of t he method. We start with the coupling of two different discreti zation schemes. The matching at the int erface is based on the du al role of Diri chlet and Neum ann boundary condit ions. Two different equivalent formulations are given for the coupling of mixed and standard conforming finit e elements . In our second example, we rewrite the nonconforming Crouzeix- Raviart finit e elements as mortar finit e elements. We consider the extreme case t hat the decomposition of th e dom ain is given by t he fine triangulation and t hat t here fore t he number of subdomain s t ends to infinity as the discreti zation par am et er of t he t ria ngulat ion t ends to zero . Finally in Sect . 1.5, we pr esent several series of num erical results. In part icular, we st udy t he influence of t he choice of the Lagran ge multiplier space on t he discreti zation erro rs . Examples with severa l crosspoints, a corne r singularity, discontinuous coefficients, a rotating geomet ry, and a linear elast icity probl em are considered. A second t est series concerns th e influence of the choice of the non-mortar side. Adaptive and uniform refinement t echniques are applied. In our last te st series, we consider the influence of jumps in th e coefficient on an adapt ive refinement process at the int erface. Chapter 2 concerns it erative solut ion techniques based on dom ain decompos it ion . A bri ef overv iew of general Schwar z meth ods, including multigrid te chniques, is given in Sect. 2.1. Examples for t he standard H I-case illust rat e overlapping , non- overl apping, and hierar chical decomposition techniqu es. The following sect ions contain new results on non-st and ard sit ua t ions; we discuss vector field discretizations as well as mor t ar methods. Secti on 2.2 focuses on an iterative subst ruct ur ing and a hierarchical basis method for Raviart-Thomas finit e elements in 3D. We st art with the definition of th e local spaces and th e relevant bilinear forms and subspaces. The central result of t his sectio n is established in Subsect. 2.2.2; it is a polylogarithmical bound ind epend ent of t he jumps of t he coefficients across the subdomain boundaries of our ite rat ive substructur ing method. The tec hnical tools are discussed in det ail with par ti cular emphas is on t he role of t race t heorems, harmonic extensions, and du al norms applied to finit e element spaces . As in t he 2D case for standard Lagran gian finit e elements , we introduce three different types of subspaces called VII , Vp , and VT . We cannot avoid t he use of a global space to obt ain quasi-optimal bounds. But in contrast to th e standard Lagrangi an finit e elements in 3D , the low dimensional Raviar t - Thomas space associated with t he macro-triangulation formed by th e subregions can

VIII

Preface

be used to obtain quasi-optimal results where the constant does not depend on the jumps of the coefficients across the subdomain boundaries. Sections 2.3-2.5 concern different iterative solvers for mortar finite element formulations. In Sect. 2.3, we combine the idea of dual basis functions for the Lagrange multiplier space with standard multigrid techniques for symmetric positive definite systems. The new mortar formulation, analyzed in Sect. 1.3, is the point of departure for the introduction of our iterative solver. We define and analyze our multigrid method in terms of level dependent bilinear forms, modified transfer operators, and a special class of smoothers which includes a standard Gaufl-Seidel smoother. Convergence rates independent of the number of refinement steps are established for the W-cycle provided that the number of smoothing steps is large enough. The numerical results confirm the theory. Moreover asymptotically constant convergence rates are obtained for the V-cycle with one pre- and one postsmoothing step. Section 2.4 concerns a Dirichlet-Neumann type algorithm for the mortar method. It turns out to be a block Gaufi-Seidel solver for the unsymmetric mortar formulation on the product space. Numerical results illustrate the influence of the choice of the damping parameter. The transfer of the boundary values at the interface is realized in terms of a scaled mass matrix. This matrix is sparse if and only if dual Lagrange multiplier spaces are used. In Sect. 2.5, we study a multigrid method for the saddle point formulation . Two different types of smoothers are discussed; a block diagonal and one reflecting the saddle point structure. In the second case, the exact solution of the modified Schur complement system is replaced by an iteration, resulting in an inner and an outer iteration. This multigrid method is given for the standard mortar formulation as presented in Sect. 1.1. In contrast to the two previous sections, the use of dual Lagrange multiplier spaces does, in general, not reduce the computational costs for one iteration step. Acknowledgments: It is a great pleasure for me to thank my colleagues, friends, and parents for their support. In particular, I wish to thank Prof. Ronald H.W . Hoppe, University of Augsburg, for his support throughout this work, Prof. Yuri A. Kuznetsov, University of Houston, who introduced me to work in this area, and Prof. Dietrich Braess, Ruhr-University Bochum, for his interest in my work since we first met at the annual GAMM meeting in Braunschweig, 1994. This work was supported in part by a Habilitandenstipendium of the Deutsche Forschungsgemeinschaft and in part by the National Science Foundation under Grant NSF-CCR-9732208. In particular , I would like to thank my coworkers, Rolf Krause, PhD Andrea Toselli and Priv.-Doz. Christian Wieners for their support and fruitful discussions . Finally, it is a great pleasure for me to thank Prof. Olof B. Widlund of the Courant Institute, New York University. His criticism, his encouragement, and his preference for simplicity influenced my work. Augsburg, October 2000

Barbara Irmqard Wohlmuth

Contents

Preface .. . . . . . . . . . .... .. . ... ... ... ... ... ... ... .. . .. . . . . . . . . . . .

V

1. Discretization Techniques Based on Domain Decomposition

1.1 Introduction to Mort ar Finite Element Methods. . . . . . . . . . . . . . 1.2 Mortar Methods with Alt ern ative Lagrange Multipli er Spaces . . 1.2.1 An Approximation Property. . . . . . . . . . . . . . . . . . . . . . . .. 1.2.2 The Consiste ncy Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.2.3 Discret e Inf-sup Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Ex amples of Lagrange Mult iplier Spac es 1.2.4.1 The First Ord er Case in 2D . . . . . . . . . . . . . . . . .. 1.2.4.2 The Fir st Ord er Case in 3D . . . . . . . . . . . . . . . . . . 1.2.4.3 The Second Ord er Case in 2D . . . . . . . . . . . . . . .. 1.3 Discretization Techniques Based on the Product Space . . . . . . . . 1.3.1 A Dirichlet-Neum ann Formul ation. . . . . . . . . . . . . . . . . . . 1.3.2 Vari ational Formul ations . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.3.3 Algebraic Formul ations 1.4 Examples for Special Mortar Fini te Element Discretizations 1.4.1 The Coupling of Primal and Dual Fini t e Elements. . . . . . 1.4.2 An Equivalent Nonconforming Formul ation. . . . . . . . . . .. 1.4.3 Crou zeix-Raviart Finite Elements. . . . . . . . . . . . . . . . . . . . 1.5 Num erical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.5.1 Influence of the Lagr ange Multiplier Spaces . . . . . . . . . . . . 1.5.2 A Non-op timal Mortar Method . . . . . . . . . . . . . . . . . . . . .. 1.5.3 Influence of t he Choice of t he Mortar Side . . . . . . . . . . . . . 1.5.4 Influence of the Jump of th e Coefficients . . . . . . . . . . . . ..

1 3 11 15 19 24 27 29 33 36 37 40 43 47 50 51 54 58 61 64 71 74 83

2. Iterative Solvers Based on Domain Decomposition. . . . . . . . . 2.1 Abstract Schwarz Theory 2.1.1 Additive Schwarz Methods 2.1.2 Multiplicativ e Schwarz Methods . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Multigrid Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2 Vector Field Discretizations 2.2.1 Raviart-Thomas Finite Elements 2.2.2 An It erative Substructuring Method

85 87 88 95 97 99 101 103

X

Contents

2.2.2.1 An Interpolation Operator onto V H 2.2.2.2 An Extension Operator onto V F 2.2.2.3 Quasi-optimal Bounds 2.2.3 A Hierar chical Basis Method 2.2.3.1 Horizont al Decompo sition 2.2.3.2 Vertical Decomposition 2.2.4 Num erical Results 2.2.4.1 The 2D Case 2.2.4.2 The 3D Case 2.3 A Multi grid Met hod for t he Mort ar Product Space Formul ation 2.3.1 Bilinear Forms 2.3.2 An Approximation Property 2.3.3 Smoo thing and St ability Properties 2.3.4 Impl ement ati on of th e Smoothing Step 2.3.5 Numerical Results in 2D and 3D 2.3.6 Extensions to Linear Elasticit y " 2.3.6.1 Uniform Ellipticity 2.3.6.2 Numerical Result s 2.3.6.3 A Weaker Interface Condition 2.4 A Dirichlet-Neum ann Type Met hod 2.4.1 The Algorithm 2.4.2 Numerical Results 2.5 A Multigrid Method for th e Mortar Saddl e Poin t Formul ation . 2.5.1 An Approximation Property 2.5.2 Smoothing and St abili ty Properties 2.5.2.1 A Block Diagon al Smoother 2.5.2.2 An Ind efinite Smoother 2.5.3 Numerical Results

105 107 113 114 115 118 121 122 123 125 126 129 131 136 137 142 145 149 151 155 155 158 162 164 167 168 171 174

Bibliography

177

List of Figures

187

List of Tables

189

Notations

191

Index

195

1. Discretization Techniques Based on Domain Decomposition

This chapter concerns domain decomposition methods as discretization techniques for partial differential equations. We present different approaches within the framework of mortar methods [BMP93, BMP94]. Originally introduced as a domain decomposition method for the coupling of spectral elements, these techniques are used in a large class of nonconforming situations. Thus, the coupling of different physical models, discretization schemes, or non-matching triangulations along interior interfaces of the domain can be analyzed by mortar methods. These domain decomposition techniques provide a more flexible approach than standard conforming formulations. They are of special interest for time dependent problems, rotating geometries, diffusion coefficients with jumps, problems with local anisotropies, corner singularities, and when different terms dominate in different regions of the simulation domain. Very often heterogeneous problems can be decomposed into homogeneous subproblems for which efficient discretization techniques are available. To obtain a stable and optimal discretization scheme for the global problem, the information transfer and the communication between the sub domains is of crucial importance; see Fig. 1.1.

Fig. 1.1. Decomposition of a global problem into homogeneous subproblems

One major requirement is that the interface between the different regions is handled appropriately. Very often suitable matching conditions at the interfaces are formulated as weak continuity conditions. One possibility is to use a dual set of boundary conditions. Then, the coupling can be realized in terms of a Lagrange multiplier. The analysis of the resulting jump terms across the interfaces plays an essential role for the a priori estimates of the B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition © Springer-Verlag Berlin Heidelberg 2001

2

1. Discreti zat ion Techni ques Based on Dom ain Decompositi on

discretization schemes . In particular, optimal methods can only be obtained if the consistency error is small enough compared with t he best approximatio n error on t he different subdomains. The consistency error measur es the nonconformity of the method and cont rols t he information transfer at t he int erface. In Sect. 1.1, we review the definition of a standard mor t ar finit e element method. For an overview of mortar tec hni ques including spectral elements, we refer to [BD98, Ben99, BM97, BMP93 , BMP 94, LSV94]. Both standard mortar formulati ons - t he nonconfo rm ing positi ve definite pro blem and t he saddle point problem based on t he un const rain ed product space - are given. Bas ic ideas and tec hniques are explained in a standard context. In the following sections, new ideas res ult ing in alte rnative spaces are st udied, and different modificati ons and aspects of t he methods are conside red. We also introduce a new p ositi ve definite mortar formulati on base d on the un constraine d pr oduct space. In Sect . 1.2, a general framework for the const ruction of new Lagran ge mul tiplier spaces is given. Sufficient condit ions on t he Lagran ge multiplier space are introduced to obtain an optimal ord er global method. So far t he Lagrange mul tiplier spaces have bee n defined as finite element trace spaces. We focus on a special type of Lagran ge multi plier spaces based on a du al basis. T his idea is new and simp lifies t he impl ement ati on as well as t he it erat ive solution of t he arising problem . T he nodal basis functi ons of t he Lagran ge mu ltiplier and t he trace spaces satisfy a biorthogonality relation. As a consequence t he constraints at the interfaces can be eliminated locally, and t he local ity of t he supports of t he bas is functi ons of t he constrained space can be guaranteed. We focus on t he int ro du ction of a new equivalent mortar formulati on based on the unconstrained pr odu ct space in Sect. 1.3. T he biorthogo nality relat ion yields a diago na l mass matrix on t he non-mortar side. Then , the Lagran ge mul ti plier can be obtained in a local post pro cessing ste p, involving t he inver se of t hat diagonal matrix. T his observation serves as our starting point for t he int roduction of our new mor t ar formulation on t he un constrained pr oduct space. In a first ste p, we show t hat t he resulting unsym metric form is nothing else t han a Diri chlet-Neum ann coupling between t he different subdomai ns . A symmet rized form gives rise to a positiv e definite variational formulation on t he un constrained product space. T his third mortar formulation can be implement ed mu ch easier and solved more efficient ly than the standard nonconforming and saddle point formul ations. Section 1.4 concerns several nonconforming situations which are analyzed wit hin t he fram ework of mor t ar finite element methods. T he examples illustrate t he lar ge variety and flexibility of mort ar tec hniques . A first example is given in Sub sect. 1.4.1. T he coup ling of primal and du al methods is ana lyzed , using the du ality of essential and natural bound ary condit ions as our point of departure. T wo different formulations are given for t his coup ling, includ-

1.1 Introduction t o Mortar Finit e Elem ent Methods

3

ing a sa ddle point problem where the Lagran ge multiplier space is defined by piecewise constants. The second exa mple is based on Cr ouzeix-Ravi ar t elements . In contrast t o the pr evious example, t he number of subdomains t end s to infinity when the meshsize t ends to zero. Each element is regarded as a subdom ain and the Lagr an ge multiplier space is t he product space of one dim ensi on al spaces associate d with the edges of t he t riangulat ion. F inally in Sect. 1.5, num erical results are given which illustrate the perform an ce of mort ar discreti zation t echniques. We st art with a comparison of t he discretiz ation err ors for t he different choices of Lagran ge multiplier spaces given in Sect . 1.2. In a second part , we focus on t he choice of t he mortar side in case of discontinuous coefficients and highly non conforming triang ulations . Finally, the use of a pos t eriori error est imat ors shows that two completely different sit uat ions arise at the int erfaces depending on the choice of t he mortar side. Throu ghout t his chapte r, we emphas ize the role of stable proj ections for t he best approxi mation prop erty of t he constrained space, the approx imati on prop erty of t he Lagran ge mul tiplier space for t he consiste ncy err or, discret e inf-sup condit ions for t he a pr iori estimates for t he Lagran ge multi plier , mesh depen dent norms for measuring t he nonconformity, and dua l basis fun ctions as Lagrange mul tipliers for t he locality of the support of the nodal basis functions of the constrained space. The followin g elliptic second ord er boundar y valu e problem -div( a'Vu)

+ bu = f

in fl ,

u=O on afl

(1.1)

will serve as our mod el problem. Here, a is a uniformly positive definite matrix , aij E LOO(fl) , 1 i ,j d, f E L 2(fl) , O:S bE LOO(fl), and o C ]Rd , d = 2, 3, is a bounded polygon al dom ain. For simplicity, we assume that the coefficients aij and b are constant on each element of t he t riangulat ions. Alt hough the det ailed analysis is given in 2D, most of our result s also hold in 3D. In the following, we only point out differences between the 2D and 3D case if different t echniques for t he proofs are requ ired or if different qu alit ative results are obtained . In t hose cases, t he analysis of t he 3D case is carr ied out separately. All constant s 0 < c C < 00 throu ghout t his work are generic and might dep end on t he coefficients a and b, the aspec t ratio of the element s and subdomains, and t he order of the discreti zation method but they do not dep end on t he meshsi ze.

:s

:s

:s

1.1 Introduction to Mortar Finite Element Methods In t his secti on , we bri efly review the standa rd mortar method for t he coupling of Lagran gian finit e elements. We recall the non conforming positiv e definit e formul ation as well as the saddle point pr oblem and t he a pr iori est imates .

4

1. Discretization Techniques Based on Dom ain Decomposition

An examinat ion of th e mort ar pr ojection shows t hat t he support of a bas is functi on on t he non-mort ar side is, in genera l, non-local. Let D be decomposed int o K non-overlapping polyhedral sub dom ains Dk such t hat K

D=

U a, .

k =l

We restrict ourselves to t he geometrical conforming sit uation where the int ersectio n bet ween t he boundari es of any two different sub doma ins aDI n aDk, k i= I , is eit her empty, a vert ex, a common edge or face in 3D; see Fig. 1.2. We call it an inte rface only in t he latter case .

/1,'~ V~Yj(~,;

Q6

l~oocl"~P~--"'--,

ro9

~:cL_L-.L-J"-.:

Q y

= 1,.

Fig. 1.2. Geo metrically conform ing (left ) and nonconforming (right) sit uation

Geometrically nonconforming sit uations are technically more difficult to han dle. A possibility t o reduce t hese complications is to require t hat each vert ex of t he decomp osition is also a vertex of each adjacent triangulation; see t he right part of Fig. 1.2. We define for each subdomain a simplicial t riangulation 0. ;h k ' the meshsize of which is bounded by h k . The finit e element space of conforming Pn k elements on Dk associated wit h 0. ;h k ' nk 2:: 1, which satisfy homogeneous Dirichlet boundary condit ions on aDn aDk , is denot ed by Xh k ;n k ' No boundary conditions are imposed on X h k ;n k in t he case t hat aD n aD k = 0. The results can be eas ily generalized to ot her ty pes of t riangulations; see Fig. 1.3.

Fig. 1.3. Decom posit ion and non-m at ching triangulations in 2D and 3D

Each int erface aDI n a[h is associated wit h a (d - I )-dimensional triangulation, inherited from eit her 0. ;h k or Ti;h l ' In general, t hese t riangulations

1.1 Introduction to Mortar Finite Element Methods

5

do not coincide . The interfaces are denoted by I'm, 1::; m ::; M. For each int erface, there exists a couple 1 ::; l < k ::; K such that 1m = 8D l n 8D k . Since we are working with finit e dimensional Lagr an ge multiplier spaces on t he int erfaces, we have to define appropriate triangulations on I'm. The triangulat ion on I'm is called Sm ;h m , and its element s are boundar y edges in 2D and boundary faces in 3D of eit her Ti;hl or ~ ;hk ' The choice is arbit rary but should be fixed. Then , by definition, the Lagran ge multiplier space inh erits its triangulation from t he non-m ortar side. The adjacent side is called t he mortar side . We denote th e subdomain associated with t he non-mort ar side by D n( m) and t he one associated with t he mortar side by Dn( m) ' I

I

II1I1I1I1IIII11111111

h1«h k

Qk Q I

Fig. 1.4. Different non-matching t riangulat ions at an interface in 2D

In general ~ ; hk and Ti ;hl do not match across t he common interface. We remark t hat no condi tions on t he triangulations ar e imposed at the common int erfaces. Figure 1.4 shows two characteristic situations for non-matching trian gul ations at the interface. The situation on t he left typically arises for t ime dependent problems, e.g., in th e case of sliding meshes. In the case of highly discontinuous coefficients a, global triangulations having a jump in t he meshsizes on th e different sub dom ains might yield better results; see t he right of Fig . 1.4. For both situations, th e mor t ar finit e element method yields optimal a priori bounds for th e discreti zation erro r in the energy norm [BMP93, BMP94]. The constant in th e a priori est imate in 2D depend s neither on the ratio of the meshsize of two adjacent subdom ain s nor on th e distortion. In contrast to t he 2D case , the ratio hmor/hnon ente rs, in genera l, into the a priori est imates in 3D; see, e.g ., [BM97, BD98]. Here, h m o r denotes the meshsi ze on the mor tar side and h non t he one on the non-mortar side . Und er some addit ional assumptions on the trian gulation, the factor hmor/hnon can be avoided. We refer t o [KLPVOO, WohOOa] for a mor e pr ecise analysis of the constant s in the a priori est imates and to Subsect . 1.5.3 for some num eric al results illustrating the influence of t he choice of th e non-mortar side. In the rest of t his secti on, we restrict our selves to the 2D case . The discussion of t he 3D case is included in Sect . 1.2. To obtain t he mortar approximat ion Uh , as a solution of a discret e vari ati onal problem , there are so far two main approaches. The first one has been int roduced in [BMP93, BMP94] and gives rise to a positive definit e nonconforming vari ational problem. It is defined on a subspace Vh of the product space , th e element s of which sat isfy weak conti nuity condit ions across t he int erfaces. The const rained finit e element space Vi, is given by

6

1. Discr et izat ion Techniqu es Based on Dom ain Decomposit ion

Vh := { v E L 2 (D ) I Vl nk E X hk;nk' 1 ::; k::; K , I [v ]J.Lda = 0, J.L E Mh mbm) , 1 ::; m::; M} , 1m

where t he test space M hmbm) is given by 2

M hm b m) := { J.L E L b m ) I J.L = w l.,m' W E Xh n(m);nn(m) , J.L le E Pnm-I(e) , if e E S m;hm contains an endpoint of 1m } , (1.2)

where n m := n n(m)' T he indices 1 and k are rese rved for the subdomains wher eas the index m is used for an interface, an d we have to un derst and t he definit ion of n m in t his sense. In 3D, t he definition of the Lagrange mul tiplier space has to be modified in the neighb orh ood of 8 We remark t hat Mh m bm) is a mod ified trace space of codimension two , associated with t he 1D t riangulation on t he non-m ort ar side which is inh erited from In (m);hn(m)' T hus , t he space M hm bm) dep ends on t he choice of the non-mor t ar side . The global product Il~= 1 Mh m bm) is denot ed by M h , and is a subs pace of L 2 (S ), where S := U;;;=I"Ym' In pri ncip le, it is also possible to introduce a new independent 1D t riangu lation on each interface "[m. which is inherit ed neither from In (m);hn(m ) nor from 'Tf. (m);hn(m)' T hen , we have to impose suitable conditions on the meshsize or ad d adeq uate bubble functions to the finit e element space to obtain a discret e inf-su p cond ition. Such stabilization t echniques are discussed in [BMOO, BFMR98] for three-field approaches. The nonconforming formu lation of the mortar method can be given in te rms of t he constrained space Vh : Find Uh E Vh such that

,m,

(1.3) see [BMP 93, BMP94]. Here , the bilinear form a(·,·) is defined as J(

a(v,w) :=

L

J

k=l [ h

a\7v· \7w

J(

+ bv w dx ,

v ,w E

II HI(D

k)

k= 1

We rem ark t hat continuity was impose d at t he verti ces of t he decomposi t ion in t he first pap ers about mortar methods. However , t his condit ion can be remov ed wit ho ut loss of stability. Both t hese set tings guarantee uniform ellipticity of t he bilinear form a( ·,·) on Vh x Vh , as well as a best approximation err or and a consiste ncy error of optimal order; see [BMP 93, BMP94]. Combining Lax-Milgram 's and Strang's Lemm as, it can be shown that a unique solut ion of (1.3) exist s. If the weak solution U of (1.1) is smooth enough and H 2 -regularity holds, then we have the following a priori est imates for the discretization error in the broken HI-norm an d in the L 2 -nor m

1.1 Introduction to Mortar Finite Element Methods

7

(1.4)

see [BMP93 , BMP94, BDW99]. Here, we use a standard Sobolev notation for norms which are not explicitly defined and set IlvilI := ~~=I IlvllLtk ' v E n~=1 HI([h). 11 ·lls;D st ands for the Hs-uoru: on t he open set Den , and 1·ls;D is t he corresponding semi norm. In th e case that D = n, the index n is suppressed . Fur thermore, we have ellipticity of the bilinear form a(' ,') on Y x Y a(v , v) ~

c Ilvlli,

(1.5)

vEY ,

where Y is defined as K

Y := {v E

II HI(n

k)

I

vlen = 0, ![v] da = 0,1 :::;

k=1

m:::; M}

;

(1.6)

1m

see [BM95]. We not e t ha t Vh c Y . Then, the energy norm I I vll1 2 := a(v ,v) , v E Y is equivalent to t he broken HI-norm. In [Gop99, Theorem IV.l], it has been established t hat the ellipticity constant is independ ent of the number of subdomains. A similar estimate is given for th e three field approach in [BMOO] . T he proof of the a priori bounds for t he discretiz ation error (1.4) is based on a best approximation error result and an a priori estimate for th e consistency err or; see [BMP93, BMP94]. We note th at very often the a priori est imate in t he broken HI-norm is given in the weaker form K

Ilu -

uhli l :::; C L h~k lulnk+l ;!h

.

k=1

However , in [WohOOa] it has been shown t hat (1.4) also holds with a const ant ind ependent of the number of subdomains. The const raints at t he interfa ces guar antee th at the consiste ncy err or is at least as good as the sum of the best approximations erro rs on the different subdomains. Replacing Vh in (1.3) by the unconstrained product space

yields a consiste ncy error t hat is not bounded in terms of t he meshsiz e. To prove the best approximation err or of Vh , t he mortar projecti on, lIm : C (')'m) ----t W h m (')'m ), plays an important role. Here, the trace space W h m (')'m ) is given by

W h m (')'m) := {fl' E C(')'m)

I

fl = w l-,.m'

W

E

Xh n (m) ;n n( mJ '

8

1. Discretization Techniques Based on Domain Decomposition

and llm is defined in terms of the Lagrange multiplier space tt;» - v E CoCrm),

J

(v - llmv)p,du

= 0,

P, E

M h= Crm)

(1.8)

1m

We have just recalled the definition of the mortar projection as in the original mortar papers. In Subsect. 1.2.1, we will consider a more natural way in defining it. It can be easily seen that the operator llm is well defined; see [BMP93, BMP94]. For the analysis of the approximation error, it is sufficient to show that the mortar projection is uniformly stable in suitable norms. The HJ-stability of the mortar projection is proved in [B~P93, BMP94]. This operator can be extended in a stable way to a linear and continuous 2Crm) operator from H66 onto WO;hmCrm) := HJCrrr,) n Wh=Crm); we will still denote the extended operator by llm. We find the following inclusions x; n HJ(fl) c Vi, c x; and observe that replacing Vi, in (1.3) by Xh n HJ(fl) or Xh does not provide a good discretization scheme. It is obvious that the quality of the nonconforming approach (1.3) and the properties of Vh depend on the space M h . Let us consider the structure of Vh in more detail. In general, Vi, is not a subspace of HJ (fl) and thus (1.3) is a nonconforming finite element method. Even for a nested sequence of global triangulations, the corresponding finite element spaces are non nested. Secondly in general, no basis of Vi, with local support can be constructed. The constraints are given in terms of a L 2-orthogonality of the jumps, and an element Vh E X h belongs to Vi, if and only if

J

llm([Vh])P,h do

= 0, P,h

E

Mh m Crm), 1 < tri < M

1=

In particular, if Vh E Vi, and [Vh] E CoCrm), 1 ~ tri ~ M, then llm([Vh]) = 0, 1 ~ m ~ M. A nodal basis function on the mortar side has to be extended to the non-mortar side such that the matching conditions are satisfied.

Fig. 1.5. Structure of the support of a nodal basis function in Vh, (standard)

Figure 1.5 shows the typical support of a basis function in Vi, associated with an interface, where the non-mortar side is on the left. The support of such a nodal basis function on the non-mortar side is a strip of length liml and width h m , and the locality of the basis functions is lost.

1.1 Introduction to Mortar Finite Element Methods

9

Figure 1.6 illustrates the trace of a basis function of Vi, on the two different sides of an interface for the two choices of the mortar side. In the left part of Fig. 1.6, the mortar side is associated with the finer triangulation whereas in the right part, it is associated with the coarser mesh. Although the basis functions on the non-mortar side have a global support, their values decrease exponentially.

'.~

1\ '[~}J"'"

.20.6 c 0.8 iJl a 4

~

.~

'1'

.20.6 r c; 0 . 8 . .~ 0.4 t

\ \

~o;J

iot~i\/ _02~ d

c_O.2

§

1

I\'

! I,

~O.6

2°.4

I

~O.2 ~

..

/

a

-I'

~O.6

2°·4

\

:@O.2

\

\

s °l=..:=:=::::'~~'==:J

-

non-mortar side

mortar side

non-mortar side

mortar side

is 1 ~O.8

(I

i o.s

Ij

Fig. 1.6. Nodal basis function on a mortar and non-mortar side, (standard)

These observations motivate a second approach which was introduced in [Ben99] and further studied in [Woh99a]. The matching conditions on the interfaces are not imposed on the global space but realized by means of Lagrange multipliers. The starting point is a constrained minimization problem leading to the following saddle point formulation: Find ('Uh' Ah) E ix; M h ) such that

a(uh' v)

+ b(v, Ah) =

(1, v)o,

v E Xh ,

= 0,

b('Uh,/~)

ME

(1.9)

Mh ,

where the bilinear form b(·,·) is given by the duality pairing on the interfaces M

K

b(V,M):= L([V],P,),m,

v E

II H

(fh ), ME

k=l

m=l

and [v] := via n(m) - vla_n{m) . Here, (H

H

M

1

II (H~(rm))

,

m=l 1 2(rm))' /

denotes the dual space of

1 2(rrn), /

and (', '),m stands for the duality pairing. Since the solution of the positive definite variational problem (1.3) and the first solution component of the saddle point problem (1.9) are equal, we use the same notation. The discrete Lagrange multiplier Ah approximates the flux. A priori estimates for the error A - Ah can be obtained, using the approximation property of M h and a suitable inf-sup condition. Here, Al,m := a ~~, where n is the outer normal on f!n(m)' This issue was first addressed in [Ben99], where a priori estimates in the H~t2 -dual norm have been established. The H~t2 -dual norm is defined by

L M

:=

m=l

IIMI1 2

L M

1 2

( H 0 0 h m ))'

:=

rn=l

sup

V

E

()2 V;M 1m

H~00 ( 1m ) IIVII IIoohm) .~

10

1. Discr etizati on Techniques Based on Dom ain Decomposit ion

where f-L E (H~t2(S)) ' and (H~t2 (S) )' := TI~= l (H~t2bm)) ' . The space H~t\'fm ) can be interp reted as an int erpolation space between £ 2b m) and HJb m). Working with a post eriori est imates , it is ofte n more convenient to deal with mesh dep end ent norm s. Here, we consider a mesh dep end ent £ 2-norm given by M

1If-LI I ~t- ! ;S

:=

L L

m=l eE S "" h",

h ellf-L 116;e'

f-L E £2(S) ,

wher e h e is the diam et er of t he element e; see [AT95]. A pr iori bounds for t his mesh depend ent £ 2-norm ar e der ived in [Woh99a]. The qu ality of the a priori est imat es in the H~t2 -dual and this mesh dep end ent £ 2-norm is t he same . 2 We use II . 11 M' to mean eit her t he HU -dual norm , II . II (H~62 (S) )" or t he mesh depend ent £2-norm, II . IIh-t /2;s . As in th e general saddle-point approach; see, e.g., [BF91], the essent ial point is to establish adequate inf-sup condit ions; such bounds have been est ablished with constants ind ependent of t he meshsize for both these norms; see [Ben99, Woh99 a]. Thus, there exists a constant such that (1.10) We not e t hat , so far , no inf-sup condit ion of t he form (1.10) has been established for the H 1 / 2 -dual norm , and t hat no a pri ori est imates are available in t hat norm. If t he solut ion u is regular enough, we find t he following a priori estimat e for t he Lagran ge multiplier by mean s of (1.10) and t he approximation pr op er ty of M h K

11..\ - ..\hll ~, ::; C L h~nk luli+nk;!h

(1.11)

k=l

T he proof for t he du al norm is given in [Ben99] and t hat for t he mesh dependent £ 2-norm in [Woh99a] . We remark t hat t he bilinear form b(·, '), defined on X h x M h is not uniformly cont inuous. To see this, we consider t he following exa mple. Let Vh E X h be const ant one on one subdomain rho and zero elsewhere, and f-Lh E M h be const ant one on one interface 'Ymo C 8D ko and zero elsewhere . Then , if the triangulation on the non-mortar side of 'Ym is qu asi-uniform with meshsize h m o , we find (1.12)

1.2 Mortar Methods with Alt ernative Lagrang e Mul tiplier Spaces

11

However , b(·, .) is uniforml y cont inuous for both II ·IIM,-n orm s if X h is replaced by a suitable subspace. For th e pr oof of (1.11) , it is imp ort an t that (1.10) also holds if t he supremum is taken over this subspace . Det ails are worked out in Sub sect. 1.2.3.

1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces In Sect. 1.1, the const ra ints at the int erfaces 'Ym are realized by means of a global L 2-projection , with test and t rial spa ces almost the sa me. In t he nonconforming vari ational formulation (1.3) , we have to face the problem th at the basis fun cti ons of Vh cannot be given easily as linear combinat ions of t hose of X h , since th e const ruc t ion of a basis of Vi, involves the solution of mass matrix syst ems for each interfac e. Furtherm ore, th e supports of th e basis functions associated with th e interfaces ar e non-lo cal on the nonmortar sides. Following th e second approach, (1.9) , we can work with th e un constrained product space X h . However , this formul ati on gives rise t o an ind efinit e problem. We note t hat efficient iterative solvers for saddle point pr oblems are often more complex th an those for positive definit e problems. In [SS98], it is shown t hat suit able lower dimensional Lagran ge multiplier spaces also yield optimal discreti zation schemes and t hat, without loss of optima lity, t he ord er of the Lagran ge multiplier space can be redu ced by one compa red wit h t he standard approach given in Sect. 1.1. One characteristic of t he Lagran ge multiplier spaces int rodu ced in [SS98] is t hat t hey are modified trace spaces of lower order conforming finit e element discretizations. Here, we pr opose different , more flexible, spaces for t he Lagran ge multiplier , in par ticular , spaces based on a du al basis. We recall t hat th e Lagran ge multiplier Ah in th e saddle point approach prov ides an approximation of t he flux. A fun ction v E H 1(f h ) has it s trace in H 1/ 2 (8fh ) and th e normal compo nent of its flux is in t he dual space H - 1 / 2(8[h). This observat ion is t he starting poin t for t he const ruction of a new ty pe of discret e Lagrange multiplier spaces. A special exa mple of a du al basis in t he first ord er case, nk = 1, in 2D has already been st udied in [WohOOa]. Here, we pr esent du al basis space s for the qu adratic case in 2D and the first ord er case in 3D. Before we define our new du al bases, we develop a general fram ework for Lagrange multiplier spaces which yield optimal results. We introduce a subspace of L 2 b m) of dimension N m :S dim WO;h = b m), and give appropriate assumpt ions und er which this space can replace the standard Lagr ang e multiplier space defined in (1.2). The spaces W h = bm) and WO ;h= bm ) = W h = bm) n HJ bm ) ar e introdu ced in Sect. 1.1 as finit e element t race spaces on t he non-mortar side . For convenience, we keep the same not ati ons as before, and denot e each element of t he new abstract class of Lagr ange multiplier spaces .by M h = bm) . We assume that th ere exists a bas is {1/Ji I 1 :S i :S N m } of M h Tn bm) satisfying t he following pr operties:

12

1. Discret ization Techniques Based on Domain Decomposit ion

(Sa) Locality of t he support : #(SUpp'¢i) ::; C,

1 ::; i ::; N m

,

#(p) ::; C, p E 'Ym , where # (SUpp'¢i) is t he number of elements in S m ;h m havin g a non-emp ty int ersection with t he simply connected supp ort of '¢i , and #(p) is t he number of functions '¢i such t hat the point p is contained in t he support of '¢i. (Sb) Approximation property of Mhmb m): For each f-t E H nm-I / 2b m) , t here exists a f-t1/J E M hmbm) such t hat

L

eES rn j h

hel/li -

f-t1/J/1 5;e::; Ch;;:m 1f-tI;'m- I/2;'Ym

1n

In 2D, he is t he length of t he edge e and in 3D, he denotes t he diam et er of t he face e. For a given basis, condition (Sa) is easy to verify and nat ur al in t he finite element context. Assumpti on (Sb) requires t hat the constants are contained in t he Lagrange multiplier space . As in t he standard mortar sit uation, a L 2 -pro jection-like operat or plays an essent ial role in establishing t he approximat ion property for th e constrained space . We base t he more general mort ar proj ecti on on a second set of linearl y independe nt functi ons 8i E WO;hmbm ), 1 ::; i ::; u.; having simply connected local suppo rts . The space WO;hmb m) := span {8i I 1::; i::; N m } is a subs pace of t he trace space WO;h mbm) . It is a prop er subspace if N,« < dim WO;hmbm) . Obviously t he space WO ;hmbm ) cannot satisfy an approximation property for H I bm) ' To establish an optima l order upp er bound for t he consiste ncy erro r, we use the following modification

e

where i E W hmb m) form a set of linear ind ependent funct ions havin g simply connected local sup ports. By const ruction, t he t hree spaces M hmbm), WO;hmbm ) and W hmbm) have the same dimension. The following t wo assumpt ions concern t he discret e spaces WO;hmbm ) an d W hmbm) and their relat ion to M hmbm): We assu me t hat for M hmbm ), there exist two sets of bas is functions defining WO;hmbm ) an d W hmbm ) such t hat (Sc) an d (Sd) holds: (Sc) Ap prox imation pr operty of WO;hmbm ) and Wh mbm): For each v E Hob m) and H Sb m), 0 ::; s ::; 1, there exists a Vo E WO;hm bm ) and W hmbm), respectively, such t hat

2: ;;hllv - vo l16;e ::; Clvl;;'Ym ' Ivoll;'Ym::; C lvk'Ym'

eESrn ;hm

e

v E H Ib m) .

1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces

c D1/J :S u .o;'

(Sd) Spectral equivalence:

c D1/J :S u.ir; -

--1

Mr :S C D1/J u,T

(ddij := 6ij IIBiI16;'Ym' zIIB iI16;'Ym' (dd ij := 6ij h

o.

(d1/J)ij :=

,

-

:S C D1/J'

where the elements of the diagonal matrices D 1 , D1/J' are given by

b,

6ijll1/JiI16;'Ym'

(d1/J)ij := 6ijh~ill 1/JiI16;'Ym'

13

i E {I, 2} ,

and D1/J E lRNm «n.; 1 :S i ,j:S N m 1:S i,j:S

,

u.; ,

where hOi and h1/Ji is the diam et er of supp Bi and supp 1/Ji' respectively, and the mass matrix M 1 E lRNm xNm is defined by

(md ij

:=

J 'Ym

1/Ji Bj da,

1:S i,j :S ».;

To obtain D z , Dz , and M z , we replace the basis functions Bi of WO ;h m (')'m), in t he definitions of D 1 , D1 , and M 1 , by the basis functions i of W h m ('Ym) . The mass matrices M i , i E {I, 2}, are sparse due to the locality of the supports . If N m = dim WO ;h m (')'m) , the approximation property (Sc) for WO ;h m (')'m) is automatically satisfied.

e

Remark 1.1. The two approximation properties (Sb) and (Be) are of different nature. (Be) is a low order approximation property which does not depend on the order of the finite eleme nt approximation on the subspaces . In contrast, the order of th e approximation property (Sb} depends on the order of th e finite element approximation on th e non-morta r side. We now use as our new Lagrange multiplier space M h = n~=1 M h m (')'m), wher e Mh m (')'m) is th e abstract discr ete space given by

The alternative non conforming finite element space Vh is defined as before in terms of M h

Vi,:= {v E LZ(fl) I

J [v]Jldo =

'Ym

Vln k E Xhk ;n k ' 1:S k:S K , 0, Jl E M h m (')'m) , 1 :S m :S M}

(1.13)

We rem ark t hat formally (1.13) is exactly the same definition as in (1.2). For simplicit y, we use the sam e notations as before , but we point out that now Vh , Uh, and Ah do not depend only on the order of the discr etization and the triangulation but also on the special choice of M h . We bri efly recall the two different mortar settings. The symmetric positive definite one can be written as: Find Uh E Vh such that

14

1. Discretization Techn iqu es Based on Domain Decompositi on

(1.14) In t he saddle poi nt ap proach, t he unconstrained pr odu ct space X h is independe nt of the choice of M h . The abstract saddle point formulat ion has exactly t he same st ruct ure as in (1.9). The only difference is that now t he special Lagran ge multiplier space (1.2) is replaced by t he genera l one: Find (U h' A h) E (X h , M h ) such t hat (1.15) Within t he genera l saddle point framework, the approximation property of

Vi" is a conseq uence of th e approxima t ion property of X h , t he cont inuity of

the bilinear form b(·, ') , and an inf-sup conditi on; see, e.g., [BF91]. A discret e inf-sup condition is necessar y in ord er t o obtain a prio ri est ima tes for the Lagran ge multiplier. In t he mortar sit uat ion, t he bilinear form b(· ,·) is not uniformly cont inuo us on Xi; x Mh ; see (1.12). Thus one has to be very careful in the a priori analysis. In t he following t hree subsectio ns, we show th at we have t he sa me quality of a pri ori est imates for t he solut ions U h and (U h' A h) as before for the standard case . Sub sect ion 1.2.1 is devot ed to t he analysis of t he best approxim at ion pr op erty of t he constrained space. It is based on t he stability est imate in t he H~{/ -nor m for t he genera lized mort ar projecti on. To establish t his stability, assumpt ions (Sa), (Sc) , and (Sd) are required . In Sub sect . 1.2.2 , we consider th e consiste ncy error of t he abstract nonconforming mort ar formulati on (1.14). In order to obtain a consiste ncy error of at least t he same order as t he best approximation error, assumptions (Sa)(Sd) are necessar y. These are t he two basic tools t o obtain a pri ori est imat es for U - U h . To obtain t he required ord er of t he consiste ncy erro r, t he assumption (Sb) is of crucial imp ortan ce. Roughly speaking, we have t o require t hat t he space M h = hm ) cont ains t he space of polynomials of ord er j, n m - 1. The stability of the sad dle point probl em relies on a suitable inf-sup condition . We establish a discrete inf-sup condit ions in Subsect . 1.2.3. Based on t hese preliminar y considerations, we obt ain a priori estimat es for t he Lagrange multiplier which are of the same ord er as t he a priori est imates for U - U h in t he energy norm . In Sub sect. 1.2.4, we pr esent an d analyze several exa mples for Lagran ge multiplier spaces wit h particular emphas is on dual basis spaces . The advantage of t hose space s satisfying a suitable biorthogonality relation is th at t he nodal basis fun ctions of Vh have a local support . Examples are given in 2D and 3D and for piecewise quadrat ic finit e elements . For higher ord er elements, we refer to [OWOO] .

1.2 Mortar Met hods with Alt ernative Lagrange Mult iplier Spaces

15

1.2.1 An A p proxim ation P r op er t y The essent ial tool in t he pr oof of t he approximation property of t he const rained space Vh is the stability of t he mort ar proj ecti on JIm . In t his subsection , we introduce a mod ified mor t ar pr ojecti on which depends on the choice of t he spaces WO ;h m (,m) and Nh m (,m) . The modi fied mortar projectio n, JIm : L 2 (,m ) --7 W O;h m (,m), is defined by

J

tt; » /-l do =

J

V/-l da ,

/-l E M hm(,m)

(1.16)

1m

1m

We observe that t he dimension of W O;h m (,m) and M h m (,m) is t he same by construction. However, t he mor t ar pr ojection JIm will not be well defined for arbit rary choices of WO ;h m (,m ) and M h m (,m).

R emark 1. 2. I n th e original papers about mo rta r m ethods, th e mo rt ar proj ection fo r the stan dard Lagrange m ultiplier space is given in a different way; see S ect . 1.1. It satisfies an approximation property for H I-functions, and 2 can be exte nded to H 66 -functions in a stable way . H ere, we us e a different f orm which does not tak e the values at the endpoints into account. However, both definitions give rise to operators which are identical when restrict ed to 2 H 66 (,m ). W e note th e follo wing difference: Th e operato r defin ed by (1.16) is in contrast to th e one given by (J.B) n ot H I (,m) -stable but L 2 (,m )-stable. Figure 1.7 illustrates the st abilit y properties of the two m ortar projection s.

\ I VIO-,

-8

j_nw L2 - stability

(1.8) IIn vll o- h (1.16) Ilnvll - 8 o

1

H - stability

Iv1

=0

1

I

_n~

(1.8) Inv11=0 (1.16) Inv1 !;: 0 1

F ig . 1. 7. St abili ty prop erties of (1.8) and (1.16)

In the following, we work with t he mortar project ion JIm defined by (1.16). The stability of t he mortar pr oj ection plays an essent ial role in the analysis of t he best approximation erro r. The following lemma provides uniform stability in t he L 2 _ and H I- norm s.

16

1. Discret izati on Techniques Based on Dom ain Decomp osit ion

Lemma 1.3. Un der the assumpti ons (Sa) , (Sc), and (S d), th e m ort ar projection (1.16) is well defin ed and is £ Z-st able and HJ- stabl e:

IIllmvII O;I~ IllmviI: l~

:::; Cll vll o;I~ ' :::; C l v h ;I~ '

v E

£2b m) ,

v E H J b m) .

P roof. The spectral equivalence (Sd ) shows that M I is non-sin gul ar and t hus t hat llm is well defined. Using the explicit rep resent ation llmv = 2:~1 ai Bi, we find M Ia

wher e r E IE.N~ , with ri := simply connec te d su pport we have N~

IIllmvI1 6;1~

:::; C L

J

=r

v'l/Ji da , 1 :::; i :::; N m . Due to t he locality of t he 1 :::; i :::; s.; and the linear indep enden ce,

OlOi'

a; I I Bi I 16;1~ = C a T o, « = CrT M I- T D IMI-I r

i =1

Finall y, t he ass um ptions (Sa) and (Sd) yield the £ Z-st ability

Reasoning as before, we get the stability of t he general mort ar pr oject ion in a weight ed £ z-n orm

(1.17) Here, we have used t he locality (Sa ) and t he spectral equivalence (Sd) . For t he pro of of t he H J -st ab ility, we use t he approximation property (Sc) of t he space WO ;h ~ (I'm ) an d an invers e estimate for polynomials. By means of t he best approximation ve E WO ;h~ bm ) and (1.17 ) , we find for v E HJ b m) Illm viI: l~

:::;

Illm( v - ve)h ;l~

:::; C (

2:

eES1T1 jh11l.

+ I veiI: l~

; 2 1Illm( v - ve )116;e) e

1

'2

+ IveiI:l~ :::; ClviI:l~ .

0

1.2 Mor tar Methods with Alt ern ative Lagrange Multiplier Spaces

17

H6/

The -stability of the mortar projecti on is guaranteed by t he £ 2_ and HJ -stability and an interpo lation arg ume nt. Following t he lines of [BMP 93, BMP94], it is easy to establish an approximati on pr op er ty for V h in the 2D case provid ed t hat the mor t ar projecti on 2 IIm is -stable, For eac h subdomain fh , we use t he Lagr an ge interpolat ion ope rator h , and define Wh E X h by Wh !n k := h u. We not e t hat Wh , in gene ral, will not be contained in V h . To obtain an element in V h , we have to add appro priate corrections. We obser ve that t he jump [Wh] is in HJ bm) for each interface "[m. in 2D . This is not t rue for 3D , and t he proof of t he approximat ion pr op er ty has to be modified . In the 2D case , we apply t he mor t ar projection t o [Wh] . The resul t llm[Wh] is extended by zero onto ofln (m ) \ "[m. and the exte nsion is still denoted by llm[Wh]. Then , the defini-

H66

t ion of WO;h ~ b m) yields llm[Wh] E H 1/ 2 (o fl n (m ) ) ' Now , llm[Wh] is extended as a discrete harmonic fun cti on into the interior of fln( m )' Finally, we define M

Vh := Wh -

L

H n (m ) (llm[Wh]) ,

(1.18)

m=l

wher e H n (m ) den ot es the discret e harmonic exte nsion operato r sa t isfying

see [Bra66, Wid88]. Here, the extens ion H n (m ) (llm [Wh ]) vani shes outside fln (m ) ' By construct ion , we have

and t hus Vh E V h . Repl acing t he test space M h in t he definition of the nonconforming sp ace, we find t hat t he approximati on prop er ty is pr eserved 2 as lon g as t he modified mort ar proj ection is -st abl e.

H66

Lemma 1.4. Under th e assumption s (Sa) , (S c) and (S d) on Mh ~ bm), 1 :S m :S M , th e nonconforming space Vh satisfies the approximation property in 2D, (1.19 )

if u is regular enough. P roof. We set V h E V h as in (1.18) and find by t he and a coloring argument

H662-st ability of llm

18

1. Discreti zation Techniqu es Based on Dom ain Decomposition

o We rema rk t hat t he constant depend s on t he shape regulari ty of t he t riangulations and t he decomp ositi on but not on t he number of subdomains. In par ti cular , t he ratio of t he length of adjacent edges on t he non-mort ar sides ente rs in the boun ds. But t he ratio between the meshsizes on mort ar and nonmor t ar sides does not enter. For a more det ailed analysis of th e constants in a special case, we refer to [WohOOa].

The 3D ca se . In the 3D case, we have t o face t he fact t hat t he boundar y of 81m is a closed one dimensional cur ve. For th e ana lysis of t he standard mortar situation, we refer to [BM97, BD98 , LSV94]. We cannot expect t hat [wh]1"In> = 0 on 81m in t he case of non-m at ching meshes on 81m, where Wh is defined as before. Since t he mor t ar project ion given by (1.16) is L 2- st abl e bu t not H I-st abl e, no uniform stabilit y results are available for the H~!/­ norm . T hus, we can not work with the discret e harmonic extension H n(m) from H I / 2 (8 Dn (m)) onto HI(Dn(m)) as in t he 2D case . We replace H n(m) by iin(m) supporte d only in a small st rip in t he neighb orh ood of "[m. on t he non-mort ar side . It is defined by its valu es at the Lagran gian int erp olation points x p { v(x p ) , X p E "[m , ( H~ n(m)v ) (x ) p := 0, elsewhere . Now, an inverse inequalit y yields, for v E Il ~= 1 M

Wh ",

M

L Iliin(m)Vlli;S]n(",) < e L L

m= 1

m =1 eES",;h",

(')'m ),

~e Il vl1 6;e.

Proceedin g as in t he 2D case and observing t hat JIm is stable in t his weight ed L 2 -nor m finally yield an a pr iori est imate which depends on (1 + hmor/hnon)

where hmor/hnon stands for the maximum ofthe local ratio between meshsizes on t he mort ar and non-m ort ar sides, and S m C Dn(m) is a st rip of width hn(m ) and area 1 1ml on the mor t ar side. We not e that ISml /IDn(m)1 te nd s t o zero if t he meshsize te nds to zero. Alt hough the constant in t he a priori est imate depend s on (1 +hmor/hnon), it might be advantageous to associate t he non-mortar side with t he finer mesh.

1.2 Mortar Methods with Alt ern ative Lagran ge Multiplier Spaces

19

We refer t o Sect . 1.5.3, for num erical results illustrating the influence of t he choice of the mort ar method. In general , using very different meshsizes on t he mortar and non-mort ar sides is only appro priate if the coefficients are st rongly discontinuous. Then , this jump of the coefficient s is also reflect ed in the constants of t he a priori est imate s and t his effect might cancel the factor h m or / h n on . For a more det ailed analysis of the constant in te rms of the coefficients, we refer to [WohOOa]. In par ticular , we find t hat t he ratio a non / a mor ente rs int o t he constant s of the a pri ori estimate in the energy norm. Here, a non and a m or stand for un upper and lower bound for the eigenvalues of the coefficient a on t he non-m ort ar and mor t ar sides, resp ectively. Remark 1.5. W e note th at in th e special situation th at th e triomqulations 'T",( m );hn (", ) and'Tn(m);hii(m) coincide on o"(m; see Fig. 1.8, an d nn(m ) = n n(m)' 1 ::; m ::; M , we can pro ceed as in th e 2D case . Then , applying th e Lagrange in terpolati on yields [wh]I )'", = (In(m)U - I n(m)n)l )'m E H~t2 bm). W e refe r also t o [KLPVOOj for th e special cas e n k 1, 1 ::; k ::; K.

=

Fig. 1.8. Trian gulations on mortar and non-mortar side of "(m in 3D

1.2.2 The Consistency Error T he definition of Vh guarantees t hat t he jump of an element is orthogonal to t he Lagran ge multiplier space M h . However , in general, t he £ 2-norm of t he jump will not vani sh . Thus, we are in a non conforming set t ing and Vh HJ(fl). In t his case, the approximat ion prop erty (1.19) is not sufficient to obtain opt imal a pri ori est imate s for the finit e element solution of (1.14) . According to Stran g's Lemma, the consiste ncy err or

ct

has to be considered ; see, e.g., [Bra97]. The weight ed £ 2-norm of t he jump of an element in Vh measures it s non conformi ty. Using t he definit ion of the mort ar pro jecti on, we find t hat for an eleme nt v E Vh ,

20

1. Discreti zati on Techniques Bas ed on Domain Decomp ositi on

(1.20) where vlnii (m ) is t he t race of v on t he mort ar side and vlnn( m) is t he t race of on t he non-m or t ar side. However , JIm satisfies an approx imation proper ty on ly in HJCtm) bu t not in H ICtm). Since no cont inuity is imp osed on t he space Vh a t t he cross po ints , the jum p [v]l-Y m' v E Vh , will be in general not belon g to HJCtm). In t he first mortar pap ers, t he cont inuity at t he cros spo ints was requ ired and bot h proper ti es, t he approx imation proper ty on H I(im) and (1.20), could be obtained at the sa me time for t he mor t ar proj ecti on. For the pr oof of t he approximation proper ty of the constrained space in 2D , it was impor t an t t o work with an operator guarante eing t hat t he result is zer o on 8i m and can be extended by zero onto 8f2n(m) \ "[m. resul ting in an eleme nt in H I / 2(8f2 n(m))' Her e, we introduce a new pr ojecti on having both pr op er ti es even without cont inuity at t he crosspo ints, bu t which does not guarantee that t he result is zero on 8i m. In the definiti on oft he mortar projection , we repla ce WO ;h mCtm) by WhmCtm) and define Pm : L 2Ctm) ~ W hmCtm ) by V

Jr ; »

u da =

"Ym

J

JL E MhmCtm)

VJL du,

1m

T he pr oj ecti on Pm is well defined under t he ass umption (Sd) . By definiti on , we find , for v E Vh, Pm[V] = 0, 1 ~ m ~ M . We remark t hat Pm cannot replace JIm in t he proof of the approximation pr op er ty. For t he construct ion of an element in Vh sa ti sfying t he approximat ion prop er ty , it is impor t an t t hat H~62 -fun ctions are mapped onto H~62­ fun cti ons. This is t he case for JIm but not for Pm. In addit ion to Pm, we now introduce a du al oper ator Qm: L 2Ctm) ~ MhmCtm ) by

"1m

"1m

The followin g lemma shows t hat Pm and Q m sa t isfy cert ain approximat ion pr op er ti es. Lemma 1.6. Under the assumpti ons (Sa) , (Sc), and (Sd), there exist con-

stants such that

L:

e ES m ;h

L:

e ES m ;h

m

m

~

e

Ilv-

Pmvl16·e ~ I

Civil. ,

v E H ~ Ctm) ,

2 ,Tnt

hellv - Qmv l16-e ~ Ch~m l vl 2 m _ I

n

1' 2 ,1'm 2 Vl 1

L: hellv - Qmvl 16;e2 c Ilv - Qm 1

eESm;hm

(H o20 C'Y m))'

V ,V

E Hnm- ~ Ctm ) , E

L 2Ctm) .

1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces

21

Proof. We st art with the proof of t he approximation prop erty of Pmv . It is an easy consequence of the locality of th e supports, (Sa) , th e approximation prop erty, (Sc) , of Wh ", (')'m) , and th e spect ral equivalence, (Sd). As in the proof of Lemma 1.3, (Sa) and (Sd) gua ra ntee t he stability of Pm in the L 2 _ and a weighted L 2 -nor m . Finally, (Sc) yields for s = 1/2

2:

eESm jh m

1 Ilv- Pmv l16;e = e

2:

e E Sm ;h m

1 Ilv- Vo e

:::; Ce E Sm 2:;h 111 v m

e

Pm(v -

Vo 116'e:::; I

vo)116;e

Civil.

2 ' ''''(711

To prove t he second inequality, we first consider the stabili ty property of Qm in a weight ed L 2-nor m . Observing that Jrm Qmv w da = JTTn v Pmwda, v , w E L 2(')'m) and using the stability of Pm in th e weight ed dual L 2-nor m , we get


'I~m- kfm) :s c 2:: h~n k lul;dl;[./k . K

k= l

Here, we have used th at>. restricted to "1m is a\7u · n and a trace t heorem.

o

There is a structural difference between t he 2D and 3D case only in th e proof of the approximat ion prop erty. The proofs of the consiste ncy error and t he discret e inf-sup conditi on are exactly th e same. Remark 1.11. Th e a priori esti m ate s (1.22), (1.23) and (1.27) can be weakH Sk(n k ) , S k ~ t > 3/2 . ene d fo r the more gen eral case U E Ht(n) n

rrf=l

W e refer to {WohOOa] fo r th e lowest order case.

1.2.4 Examples of Lagrange Multiplier Spaces In the previou s subsect ions, a general fram ework was given for new Lagrange multiplier spaces. Here, we focus on Lagrange multiplier spaces based on a du al basis and present concrete exa mples. The main advantage of such a biorthogonal basis is the locality of t he supports of the nodal basis functions of the const ra ined space Vh. Additionally, th e implementation of the mortar method for such a basis can be carr ied out using th e unconstrained product space; det ails are provided in Sect . 1.3. We consider t he special case where Ni ' 1 :s i :s N m } and {Bi , 1 :s i :s N m } are biorthogonal and sati sfy (Se)

J B(l/Jj da =

~~

c5ij ci

J BJ da,

c :S Ci

:s C

.

/~

Considering now t he support of an element V h E Vh , we find a st ructural difference between t he standard case and the more gener al one satisfying (Sa)-(Se) . We not e t hat WO ;h m ("(m ) C Wh m ("(m), t hus {Bi 11 :s i :s N m} can be easily exte nded to a basis of Wh m ("(m ) with local support; {Bi 11 :s i :s v m }, V ni := dim W h m ("(m )' We recall that N m < t/-« . Furthermore in 2D , we find that N m :s V ni - 2. In the 3D case, we have N m :s V m - N p , where N p is the number of nod es on EJ"fm . Each v E X h , rest ricte d to a non-mortar side "[m , can be written as Vm

vI17n = ""' L...J a ·B·1-

i=l

1,

(1.28)

Let v E Xh rest ricted to "[m be given as in (1.28) . Then , v E Vh if and only if for each non-mortar side "[m.

(1.29)

28

1. Discretization Techniques Based on Domain Decomposition

The proof follows from (1.28) and the biorthogonality relation (Se). Using (1.29) and taking (Sa) into account, it is easy to construct nodal basis functions of Vh that have local support. As in the standard finite element context, nodal basis functions can be defined such that the diameter of the support is bounded by Ch. Here, C depends on the maximum number of edges on the mortar and non-mortar sides that have a non-empty intersection with the supports of'l/)i and e·i , 1 :::; i :::; N m , respectively; see Fig. 1.9. We recall that this is not possible in the standard mortar case; comparing Fig. 1.5 with Fig. 1.9 shows the structural difference. In contrast to the standard case, the value of an element v E Vi, at a point p on the non-mortar side is determined completely by its values in a small neighborhood of p on the mortar side. As a consequence, the constraints at the interfaces can be locally satisfied.

Fig. 1.9. Structure of the support of a nodal basis function in Vh, (dual)

Working with a biorthogonal basis reflects the duality between the trace space of the weak solution and the one of the flux. The basis functions of such a dual basis are, in general, not continuous and cannot be defined as a trace of conforming finite elements. The approximation property, well known for trace spaces, has, in general, to be checked by hand. The following lemma provides a simple tool for verifying the approximation property (Sb) of a Lagrange multiplier space satisfying the biorthogonality relation (Se). Lernrna 1.12. Under the assumptions (Sa), (Sd) , and (Se) , the Lagrange multiplier space M h = (rm) satisfies (Sb) if and only if Pn m - 1(r m) cMh m (rm).

Proof. The proof is based on arguments similar to the Bramble-Hilbert Lemma. Obviously, P n= -1 (rm) c M h= (rm) is a necessary condition for the approximation property (Sb). We define 7/Jv .- ~~"i ai (v )7/Ji for v E H- 1j2(r m) by

I

ai V := ()

"1m

.

c."

veid(J

r e2 da(J ' i

l' N :::; 'l:::; m '

"1=

Then, the spectral equivalence (Sd) and the fact that M 1 is a diagonal matrix yield af(v)II'tPiI16;'Y= :::; Cllvl16; suppe', for v E L 2(rm). Considering an element e E Sm;h= at a time and using the locality of the supports of ei and 'I/)i, we obtain the following stability estimate

1.2 Mortar Methods with Alt ernative Lagrange Multi plier Spaces

II 1Pvll~;e : : ; C

N~

L

i= ] e n suPPVJi #0

29

N~

L

a7(v)ll 1Pill~;l'm::::; C

Il vll ~;suPP8i::::; Cllvll~;De

i=l

e n s u PPWi #0

Here, e C D e is a simpl y connect ed union of at most a fixed number of elements e' E Sm ;h ~ . T he maximum number of elements contained in D e depend s on the number of elements contained in th e supports of Bi and 1Pi but not on the meshsi ze. If P nm-1 ("(m) C Mh m((m), then 1Pv = v for v E Pn~- l ({m) , and we find

Ilv - 1PvIlo;e:: :; Ilv - lInm-1 vllO;e + 111P(v-lInm v)IIO;e : : ; Cllv -lInm- 1 vIlO;De ::::; Ch~:-2I vl nm_ ! ;De - l

1

'

(1.30)

where bo, is t he diam et er of D e, and lIn~-l is a locally defined L 2 -projection onto Pnm- 1 ({m)

JlIn~-l V

De

ui

do

=

J

De

vw da,

wE Pnm- 1 ({m) .

T he local quasi-uniformity of t he triangulation yields ehe' ::::; he ::::; C he',

e' E D e .

Together with th e fact t hat the number of elements in De is bounded by a const ant, we obtain (Sb) by summing over the elements and using (1.30) . D

The rest of t his subs ection is devot ed t o the const ruction of alte rnat ive Lagran ge multiplier spac es, which provide optimal finit e element solut ions . We rest rict our selves to low ord er finit e elements and refer to [OWOO] for t he general order case in 2D. As has been shown earlier, it is sufficient to verify (Sa)-(Sd) . Furthermore if, in addit ion, th e biorthogonalit y relation (Se) is satisfied , a basis of Vh having local support can be const ruc te d. The idea of using du al spaces for t he definition of th e Lagrang e multiplier space can also be carr ied over to 3D. As in the st andard mortar approach [BM97, BD98], the analysis and the definition of t he Lagran ge multiplier space is mor e technical than in 2D. 1.2.4.1 The First Order Case in 2D. We st art with the first ord er case, i.e., nk = 1, 1 ::::; k ::::; K , in 2D. Figure 1.10 shows basis functions of four different types of Lagrange multiplier spaces. All four typ es satisfy the assumptions (Sa)-(Sd) . T he two pictures on t he right represent elements of a du al basis which also satisfy (Se). In each case , we t ake WO ;h ~ ({m) as WO ;hm ({m ) and choose W hm ({m ) equal to th e standard Lagrange multiplier space defined by (1.2) . The basis function s und er considerat ion are the st andard nodal ones. Then , the approxima t ion property (Sc) is sat isfied without any further assumptions.

30

1. Discretization Techniques Based on Domain Decomposition

We consider four different Lagr ange multiplier spac es for the piecewise linear case denoted by Mh, 1 :::; i :::; 4, each defining a constrained finite element space The corresponding mortar finite element solutions of (1.14) and (1.15) are denoted by uh and (uh ,Ah) , respectively. The nodal basis functions of Mh(f'm) ar e denoted by 'ljJf, 1 :::; l :::; N m, where N m is the number of vertices X I in the interior of "[ m - For simplicity, we suppress the ind ex m in the case of the basis functions and the vertices. The enumerat ion of the vertices XI is lexicographically, and the two endpoints of ar e denoted by Xo and XN=+l' Furthermore, the length hi, 1 :::; l :::; N m + 1, is defined by hi := IlxI-l - xLII, and we define the diagonal matrix D by d ii := hi ,

Vi: .

,m

1:::; i:::; s.;

standard Lagrange M~

dual basis (piecewise linear) M~

piecewise constant M~

dual basis (piecewise constant) M~

-1 -------------------------------------------------

Fig. 1.10. Different types of basis functions for Lagrange multiplier spaces

We set M~ equal to t he standard Lagrange multiplier space; see [BMP93 , BMP94], and observe that the nodal basis functions are cont inuous and piecewise linear. The nodal basis functions in the interior of the interface are the st andard hat functions , and they are modified only in the neighborhood of the two endpoint s; see the upp er left of Fig . 1.10. It is then easy t o see that the st andard Lagrange multiplier space (,m) and its set of nodal basis functions satisfy the assumptions (Sa) and (Sb) . Observing that M, is symmetric in this speci al case and that M i , D i , D,p and D,p , D~ and ii; Djl are spectrally equivalent, respectively, we find (Sd) . The second space M~ is based on piecewise constant functions . We define the nodal basis functions 'ljJ; of M~(f'm) by

ML

'ljJ; (X) :=

{

~:

X E [~(X i -l elsewhere,

+ Xi),

~(X i

+ Xi+l )],

2:::; i:::; N m -1 .

1.2 Mort ar Methods with Alt ern ative Lagrang e Multiplier Spaces

31

'l/Jr (X) and 'l/JJvm(x) are equa l one on [x o, ~(Xl + X2 )] and [~(XNm- l + XNm), XNm +l ], resp ectively, and zero elsewhere; see the lower left of Fig . 1.10. Obviously, the locality of the supports (Sa) and the approxima t ion property (Sb) are sat isfied. To verify (Sd) , we consider the mass matrices. We find that (d,p)ii = 0.5(h i + h iH) , 2 ~ i ~ N m - 1, (d,p)l1 = 0.5(2h l + h 2) , (d,p) NmNm = 0.5(h Nm + 2h NmH), (dl) ii = 1/3(h i + h iH ), 1 ~ i ~ N m . The mass mat rix M l in (Sd) is symmetric positive definit e and t ridiagonal

~ hl + ~ h2

3 ~ h2 s(h2 + h3)

~h2

~h3

~ hNm - l ~ (hNm -l + hNm) ~ hNm l8 h Nm ~8 hNm + l2 h Nm +1 The explicit represent ati on of th e matrices shows t he spect ra l equivalences of u ; D i , D,p , Di l and D as well as between D,p and D3. These equivalences gua rantees (Sd) . The next two examples sat isfy th e biorthogonality relat ion (Se) , in addit ion t o (Sa)- (Sd). Two different sets of bior thogonal functions are introduced . In the first case, the funct ions are piecewise linear whereas in th e second case they are piecewise constant. The space MK was originally introduced in [WohOOa] for t he mor t ar set t ing and is spanned by piecewise linear but discontinuous fun ctions. The main advantage of these basis functions is t heir bior thogonality with respect t o the standa rd hat functions. Here, we review the definit ion of t he nod al basis functions and define I~i (21 1x -

'l/Jr (x ):=

{

Xi- III - Ilx - XiII) , h i~ l (211x - xiHII - llx - Xi iI), 0,

x E [Xi-I , Xi] , X E [Xi, XH l] , elsewhere,

for 2 ~ i ~ N m - 1. The two basis functi ons 'l/Jr(x) and 'l/J~m (x) close to t he endpoints of "[m. are equal one on [Xo , Xl ] and [XNm , XNm+ l ], respectiv ely, while they have t he same st ructure as t he oth er basis functions elsewhere; see t he upp er right of Fig. 1.10. Our next example is also a du al basis, but in contrast to MKb m) , its basis funct ions are piecewise constant . For 2 ~ i ~ N m - 1, let

'l/Jt (X) :=

{

~ ' X E [ ~(Xi- l + Xi), ~ (Xi + XiH) ] , - ~ , X E [Xi- I , ~ (Xi -l + Xi)) U ( ~(X i + XiH ), xHd 0, elsewhere,

,

and let 'l/Jt(x ) and 'l/J1vm(x ) be equa l one on [XO , Xl] and [XNm, XNmH], respectively, while t hey have t he same st ructure as t he ot her basis functions elsewhere; see t he lower right of Fig. 1.10. T he condit ions (Sa) , (Sd), and (Se) can be eas ily verified for t he two Lagran ge multiplier spaces ML bm), I E {3, 4}, assoc iated with th e two sets of du al basis functi ons. In both cases, we find t hat

32

1. Discretization Techniques Based on Domain Decomposition

N=

1=

L7/JL

l E {3,4} ,

i=l

ML,

and thus PO(rm) C (rm), l E {3, 4}. By means of Lemma 1.12, we obtain (Sb). Moreover, we find

./7/JiBj do = 6 j'/ ¢j da = i

ITn

ITn

l) n 8(supp 4>j) contains an edge and zero elsewhere. We refer to [BM97] for an introduction of a standard Lagrange multiplier space, and to [BD98] for some more sophisticated choices of the weights in (1.31). In particular, it is possible to replace Ifnj by a weight involving the areas of the adjacent elements. Obviously (Sa) is satisfied. To verify (Sc) and (Sd), we have to specify the spaces WO ;h m ("(m) and W h m bm) and their basis functions . As in the 2D case, we set WO ;h m ("(m) := WO ;h m ("(m) and choose the standard hat functions as basis functions Bi := 4>i . The space Wh m bm) is obtained from WO ;h m bm) by modifying the standard hat functions in exactly the same way as the Lagrange multiplier basis functions before :

Bt

:= 4>1

+

L

xjE M t

~ . 4>j + J

L

alj4>j,

x; E

"[m. .

xjEM

It is now easy to see that I::Xt E'Ym 'ifJl = I:: XtE 'Ym Bt = 1. Thus (Sc) is satisfied and Pobm) C M h m bm). Furthermore, a straightforward computation shows that the following biorthogonality relation holds

This implies (Se) and (Sd) for i = 1. (Sd), for i = 2, is obtained by observing that M 2 , D 1 and D 2 are spectrally equivalent as well as b, and D2 . Applying Lemma 1.12 yields (Sb) . As in th e 2D case, the biorthogonality relation results in a mass matrix, on the non-mortar side , which has a diagonal block associated with the interior nodes . In addition, the nodal basis functions of

1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces

35

the constrained space have local support. We remark that we do not have supp 't/Jl = supp ¢l in the neighborhood of the boundary of "[niIn the special situation of a tensor product mesh on a rectangular interface, the construction of a dual basis function can be simplified. Piecewise bilinear and piecewise constant dual basis functions can be defined as a product in terms of the dual basis functions 7/Jr and 't/Jt given in the previous subsubsection, respectively. Then, the support of this simplified dual basis function 'I/Jt associated with the vertex .Tl E "[m. is the union of the four adjacent rectangles sharing the vertex Xl. Figure 1.14 illustrates the isolines of such a dual basis function if 8(supp7/Ji) n 8 rm= O. -2

-2

Fig. 1.14. Isolines of piecewise bilinear and piecewise constant dual basis functions

Comparing these simplified piecewise bilinear dual basis functions with the definition (1.31), we find that they are the same if 8(supp't/Ji) n8 rm= O. However, in the neighborhood of the boundary of "[m , we observe a difference. In particular, the tensor product dual basis functions 'IN have a smaller support than 't/Jl' We distinguish between three different types of vertices Xl. The inner ones, i.e., 8(supp ¢z) n 8 rm = 0, are marked with empty squares, the ones close to the corners, i.e., 8(supp ¢z) contains one corner of "[m , by empty circles and all other vertices are marked by filled circles; see Fig. 1.15.

Fig. 1.15. Different types of piecewise bilinear dual basis functions

Figure 1.15 shows the different groups of vertices and one piecewise bilinear dual basis function 7/Jt for each vertex type. The other ones can be obtained by local rotations. For the simplicial case, it is also possible to modify the definition (1.31) in the neighborhood of the boundary of rm; see [KLPVOO]. Then, the nodal basis function ¢l and the dual basis 7/Jl have the same support. The advantage of the simplified dual basis functions is the

36

1. Discretization Techniques Based on Dom ain Decomposition

smaller support. In contrast to supp 'l/Jl' the support of 'I/Jt contains for all indi ces 1 only the element s sharing the node Xl. The assumpt ions (Sa)-(Se) are easy to verify. 1.2.4.3 The Second Order Case in 2D. In our last example, we consider t he case n k = 2 in 2D. Again, we set WO ;h m(')'m) := WO ;h m(')'m), and W hm (')'m) is defined as the standard Lagrange multiplier space (1.2) with nk = 2. Both spaces are associate d with t he set of the corr esponding nod al basis functions. Figure 1-16 illustrat es the numbering of the nod al basis functions ()i , 1 ::; i ::; N m, in WO ;h m(')'m).

Fig. 1.16. Numbering of the nodal basis functions of

The du al basis fun ctions are now given in t erms of define 'l/Ji' for i = 21 + 1,1::; 1 ::; (N m - 3)/2 , by

'l/Ji (X) :=

{t 1 + ~()i)(X),

and for i = 2/,2::; 1 ::; (Nm

'l/Ji (X) := {

-

()i,

1 ::; i ::; N m . We

E SUPP()i ,

elsewhere,

3)/ 2 by

6:- ~()i- l +

The t wo first basis functions, ar e defined differently by

X

WO ;h m (--Ym)

()i -

~ ()i+d (x),

X

E SUPP()i ,

elsewhere .

'l/Jl ' 'l/J2 ' and the two last ones , 'l/JNm -l' 'l/JNm, X

E SUP P ()l ,

elsewhere, X

E supp ()l

,

X E supp ()2 \ supp ()l elsewhere ,

,

with 'l/JNm - l and 'l/JNm given in a similar way. A straightforward computat ion shows th at (Se) is sat isfied. In this case , th e spe ctral equivalence (Sd) follows from t he spectral equivalence of D 1j; and D 1. Fur thermore, we find L:~i 'l/Ji = 1. Observing th at 'l/J2i + O.5 'I/J2i-l + O.5 'I/J2i+l' is t he st andard piecewise linear and continuous hat fun ction associated with t he vertex Xi , 1 ::; i ::; (N m - 1)/2, th e approximation property

1.3 Discretization Techniques Based on th e Product Space

37

(Sb) follows from Lemma 1.12. Figure 1.17 illustrates the du al basis fun ctions

'l/Ji'

1:: : i s s.;

We obs erve t hat a priori est imates for the Lagran ge multiplier of order h" can be obt ain ed by using piecewise polynomials of order (n - 1) for the Lagran ge mul tiplier ; see also [8898] . Since the a priori est imates for the Lagrange mul ti plier depend s on the est imate for U - U h , the a priori est imate for t he flux cannot be improved by choosing higher ord er elements for the Lag range multiplier.

Fig. 1.17. Dual basis functions, (nk = 2)

Remark 1.13. Th e assumptions (Sa)-(Sd) also allow us to work with Lagrange multipli ers which are defined on a coarser triangulation than S m ;h m • For example if S m ;h m is obtain ed by a un iform refine ment step from S m ;2h m , th en M 2 h m bm ) sat isfies (S a)-(Sd) if M h m bm ) does. Finally, we not e t hat the concept of du al Lagran ge multiplier spaces can be gener alized to higher order elements . The const ruc t ion of higher order du al basis funct ions is very t echn ical , and we refer t he inte reste d read er t o [OWOO] . In [OWOO], dual basis fun ctions are const ructed for conforming P n-elements in 2D. The number of edges contained in the support of a du al basis fun ction is at most t hree and ind ep end ent of t he order n . Moreover , t he assumpt ions (8a)-(8e) are satisfied by construc t ion. Thus, optimal higher order mortar methods for du al Lagr an ge multiplier spaces are obtained .

1.3 Discretization Techniques Based on the Product Space In the pr evious subsections, two possibl e approac hes to t he impl ement ation of a mort ar discreti zati on have been discussed . The first gives rise t o a positiv e definit e problem , and is based on t he constrain ed space . The second works with the un const rain ed product space and a Lagran ge mult iplier space and results in a sa ddle point formulation , where t he weak cont inuity is enforce d

38

1. Discreti zati on Techniques Based on Domain Decompositi on

by means of t he Lagran ge multiplier space . In this sub section , we consider discret ization t echniques working only on the un constrained product space . Such a possi bility is also discussed in [BH99, St e98]. An idea introduced by Nitsc he in [Nit70] is rediscovered and generalized to non-m at ching trian gul atio ns . It t urns out t o be a penalty meth od where t he penalty par am et er does not depend on t he meshsize and does not influence t he condition number of t he resulting linear system. We refer to [BH99] for det ails and some num erical results including a posteriori error esti mators. Here, we develop a different idea . We note th at th e num erical solut ion of t he mortar method is, very ofte n, based on its saddle point formulat ion [AK95, AKP95, AMW 96, AMW9 9, BD98, BDH 99b, BDW9 9, EHI+98, EHI+ OO, HIK + 98, Ku z95a, Ku z95b, Ku z98, KW9 5, Lac98, LSV94, WW98 , WW99]. Work ing with t his ind efinit e formulati on has t he advantage t hat t he unconstrain ed product spaces assoc iated with a nest ed sequence of global t riangulat ions are nest ed , while the nonconforming constrained spaces are not . Our approach is based on t his observation, and we introduce an uns ymmetric Diri chlet-Neum ann coup ling and a sym met ric and positi ve definit e var iati onal pr oblem on t he unconstrain ed produ ct space . We t hen show t hat t he solut ion of t hese pro blems coincides wit h t he mortar finit e element solutio n. The idea is not rest rict ed to first order discretizations or to 2D. We assume t hat t he pair (X h, M h ) defines a constrained space Vh such t hat t he nonconforming variational pro blem (1.14) and t he saddle point problem (1.15) are well defined and yield optimal a priori bounds for t he discretization errors , see conditions (Sa)-(Sd) in Sect. 1.2. Then , we have only to ensure t hat t he nodal bas is fun cti ons of M hm(rrm ) an d WO;h m(rrm) form a biorthogonal set

J

'l/JtOk da = Ct 6tk IIOk I16;'Ym'

1 ::; i.e -;

Nm

,

(1.32)

'Ym

where C ::; Ct ::; C, 1 ::; l ::; N m, and n.; = dimMhm(rrm) = dimWo ;h m(rrm ), see condit ion (Se) in Sect . 1.2. We recall that WO;h m(rrm ) is a suitable subspace of t he finit e element t race space on t he non-mort ar side with zero valu e on t he boundar y of "[m - The inde x m is dr opp ed for t he basis functi ons and also for t he constants Ct. Examples of such pairings have been given in t he prev ious subsections for t he first order cases in 2D and in 3D , and for the qua dratic case in 2D. We recall t hat the only exa mples introduced in the previous subsections which do not satisfy t hese condit ions are t he standard Lagran ge multi plier space M~ and t he space M~ . Here, we restrict ourse lves to t he spec ial case N m = dimWo;hm(rrm) and Ot = (Pt, 1 ::; l ::; N m, where cPt are t he standard nod al basis functions . Our int erest in t he int roducti on of a new, t hir d, mort ar formulation is based on the following observation: Saddle point pro blems , like (1.15), are, very often, solved by Uzawa like algorit hms , and inn er and outer it erati ons are

1.3 Discreti zat ion Techniques Based on t he P roduct Space

39

required . Recentl y such a multigrid t echnique has been st udied in [WW99] for t he t raditional mortar approach. In t his case, a modified Schur complement system has to be solved it eratively in each smoothing ste p. This method is a genera lization of th e techniques given in [BD98, BDW99, Br aOl , WW98] where t he modified Schur complement system was solved exa ct ly. Another approac h is analyzed in [Kuz95a, Ku z95b] an d further used in [EHI+98, EHI+ OO, HIK+98, Ku z98]. A good pr econdi tioner for t he exact Schur complement is required , and t he ite rative solution of t he saddle point problem is obtained by a generalized Lanczos method. It is also true t hat standard multigrid t echniques cannot be applied for th e pos iti ve definite formulation (1.14) which is associate d with a nonconforming space. We refer to [AMW96, AMW99 , BDH99b, CDS98 , CW96 , Dr y96, Dr y97, Dry98a, Dr y98b , GPOO] for multilevel and multigrid meth od s and domain decomposition t echniques. Recentl y, multigrid methods for th e non conforming formulation have been developed [BDH99b , GPOO] which involve t he solution of a mass matrix syst em in each smoo t hing st ep . Dirichlet-Neum ann ty pe preconditioners have been proposed for t he positiv e definit e system in [Dry99 , DryOO]; see also [LSV94]. We now introduce a t hird equivalent mor t ar formul ation. It is based on t he un constrain ed product space and gives rise t o a symmetric positiv e definit e formulation on which we can apply standa rd multigrid t echniques. The rest of this secti on concerns th e const ruct ion of a symmetric positive definit e syste m defined on the product space . We present a vari ational formulation on the product space as well as its algebraic form . The st iffness matrix associated with the un constrain ed product space can easily be obtain ed by eliminating the Lagran ge multiplier. The starting poin t is the observation that t he Lagran ge multiplier Ah is given in a postprocessing ste p. But it can be easi ly evaluated in te rms of the right hand side and t he mort ar solut ion Uh only in t he case t hat t he bior thogonality relati on (1.32) holds. Otherwise, t he eliminat ion involves t he inverse of a mass matrix. We introduce an abstract fram ework for a new equivalent mortar setting based on t he un constrain ed product space . The analysis of our syste m on th e product space is carr ied out using the to ols of th e mortar framework. T he new idea is to use a biorthogonal basis for t he Lagrang e multiplier space and t o eliminate t he Lagran ge multiplier in t he saddle point formulat ion. We begin by considering, in Sub sect. 1.3.1, a Dirichl et-Neum ann couplin g for t he special case of two subdoma ins. Subs ections 1.3.2 and 1.3.3 concern t he construction of a symmet ric positive definit e syst em defined on t he pr odu ct space. It s condition number is bounded by a constant times 1/h 2 , as in the usual finit e element conte xt. We pr esent the variat ional formulat ion on t he pr odu ct space in Sub sect . 1.3.2 and its algebraic form in Subs ect . 1.3.3.

40

1. Discreti zati on Techniques Based on Dom ain Decompositi on

1.3.1 A Dirichlet-Neumann Formulation Let us consider t he following special sit ua tio n: The domain D C 1R2 is decomposed int o two nonoverlapp ing subdomai ns , D = D I U D 2 , and meas (8 D n 8Di ) > 0, i E {l , 2} ; see Fig. 1.18 .

Fig. 1.18. Decompositi on into two sub domains

On D I , we consider a Diri chlet bounda ry value problem -div(a'Vu d

= t. in DI , UI = 0, on r l := 8D n 8DI , UI = ut», on T := int(8D I n 8D 2 )

+ bUI

(1.33) ,

where so E H6// (r) · Its variational formulation is: Find UI E H~D ; rl (D I ) such t hat al(uI,vd = (f, VI)O ;{]ll VI E HJ( Dd . Here, H~D ; rl (Dd is t he subset of HI (D I) with a vani shin g t race on r l and a trace equa l to gD on r . The bilinear form s ai(-, '), 1 ::; i ::; 2, are given by

ai(V,w )

:=

J

a'Vv'Vw + bvwdx,

v,w E HI (Di ) .

{] i

On D 2 , we solve a partial differential equation with a Neuma nn boundar y condit ion on r -div (a'Vu 2) + bU2 = i, U2 = 0, afu!:L an = 9 N,

in D 2 , on r2 := 8D n 8D 2

onr ,

,

(1.34)

where n denot es t he outer uni t normal on D 2 • The weak variati onal formulation is given by : Find U2 E H h (D2) such t hat

a2(u2, V2 ) = (f ,V2 )O;{]2 +

J

YNV2 da = : !2(V2 ),

V2 E H h (D2) .

r

Here, Ht i (D i) , 1 ::; i ::; 2, denot es t he subspace of H I(Di) havin g a vani shin g t race on t he oute r boundary rio

1.3 Discr etiz ation Techniques Bas ed on the Product Space

41

Let us assume for the moment that the solution u of (1.1) and the flux on the interface r ar e known. Then, the choice gD := ulr and gN := aBu/Bnlr yields UI = Ul fl l and U2 = Ulfl2' The discrete spaces of conforming PI-Lagrangi an finit e elements on D I and D 2 , associat ed with t he simplicial triangulations Ti and 72 , are denoted by Xl C H}, (Dd and X 2 C Hh (D2 ) , resp ectively. The set of corresponding vertices are denoted by P I and P 2 , and the subs ets of vertices on the int erface by p[ and Pf. Then , (1.33) and (1.34) can be discretiz ed by means of the discret e spaces Xl and X 2 , respectively. We will now use th e same notation for an element in X i , 1 :s; i :s; 2, and its vector repres entation with respect to the nod al basis , {rPP' P E Pd, of Xi . The matrix representation of the Dirichlet problem (1.33) can be given by

(UI) ADUh;1 := (AI/AIr) 0 Id u~ I

=

(iJ) I}

.

(1.35)

Here, A I/ and A Ir are the st iffness matrices associate d with the bilinear form restricted to Xi x Xi and Xi x X}, respect ively. The space X} is spa nned by the nodal basis functions rPP' p E pf, and Xi := Xl n HJ(Dd . The right hand side is associat ed with the linear form I st, I vdx , and I} dep end s on soDirichlet boundary conditions are often realized by a pointwise equa lity at the nodes of the triangulation al (-,.)

iJ

(f})p = gD(p) ,

pE

p[ ,

if the boundary data function gD is cont inuous. Here, we use a different appro ach and sp ecify th e Diri chlet boundary condit ion by a weak int egral condit ion 'l/JpgD da , pE p{ ,

J

where e; := 3rPp- 1. We refer to [SZ90] , for a discussion of nonhomogeneous Dirichl et boundary cond iti ons , and observe that 'l/Jp is t he sam e as a nodal basis function of the du al spac e h'm) in Sect . 1.2, if the vertex p is not next to an endpoint of th e int erface. In cont rast to Dirichl et boundary conditions, Neumann boundary conditions gN ar e natural and ente r dir ectly into the variational formulation. The discret e vari ational formul ation of (1.34) can be written as

ML

(1.36) We use the sa me not ation as before ; and define

jJ.

by

42

1. Discretization Techniques Based on Domain Decomposition

Unfortunately, the solution and the flux restricted to T are, in general, unknown, and the boundary data gD and gN are not available. However, if in the continuous setting we set sr: := 7L2 and s» := aa7Lr!an, (1.33) and (1.34) form a coupled system of boundary value problems. Then, h(V2) can be rewritten, and we find that

Here, v E HJ(fl) is an extension of V2 E H r 2(fl 2 ) . We remark that the jump of the trace of v across r is zero. However, the jump of the flux of v does not have to be zero. These observations form the starting point for our discrete approach. When working with non-matching triangulations at the interface r, it is in general not possible to extend a function Vh;2 E X 2 to Xl X X 2 such that the jump of the trace is zero. However, in the mortar approach, it is standard to replace the strong continuity of the jump by a weaker one. Observing that each element in Xl x X 2 is uniquely defined by its values at the vertices Pl E Pi and P2 E P 2 , a suitable discrete extension EVh;2 of Vh;2 E X 2 is constructed in the following way: P E P2 PE PE

,

v, \ pi , pi .

(1.37)

We note that the extension operator E is not uniformly bounded in the broken Hl-norm.

Fig. 1.19. Support of the extended function

of

The shadowed region in Fig. 1.19 illustrates the structure of the support E Xl X X 2 . Using the definition of'l/;p and E, we find

EVh;2

1.3 Discretization Techniques Based on the Product Space

Xl

We are now ready to formulate our discr et e approach: Find X X 2 such that a l (Uh ;l ' V h ;l )

U h; l (P ) a 2(uh ;2, Vh ;2)

= (I, vh; d o; .l?ll = EUh ;2(p) , = (I ,Evh;2) O -

Vh ;l E

pE al(uh ;1 , E vh ;2) ,

(U h ;l ' Uh ;2)

xl ,

p{ ,

V h;2 E

X2

43

E

(1.38)

,

where EVh ;2 and EUh ;2 are the discret e ext ensions of V h ;2 and U h ;2 given by (1.37) . The solution of (1.38) dep ends on the definition of E on the inte rface bu t not on t he valu es at P, P E PI \ p{ . Thus the t rivial exte nsion E by zero into the int erior of ill could be repl aced by a discrete harmonic exte nsion wit hout influencing the solution of (1.38). The matrix formul ation of (1.38) reflect s dir ectly the Dirichlet -Neumann coupling between t he two subdomains

(1.39)

where M:= (m p 2P 1 ) P2E P f m

,PI EP [

is a sca led mass matrix defined by

2 [supp cPPI I

.- . , - - - - -

P2Pl . -

J

s UPPQ>P I

Before we analyze t he unique solvability of (1.38) and the discretiz ation error in the new setting, we consider our approac h in t he more general sit uat ion with sever al subdomains and crosspoints . We rem ark that we have to modify the approach in the neighborhood of a crosspoint . Using the same notati ons as in Sect. 1.2, we find t hat t he choice of t he Lagran ge multiplier space M h plays an essential role. We show in the next subsect ion that the discret e vari ational problem on the product space can be obtain ed from the sa ddle point problem by locally eliminat ing t he Lagran ge multiplier . R emark 1.14. W e observe that th e stiffness matrix in (1.39) reflec ts the Dirichlet-Neumann coupling between the two su bdom ains . Applying a block GaujJ-Seidel m ethod on this syst em results in a Dirichlet-Neumann type preconditioner. This me thod is well kno wn for conform ing fin it e elem ent m ethods; see [SBG96] and the references th erein , and has been also studied fo r th e stan dard mo rtar approach; see [Dry99, DryDD, LSV94]. For' details, we refer to S ect. 2.4. 1.3.2 Variational Formulations

In t his subsection, we introduce our new abst rac t vari ational formul ation on t he un constrain ed product space. The only necessary condit ion is the

44

1. Discreti zation Techniques Based on Domain Decomp osition

biorthogonality relation (1.32) . An un symmetric formul ation as well as a symmetric one ar e discussed . A car eful analysis shows that th e un symmetric one for t he special pairing (X h , M~) is the Dirichl et-Neum ann coupling described in t he pr evious subsect ion. The symmet ric one is positive definit e, and as we will see in Sect. 2.3, a standa rd multigrid method can be applied for t he resulting linear system. In t he rest of this sect ion, we assume th at the biorthogonality relation (1.32) is satisfied . At first glance the saddle poin t problem (1.15) has t he same st ructure for all Lagr an ge multiplier spaces. However , th ere is an essent ial difference. Given th e solution Uh of (1.14) , t he Lagran ge multiplier Ah l can be obtain ed by solving a mass matrix syste m. This postprocessing ste p involves t he inverse of a mass matrix which is, in general, dense on each int erface "[m - Only in th e case t hat the biorthogonality relation (1.32) hold s will the inverse mas s matrix redu ce to a diagonal one. Then , t he valu e of an element v E Vh at a point on one non-mort ar side is det ermined by it s values in a small neighborhood of that point on the adjacent mortar side. In par ti cular , t he mass matrix associate d with the non-mortar side is replaced by a diagonal one reflecting the biorthogonality. On each non-m ort ar side "[m» we can write Ah as Ah l .,~ = L:~l A m ;i "pm; i. Compared with Sect. 1.2, t he addit ional ind ex m indic at es th e corre sponding non-mort ar side "[m » The coefficients Am;i are given in te rms of Uh and f by ., ~

(1.40)

where cPm;i is t he nod al basis fun ction which has it s support in S?n (m) ' Let us now define linear functionals 91 : X h --+ M h and 92 : X h --+ X h · Both functionals are associated with the int erfaces. We define M N~ """"' """"'

91(V) := ~ Z::

m= l i==l

vI

(p.)

nn ( ~) m;t .t , . . A. .' do "pm;i,

J If/m, wm; 2

v E Xh

,

'1,

,~

where Pm ;i are the nodal points on "(m, i.e., cPm;i (P m ;j) = 6 i j . We fur t hermore use a linear func ti onal 92(-) which vani shes on Vh . It is defined by (1.41) It is easy to see that 92(V) is only supporte d in a sma ll neighb orhood of the non-mort ar side and vanishes on the mortar sides of t he interfaces. Proposition 1.15. Th e kern el of 92( ') is exactly Vh and 92 (9 2 (v)) = 92(V) . Moreover, th e L 2-no rm of 92(V) on S? can be bound ed by the L 2-norm on th e interf aces of th e jump of v

1.3 Discretization Techniques Based on the Product Space

45

M

Ilgz(v)ll& ::; eLL hell[vlll&;e · m=l e ES m ;h m

Proof. The definition of gz(-) and the biorthogonality relation (1.32) yields the first two assertions. A straightforward calculation shows

Z ~ ~( II gz (v )II 0::; mL:1 i~ f

b(v, 'ljJm'i ) ) Zll " liz 'ljJm;i¢r:.;i da 'f'm;i 0

~m

Here, we have used th e assumptions (Sd)-(Se) which guarantee the equivalence of II¢m;illo;'Ym and II'ljJm;illo;'Ym' and of h"'m:i and h m :i' Finally, (Sa) yields that the maxim al number of 'ljJm ;i having an overlapping support is bounded independently of the local meshsiz e. D There are now two possibilities to show that an element v E X h is also in Vi,. The first uses the definition (1.13) of Vh ; it is sufficient to prove that b(v, 'ljJm ;i) = 0 for 1 ::; Tn::; M, 1 ::; i ::; u.; The second possibility is based on th e projection gz(.); v E X h is an element of Vh if and only if gz(v) = O. The following lemm a shows an equivalence between the positive definite vari ational problem (1.14) on t he constrained spac e Vh and a non-symmetric and a symmetric variational problem on the unconstrained product space X h . The idea behind the introduction of t he new variational problem is to use the decomposition v = (v - gz(v)) + gz(v) . Then, v - gz(v) is an element of the constrained space Vi,. Lemma 1.16. Let Uh E Vi, be the unique solution of (1.14) . Then, Uh is the unique solution of the non-symmetric variational problem (1.42)

and also of the positive definite symmetric variational problem a(Uh - gZ(Uh) , v - gz(v))

+ a(gz(uh) ,gZ(V)) =

(/, v - gz(v))o,

v E Xh . (1.43)

Proof. In a first step, we show that the solution Uh of (1.14) satisfi es (1.42) and (1.43) . Using t hat Uh E Vh, we find b(Uh ,gl(V)) = 0 and gZ(Uh) = O. Since gz(-) is a projection, we have gz(v - gz(v)) = 0 and thus v - gz(v) E Vh. Combining these observations , we imm ediately find th at Uh satisfies (1.42) and (1.43) . To obtain t he unique solvability of (1.42) and (1.43) , it is sufficient to show that each solution of (1.42) and (1.43) , respectively, is an element of Vh and thus a solut ion of (1.14) . An element v E X h is in Vh if and only if b(v ,f..l) = 0, /1 E Mh , or equivalent ly if gz(v) = O. Let us assume that Uh

46

1. Discreti zation Techniques Based on Domain Decomp osition

satisfies (1.42) . For each 1 ::; m ::; M , 1 ::; i ::; N m , we have 92(rPm;i) = rPm; i and 91 (rPm;i) = Dm;i1/Jm ;i with Dm;i =I- O. The choice v := rPm;i yields

b(u", 1/Jm;i) = 0 ,

u"

and t herefore is t he unique solut ion of (1.14) . To prove th at E V" if is the solution of (1.43) , we use the second possibility and show th at 92 (U,,) = O. Setting v := 92(U,,) yields

u"

u"

a(92(u") ,92(U,,)) = 0 . Since t he bilin ear form a(·,·) is coercive on 92(X,,), we find that 92(U,,) = O. It is now easy to see t hat the vari ational problem (1.43) is symmetric and positiv e definit e. We define a(v, w) := a(v-92(V) ,W- 92 (w))+a(92 (v), 92(W)), v , w E X " . Then a( ·, ·) is a symm etric bilinear form satisfying a(v , v) ~ O. For each v E X " sati sfying a(v,v) = 0, we find that 92(V) = 0 and a(v ,v) = O. Since t he kernel of 92(-) is V" , an d a(·,· ) is V;,-elliptic, we get v = O. 0 Remark 1.17. Optimal a prio ri estim ates are available fo r th e variation al problem s (1.42) and (1.43) by usi tu; Lemma 1.16.

In th e rest of t his sect ion, we show t he relation between t he new mortar formulation (1.42) for t he special pairing (X" , M~) and t he Dirichlet Neum ann coupling (1.38). It t urns out th at t he non- symmetric product formul ation is almost t he Diri chlet - Neum ann formulation in t he case of two subdomains in 2D. Here, we associate the non-mortar side with t he subdomain VI . The space Xl can be dir ectly decomposed into x! + X[. Each element in X i is extended by zero on the adjacent subdomain , and we will use t he sam e notation for th e extended function. Using t he decomp ositi on for X l and observing that gl (v) = g2(V) = 0 for v E xl and g2(V) = v for v E X [ , (1.42) can be rewrit t en as a coupled system

adu" ,v) = (f ,V)O ;fh , b(U" ,g l(V)) = 0, a(u" , v - g2(V)) + b(U" ,gl(V)) = (f , v - g2(V))O ,

v E xl , v EX[ , v E X2 .

(1.44)

Comp aring (1.38) with (1.44) , we find th at the first equation is exactl y th e sa me. In a second ste p, we consider the second equation in more det ail and obtain for v = rPi

J

(u" ln, - u" ln)1/Jt do = u" ln, (Pi)

r

J

rPi do -

r

Using t he definition of 1/Jr an d t hat int erior verti ces of T

U" In, (Pi)

=

IsuP~ rPi I

Ir rPi do =

J

supp o ,

J

u" ln21/Jt da = O . (1.45)

r

0.5l supp rPi l, we obt ain for the

u" ln21/Jt da ,

1 ::;' i ::; N r ,

1.3 Dis cretiz ation Techniques Based on the Product Sp ace

47

where N r is the number of interior vertices on the non-mortar side of r . Having (1.37) in mind , (1.45) gives rise to almost the same system as t he second equa t ion in (1.38). The only difference is for i = 1 and i = N r : in these cases 'l/Jr :j:. 'l/JPi ' In the genera l mortar approach, t he analysis of t he consiste ncy error requires this modification , but only for th e vertices on t he non-mortar sides sharing one edge with a crosspoint. Fin ally, we have to compare the equa t ions for v E X 2 • St arting with the observation t hat v - g2(V) is almost equa l t o Ev , v E X 2 , and using 91(V) = 0 for v E X 2 , we find equa lity except at t he two endpoints of T of th e coup led syst ems (1.38) and (1.44) . Thus t he non-symmetric variational probl em (1.42) obtained by a local eliminat ion of t he Lagrang e multiplier is nothing else than a Dirichlet Neumann coupling of boundar y valu e problems on th e different subdomains . On t he mortar sides Neumann boundary condit ions and on t he non-mortar sides Dirichlet boundary condit ions are used . 1.3.3 Algebraic Formulations The introduction of t hese t hird mortar formul ations was motivat ed by a need to find new, mor e efficient, iterativ e solvers. The positive definite vari ati onal formul ation (1.14) gives rise t o a non conforming approach where t he const rained spaces are non-nest ed . Working with t he equivalent saddle point approac h yields an ind efinit e syste m for which efficient iterative solvers are ofte n relat ively expensive. Two special iterative solvers will be considered in Sect. 2.3 and Sect . 2.4. One is based on the symmet ric approach (1.43) and the ot her one on th e un symmet ric Diri chlet-Neum ann formul ation (1.42). We point out that t he use of du al basis functions as Lagran ge multiplier space is essent ial for the efficiency of our new iterative solvers. In t his subsection, we consider the algebra ic formul ation of t he vari ational probl ems (1.42) and (1.43) given in Subsect . 1.3.2. We show t hat t he corresponding st iffness matrices can easily be obtained by means of those of t he un constrain ed product space.

BBB B B B

.

o

Set 0

(interior nodes)

Set 1

(mortar side)

Set 3

(non-mortar side)

Set 2

(crosspoints)

Fig. 1.20. Decomposition of the nodes into sets in 2D

We start with a split t ing of the degrees of freedom into different sets ; see Fig. 1.20. All nod es in the int erior of a subdom ain have ind ex zero. T he nodes

48

1. Discretization Techniques Based on Domain Decomposition

on t he interior of th e mor t ar sides have ind ex one . The node s on the int erior of the non-m ortar sides have ind ex zero or three; each nod e corre sponding t o a basis function of WO ;h~ (')'m) has ind ex three, all other nodes have ind ex zero. In t he special case th at WO;h ~ bm) = WO ;h~ bm ), all nodes on the int erior of t he non-mort ar sides have ind ex three, this is t he situation illustrat ed in Fig. 1.20. All nod es on the boundary of the mortar and non-mort ar sides have ind ex two . In 2D, t hese are the vertices of th e triangulations coinciding geomet rically with a crosspoint. There are mor e nod es of this typ e than crosspoints, since each crosspoint p is associated with n p nod es, where n p is the number of subdom ains sharing the crosspoint p. In 3D , th ese are the nodes on the wirebasket of the decomposition, i.e., on U~=l fh m\ see Fig. 1.21. In the right of Fig. 1.21, the sit ua tion of four subdomain s shar ing one edge of the wirebasket is illustrat ed . The dashed lines symbolize the edges of t he t riangulations of the different subdomain s.

an;

..' , :

---_.- -----.......:: =.. i·····

..... :

/1"»tl l

>

--- -- f ~ ~ :::;·..:s:: :-·

------

Fig. 1.21. Wirebasket (left) and detail (right)

T he vari ational problems (1.14), (1.15) , (1.42) , and (1.43) are equivalent, and, in addit ion, the Lagran ge multiplier Ah can be obtain ed from Uh by local postprocessing. However, th e choice of the underlying mortar formu lation will make a big difference for t he it erati ve solut ion. Here, we only remark t hat t he natural setting for man y efficient iterative solvers such as preconditioned conjugate gradient metho ds or multigrid methods is a symmet ric positive definit e problem based on nest ed discret e spaces. In th e case t hat one of these condit ions is violated , standa rd t echniques, very ofte n, have to be modified . The new formulation (1.43) appears t o offer a real advantage since it results in a symmet ric positive definit e problem associate d with nest ed spaces. Formulations (1.14) and (1.43) give rise t o symm etric and positive definit e problems. However in t he case of (1.14) , the discret e const rai ned spaces are non conforming and, in general, non-nest ed . The un constrain ed product spaces are nest ed , and (1.15), (1.42), and (1.43) are associated with th ese spaces. But only (1.43) yields a symmet ric and positiv e definit e system . The saddle point formulat ion (1.15) yields a symmet ric but ind efinite system , and (1.42) gives rise to a non symm etric formul ation. In t his subsection, we discuss some algebra ic aspects of (1.42) and (1.43). In par t icular , it is shown how we find (1.42) and (1.43) from (1.15) by 10-

1.3 Discr etiz ation Techniques Bas ed on the Product Space

49

cal elimination. We consider the corresponding algebraic system, first for the saddle point formulation. Using nodal basis functions and grouping the vee. diIces UT T ) , were h T := (T T) an d tors accor dimg to t hei eir in UR, UN UR Uo, U T 1 ' U2 h = (T UN := U3, (1.15) results in the following indefinite system

G:)

U~) - (0

(1.46)

All indices related to the meshsize ar e suppressed for the stiffn ess matrices and subvectors. In addition, the same notation is used for functions in the discrete spaces and their coefficient vectors with resp ect to the nodal basis fun ctions. A RR, A RN , ANR and ANN ar e the stiffness matrix associated with the bilin ear form a(', ') restricted to the different subsets, and M R and D are associated with b(·, .). We remark that M R is a block mass matrix with many zero blocks. Moreover D is a positive definite, diagonal matrix. This is an essential point, and is, as already pointed out, only true if dual basis functions for the Lagrange multipliers are used . Ex amples of dual basis functions have been discussed in Subs ect. 1.2.4. The non-symmetric formul ation (1.42), based on the product space, is obtain ed by choosing Ah according to (1.40) and setting f-l := 91(V) in (1.15) . This can be expressed in a very simple way in the algebraic setting Ah = WT(f - AUh) , f-l=WTv ,

where W is defined as

W T := (0, D- I )

Then , local elimination of Ah and f-l yields (Id , W)

( AB) ( BT 0

Id A ) Uh = (Id - BW T )f =: f _ WT

A

(1.47)

It can be easily shown that (1.47) yields ((Id - BWT)A + W BT)Uh = l, and that (1.47) is the algebraic form of (1.42) . A different possibility to obtain a reduced system for Uh is to define f-l by f-l := W T A(WB T - Id)v ,

and repl ace Ah = W T (f - AUh) by the equivalent formula

We note that the mortar solut ion satisfies BTUh = O. This choice gives rise to a symmetric positive definite system

50

1. Discretization Techniques Based on Domain Decomposition

Au" := (Id, (B W T - Id) AW ) A

(

AB) (W T A( WIdBT _ Id) ) u"

BT 0

= f

A

(1.48) T he following lemm a establishes a relation betwee n the algebraic formulation (1.48) and the vari at ional pro blem (1.43). Lemma 1.18. The algebraic fo rmulation of the variational problem (1.43) is given by Au" = j . Moreover, th e Lagrange m ult iplier can be obtaine d, once u" has been computed, by A" = W T(j - Au,, ) .

Proof. To start, we rewrit e t he linear funct ional 92 (v) in its algebra ic form

W BTv . T he multiplicat ion with W can be interpreted as a scaled map ping from M" onto X " such t hat t he result ing element of X" is supported in a sma ll st rip of width h on t he non-m ort ar side, whereas t he mult iplicat ion with B T maps each element in Vi, to zero . Using t hese Observations, t he algebra ic form of variational problem (1.43) can be written as (Id - B W T )A (Id - W BT)u" + BW T AWB T = (Id - BWT )f .

U"

A st raightforward comp utation completes t he proof by comparing t his system wit h (1.48) . 0 The matrix A can be easily assem bled from A, M := M RD- 1 , and is sparse. Using t he indices given at t he beginni ng of t his sect ion, it can be written as a 2 x 2 matrix as

A=

(

ARR + 2MA NNM T - M A NR - A RN M T ANNM T

)

(1.49)

1.4 Examples for Special Mortar Finite Element Discretizations In t his subsection , we consider severa l mort ar sit uations. Each of t hem pr ovides an int eresti ng insight into t he abstract general fra mework. The flexibility and t he wide ra nge of app lications is illustrat ed. Duality arg uments play an essential role in our first example. Essenti al an d natural boundary conditions are used to realize t he coup ling of mixed and standa rd finite elements . In our second exa mple, we conside r t he special case t hat each element is one subdomain , and thus t he number of sub domains te nds to infinity as t he meshsize te nds to zero . The set ti ngs are given for simplicial t riangulations in 2D, but can be generalized wit hout any pro blem t o 3D.

1.4 Ex amples for Special Mortar Finite Elem ent Discret izations

51

1.4.1 The Coupling of Primal and Dual Finite Elements

In this subsection, we focus on th e coupling of two different discreti zation schemes . We use a mixed finit e element discreti zation on one sub dom ain and a standard conforming one on the ot her subdomain . The coupling of mixed finite element discreti zations on non-m atching grids has been introduced and analyzed in [WY9S, AY97, Yot97) . In cont rast t o standa rd mortar methods , t he finit e element trace space of th e flux in norm al dir ection at th e int erface defines a Lagr ange multiplier space which does not yield optimal results. In par ti cular, t he consiste ncy error is to o lar ge compared wit h the best approximation error. To obt ain optimal results, t he ord er of the Lagran ge multiplier space has t o be increased by one. In t he lowest order case, the Lagrang e mult iplier space has to contain piecewise linear functions and not only the constants . We observe t hat t he trace space of HI -fun cti ons and th e Lagran ge mult iplier space reflect the du ality between H I / Z and H- I / Z on th e interface. In our approach, we use t he du alit y of t he roles of Diri chlet and Neumann boundary cond it ions in the primal and du al setting. Using t his du ality, we can work on the pr odu ct space without int rodu cing a Lagran ge multiplier. The idea of combining mixed finit e element methods with prima l ones was originally introduced in [WW9S). Recently, efficient iterative solvers were st udied in [LPV99). For simplicity, we restrict our selves t o th e case of two subdomains. Here, il c JRz is decomp osed into two nonoverlapping polyhedral subdoma ins ill and ilz , and we assume that meas(ailz nail ) > O. On ill , we use du al discret ization techniques, whereas st andard primal approaches are used on ilz . The coupling at t he inte rface T := ail l n ailz is realized without a weak cont inuity conditi on between t he spaces and without a Lagran ge multiplier enforcing the orthogonality of t he jump. In st ead we t ake the flux of the solut ion on ill to define a Neuma nn boundary condit ion on T for t he boundary valu e probl em on ilz , and t he solut ion on ilz as a Diri chlet boundar y condition on T for t he bo unda ry valu e pr oblem on ill ; see Fig. 1.22. We show t hat pro ceedin g in this way yields a suit abl e coupling between t he two doma ins. The global discreti zation error can be bounded by the sum of t he local best ap pr oximation errors, and optimal a priori bounds can be established. flux

r>.

mixed FE

primal FE

Dirichlet problem U

Neumann problem trace

Fig. 1.22. Coupling betw een du al an d pri mal finit e eleme nts

52

1. Discretization Techniques Based on Dom ain Decomposition

We introduce simplicial triangulations Th, and Th 2 on the subdomains D 1 and D 2 which do not have to match at the interface. The sets of corresponding edges are called [h , and [h 2 • On D2 , where primal t echniques are used, we work with standa rd conforming P n 2 -element s, Xh2;n2 C HJ;r2 (D 2 ) . Her e, HaIr (D 2 ) is a sub space of H 1(D2 ) satisfying homogeneous Dirichlet , 2 boundary conditions on := 8D n 8D2 · The cont inuous space on D 1 is given by H(div; D 1) x L 2(Dd , where the vector valu ed Hilb ert space H (div ; Dd is defined by H (div ; Dd := {q E (L 2(D1))21 div q E L 2(D)} . For the discretization, we use Ravi art-Thomas finit e element s

n

RTh, ;n, := {q E H(div; D 1) I qlT E (Pn, (T)) 2 + r.; (T)x, T E Th,} , of order n1 2: 0 for the flux; see [RT77], and the space of piecewise polynomials Wh, ;n, := {v E L 2(Dd 1 v lT E Pn,(T), T E Th,} of the same order , for the primal variabl e in D 1. We refer to [BF91] for an overvi ew of mixed finit e element s. We remark that no boundary condit ion has t o be imposed on the int erface T. In cont rast to the standard case, Diri chlet boundary condit ions ar e natural boundary condit ions for the mix ed formul ation, i.e., t hey appear in the definition of the right hand side of the vari ational problem but are not enforced in the const ruct ion of the spaces. For the st andard prim al approach, t he sit uation is exactly reversed . The Neumann boundary conditions ar e the natural ones and t he Diri chlet boundary condit ions are imposed on t he space. This du ality allows t he coupling between Raviart- Thomas and conforming finit e eleme nts without Lagr an ge multiplier on t he interface . Using this Diri chletNeumann coupling, we find t he following discr et e vari ational problem : Find (jh' , Uh2' Uh,) E RTh,;n, x X h2;n2 x Wh, ;n, such that

a1(h"qh) - d(qh ,Uh2) d(jh" Vh)

+ b(qh ,Uh,)

+ a2(Uh 2' Vh)

= 0,

qh E RTh,;n, ,

= (1, Vh)O; il2'

Vh E Xh 2;n2 ,

- C(Uh, , Wh) = -(1, Wh)O; il" Wh E Wh,;n, . (1.50)

b(jh" Wh)

Here, t he bilinear forms ai(-, '), i = 1,2 , b(·, '), c(·, ·) and d(·,·) are defined by a1(P 1, q d: = J a- 1p 1 ' q1 dx, P1 , q1 E H(div;D1) , il, a2(w2, V2) := J (aV'v2 V'W2 + bV2W2) dx , V2, W2 E H 1(D ) ,

b(q1,vd := c(w1, vd

:=

il2

J divqj V1 dx , J bW1 V1 dx ,

il, il,

d(q1 ,V2 ) := (qj n, V2),

2

V1 E L 2(D1), q1 E H(div; Dd , V1, W1 E L 2(D1) , q1 E H(div; Dd , V2 E H 1(D2) ,

where (". ) stands for the du ality pairing of H- 1 / 2(T) and H 1 / 2 (T) . The bilinear forms a1(., .), b(·, ') , and c(', ·) ar e associate d with the du al approach on

1.4 Examples for Special Mort ar Finit e Element Discretizations

53

.01 whereas a2(',') is a standard H6;r2(.o2)-elliptic bilin ear form. The du ality of t he boundar y condit ions on r is realized by mean s of d(·, -). Mor eover , d(·, ·) t ransfers the boundary condit ions between the subdomains. Con sidering (1. 50) in more det ail , we find that it has t he followin g saddle point st ructure ) = ( BAT -B0 ) (/l'h U hl

(h) , h

fLh :=

(jhl) , U h2

where the operato rs B and 0 are associate d with the corresponding bilinear form s. The op erator A is non symmetric and has the form

AI - D ) A:= ( D A 2

'

where the op er ators AI , A 2 and D are associate d with t he corre sponding bilinear form s. The right hand side is given by t he linear form (1, ' ) 0; [./; , i = 1, 2. The kernel of the cont inuous op erat or B'[ : H(div; .od x H6.r (.02 ) ---+ , 2 2 L (.od , whi ch is associated with t he linear form b(·, vd, is KerB'[ := {(ql , V2) E H(div ;.od x H6;r2(.o2 ) I divq, = O} . It is well known; see, e.g., [BF91], t hat we have t he following equivalence : For q l E RThl ;n l ' divq, = 0 if and only if b(ql ' vd = 0, VI E Wh l ;n l . Thus the kernel of the discret e op erator B T : RTh l ;n l X X h 2 ;n 2 ---+ Wh l ;nl is a subs pace of KerB'[We fur t her introduce t he non symmet ric bilinear form a(a, T) := a2(w2,V2 ) + d(P l, V2) + al(Pl , ql ) - d(ql, W2), where a := (ql , V2) and T:= (P l , W2) are elements of the product space H(div; .od x H6;r 2(.o2 ) . The norm II· II on t his product sp ace is inh erited by the pro du ct topology and is defined by IIal1 2 := IIqlI1 6;[./1 + Ildivq lI1 6;[./1 + Il v21I r;[./2· It is now easy to see that t he nonsymmetric bilinear form a( ·, ·) is coercive on KerB'[

Here, we have used t hat a2(' , -)l / 2, rest rict ed to H6;r2(.02), is equivalent to t he H I-norm on .02 and t hat divqj = O. An essent ial tool in establishing a pri ori bounds for t he discretization err or is pr ovid ed by the ab st ract sa ddle point t heory. A suitable inf-sup condition guarantees t hat t he discreti zation error can be bounded in t erms of t he b est approximation err or. Here, t he relevant inf-sup condit ion is nothing else than the standar d one for t he mixed finit e element scheme on .01

Choosin g a as (q l ' 0) , q l E R T h l ;nll this is a standard result for Raviar tThom as finit e element s, and it is sat isfied with a constant ind ep endent of t he

54

1. Discretization Techniques Based on Domain Decomposition

meshsiz e without any further assumpt ions; see, e.g., [BF91]. In addit ion, we obtain unique solvability of the saddle point problem (1.50) by t aking th e continuity oft he bilinear forms, t he inf-sup condition, and th e coercivity of a(·,·) on K erE T , into account; see, e.g., [BF91, WW98] . Following [BF91], an opti ma l a priori bound for the discretiz ation erro r can be established by mean s of t he best approximat ion error. Since we are workin g with un constrained standard finit e element spaces , t he approximation properties are well known ; see [BF91]. With no matching condit ion imposed on th e discret e spaces at th e int erface, the analysis as well as t he impl ement ation of t he method becomes quite simple. A priori est ima tes of the ord er of min(nl + 1, n 2) are also obtained

Ilj - h 111 ~iv; .o1 + lIu -

Uh 1116;.o1

+ Ilu - Uh211r;.o2

::; c (h~(n1 +1) (lul;'1+1 ;.01 + UI;'1+1 ;.01 + [div j 1;'1+1;.oJ + h~n2Iul ;'2+1 ; .o2) (1.51) if the problem has a regular enough solution. Here, j denotes the cont inuous flux defined by j := a\1u . n. Remark 1.19. Th e coupling of m ixed finit e eleme nts on non-matching triangulations has been work ed out in [WY98, A Y97, Yot 97]. A s in th e standard conforming setting , one has to work with Lagrang e multipliers or equivalently a suit able cons traine d space. In the lowest order case, th e Lagrange multiplier space has to be at least piecewis e lin ear, since the optimality of th e me thod is oth erwise lost.

In t he next subsect ion , we show t hat the saddle point problem (1.50) can be rewritten . Introducing a piecewise constant Lagr an ge multiplier on t he interface, we obtain a mort ar finit e element method which couples conforming Lagran gian finit e elements and nonconforming Crouzeix-Raviar t elements. The special characteristic of t his approach is t hat t he piecewise constant Lagran ge multiplier gives rise to a diagon al mass matrix on t he non-mortar sides . In Sub sect . 2.3, we discuss how standa rd multigrid techniques can be applied in such special mor tar sit uat ions . 1.4.2 An Equivalent Nonconforming Formulation It is well known that mixed and nonconforming finit e element methods are

equivalent; see, e.g ., [AB85, BF91]. Introducing Lagr an ge multipliers on the edges of t he t riangulation, th e flux vari able as well as th e primal vari abl e can be evalua t ed locally. The resulting Schur complement syste m is th e same as for t he positive definit e vari ational problem associated with a non st andard non conforming Crouzeix-Raviart discretiz ation; see [BF91]. Furthermore, th e mixed finit e element solut ion can be obtain ed from t he non conforming by local postprocessing. Here, we pr esent a mor t ar coupling of conforming and

1.4 Examples for Special Mortar Finite Element Discreti zati ons

55

nonconforming finite elements which is equivalent to t he present ed primal du al coupling . In t he rest of this sect ion, we restrict our selves to t he lowest order Raviar tThomas space, n l = O. The dimension of t he local flux space is t hree, and the local space for t he discret e primal vari able is one dimension al. An equivalence with Crouzeix- Raviart elements can be obtained if we enrich the nonconforming space by local cubic bubble fun ctions. We t herefore consider t he enr iched Crouzeix-Ra viart space

where C R h , is t he Crouz eix-Raviart space of piecewise linear functions which are cont inuous at t he midp oint s of t he t riangulation '!h I and equa l to zero at t he midpoints of any boundary edge e E Chi naD. B is t he space of piecewise cubic bubble functions which vanish on the boundary of t he elements . Restricting th e vari ati onal problem (1.50), for t he moment , to D1 by setting V h E X h2 ;n 2 equa l zero, we get

'q

a d j"ll q h)

+

b (qh ,uh ,)

bO h " Wh )

-

C( Uh l, W h)

= d(qh , Uh2) ' = -(J, W h) O;D I

qh Wh

,

E RTh , ;o E Wh l ;o

, .

(1.52)

If t he solution U h2 is known, t hen t he Dirichlet boundary value problem (1.52) for the mixed formulation on D 1 can be solved. Compa ring (1.50) and (1.52) , we find t hat both vari ati onal problems have a saddle point st ructure . Introducing a Lagran ge multipli er and eliminati ng t he vari ables j,,1 and v«, in (1.52) yield a symmet ric positiv e definite variationa l problem. The boundar y condition now ente rs into t he definiti on of t he trial space . Let N Cg;h" 9 E L 2 (r ), be t he linear manifold of N Ch , defined by

NCg;h, := {'l/Jh E NCh,

I

J

'l/Jh da =

e

J

q da, e E Chi

n r} .

e

Then , t he solut ion of t he saddl e point problem (1.52) can be obtained , equivalentl y, by solving: Find lJIh l E N CUh2 ;h l such t hat (1.53) where aNc( ¢h,'l/Jh) := L:T EThl I T P a- I (a\l ¢h)\l'I/Jh + bIIo¢h IIo'I/Jh dx. Here, IIo is t he L 2-p ro jection ont o W hl ;O, and P a- I is the weighted L 2-pro jection , with weight a-I, ont o t he local Rav iart- T homas space of lowest orde r which has t hree degrees of freedom per element; see [AB85, BF91]. We remark that t he solut ion space NCUh2 ;h l for the solut ion in D1 depends on t he solut ion in D2 , but that t he right hand side in (1.53) does not depend on t he solution. Here aga in, t he different roles of the boundary condit ions come into play. Using t he equivalence of (1.52) and (1.53) in (1.50), we find a globa l variational problem in D: Fi nd (lJIh ll U h2) E N CUh2 ;h l x X h2 ;n 2 such th at

56

1. Discreti zat ion Techniques Based on Domain Decomposition

The impl ementation of (1.54) is based on a different formul ation. Considering (1.54) , in more det ail, it can be seen t hat it is a minimization problem with constraints, where the const raints depend on th e solut ion. Introducing a Lagran ge multiplier space, we can transform (1.54) into a saddle poin t problem . To do so, we show in a first ste p that t he Dirichl et problem (1.53) can be extended t o a vari ational problem on the whole space NChl . Using that j hl = Pa- l(a'VlJfhl) and v«, = IIolJfhl and applying Gr een 's formul a to t he second equation of t he saddle point pr oblem (1.52), we obt ain

aNC(lJfhll '1/;h) - d(Pa- 1(a'VlJfh l) , '1/;h) = (j , IIO'l/;h)O;f2} ,

'l/Jh E NChl

(1.55)

From t he definition of NCg ;hl' we find that an element X E N C hl is in NCg ;hl if and only if

J

J..l (X - g) da = 0,

J..l E Mh l ,

(1.56)

r

where M hl := {J..l E L 2(T ) I J..ll e E Po(e) , e E [ hI n T} . In particular, M hl is used as a Lagran ge multiplier space. T he dim ension of M hl is equa l to th e number of edges in [ hI n T. Finally, we obt ain a mor t ar coupling between conforming and non conforming finit e element s by means of piecewise constant Lagrange multipliers. Theorem 1.20. Let (lJfhl , Uh2) E NCUh2;hl x X h2;n2 be the solution of (1.54)·

Then, UM := (lJfhl ,Uh2 ) and AM := Pa- I(a'VlJfhl )·nlr is the unique solution of the following saddle point problem: Find (U M' AM) E (N C hl X X h2;n2) X M hl such that a(UM, v) - c!(AM,v ) = j(v) ,

v E N C hl x X h2;n2 ,

c!(J..l,UM)

J..l E M hl .

= 0,

(1.57)

Here, t he bilin ear and linear forms are given by, v := (VI,V2), w := (WI ,W2):

a(w , v ) := a2(w2, V2 ) + aNc( WI,VI), v, W E N Chl x X h2;n2 , d(J..l , v ) :=

I J..l(VI -

r

V2 ) da ,

J..lEM hl ,

j(v) := (j , V2)O;.l?2 + (j, IIovdO;.l?1 Proof. The assertion is an easy consequence of (1.54) by usin g (1.55) and (1.56). 0 Theorem 1.20 states t he equivalence of (1.50) and (1.57) in t he case n l = O. We obtain t he solut ion of (1.50) by a local postprocessing from th e solut ion

1.4 Exam ples for Sp ecial Mortar Finite Element Discreti zation s

57

of (1.57), from the formul as h I = Pa- l (a'Vu MIQl) ' v« , = lIouMIQl and Uh2 = UM!Q2' The a priori bound (1.51) and Theorem 1.20 guar antee t hat t he discretiz ation error of U-UM in th e energy norm is of ord er h . Comp aring (1.57) with (1.15), we find t hat (1.57) repr esents th e saddl e point formul ation of a mortar coupling . We recall t ha t t he saddl e point problem (1.50) realizes th e coupling of du al and primal finite elements methods. Using th e equivalent non conforming approach instead of t he mixed method, we ar rive at a mortar finite element method expressing th e coupling of conforming and nonconform ing spaces. The ana lysis of the resulting discrete problem could be also don e within t he abst rac t mortar framework. The piecewise constant Lagrange multiplier AM = h I .nlr is associate d with the side of the nonconforming discretization , and the mass matrix on t he non-mortar side is diagonal. We find that the mortar side is associate d with the conforming discretization and th e nonmortar side with th e nonconforming discreti zati on. Figur e 1.23 illustrates th e relation between th e different couplings.

c Mixed:5~

= Pa- l (a'V1[thl ) Uhl = IIo1[thl

j hl

IT

La" .., .

0, is given recursively and consists of three steps - presmoothing, error correction and postsmoothing. The correction step depends on the transfer operator. • Presmoothing step: For 1 :S i :S

Zzi

:=

Zii - I

Tnl,

let

+ G-1 1 (1Z-

A ZZzi-I) ,

where Tnl ~ 1, G l is a suitable smoothing operator and • Correction step: dz := fi - Azz;'" - Defect restriction: di- l := I;-ld i

(2.6)

z? := ul .

2.2 Vector Field Discretizations

99

- Coarse grid correction : For 1 ::; j ::; p , let j . - MG (l j-1 d ql1 .- 1 , ql-1 ' l-l ) ,

where q?-l := o. - Prolongation of the correction : Z;"l +l := • Postsmoothing step: For m1 + 2 ::; i ::; m1 1 z (nt Zli := zli- 1 + G-

Z;"l + IL l qf- 1 + 1 + mz, let - A lZli- 1) ,

wher e mz ~ 0 and G z is a suitable smoothing op erator. Finally, a multigrid step is defined by M G (l, u'(, Il) := z;"1+m2+1 . T he case p = 1 is called the V-cycle and the case p = 2 t he W -cycle. In 1 pr acti ce, t he rest ricti on is, very ofte n, chose n as t ranspose of t he prolongat ion IL l ' and in the case of nest ed spaces VI- 1 C VI the natural injection is taken for IL l. The easiest typ e of a smoot her is a damped Rich ardson met hod where the damping fact or has to be smaller t han one over the lar gest eigenvalue of A l . Symmet ric Gaul3-Seidel methods and incompl et e LV-fact orizations pr ovid e more robu st smoot hers. The choice m 1 = m z , G1 = and IL l = (IL 1)T yields a symmet ric it erative solut ion scheme . One possibility t o prove level ind epend ent convergence rate s for W- cycles relies on suitable approximat ion and smoot hing properties [Bra97, Hac85). The analysis of the mult igrid case is based on the two gri d case and a perturbation argument . In t he two grid case , approximat ion and smoothing prop erties yield that t he convergence rates are ind ep endent of t he meshsize provided t hat t he numbe r of smoothing ste ps is large enough. Additionally, a stability est imate for t he smoot hing scheme is required for t he multigrid analysis.

11-

Gr,

2.2 Vector Field Discretizations In t his section, we consider a boundar y valu e pr oblem for vect or fields , associate d with t he divergence op erator . We focus on t he 3D case and t he well known Ravi art-Thomas finit e elements; see, e.g., [BF91). Applications of t hese vect or fields can be found in [AFW97) . In Sub sect . 2.2.2, we develop an it er at ive substruct ur ing method for this class of finit e elements and in Sub sect . 2.2.3 , one based on a hierar chical basis. We not e t hat t here are several inter esting differences when compared to the H 1 -case. We consider the following second ord er partial differential equat ion for vector fields

Lu := - grad (a divu)

+B

u

tr- n

=f, = 0,

in n , on an

(2.7)

100

2. It erative Solvers Based on Domain Decomposition

where D is a bounded polyhedr al domain in IR3 of unit diamet er , and nits outward norm al. We assume that f E (L 2 (D)) 3, th at the coefficient matrix B is a symm etric uniforml y positive matrix-valued function with bi j E L OO(D) , 1 :::; i,j :::; 3, and t hat a E L OO(D) satisfies a ~ ao > 0 almost everywhere . T he weak formul ation of problem (2.7) is defined in a vector valued Hilbert space of which (HI (D)) 3 is a pr oper subspace. Applying Green's formul a, we find t ha t H (div j D) is a suitable space

equipped with the inner product (., ')div (W,V)div :=

!

w·vdx+

[J

!

divwdivvdx

[J

T he corresponding norm is defined by Ilwllaiv := (w , W)div. The boundar y condition in (2.7) will be imposed on t he space. A trace theorem given in [BF91] shows t hat the normal component, v -n , of any vector v E H(div; D), on t he boundar y aD , belongs t o t he spa ce H- I / 2(oD) . The subspace of vectors in H( div ; D) with vanishing normal component on aD is called H o(div ; D). We are now prepared to formul at e the weak vari ational form of t he boundary value problem (2.7) : Find u E Ho(div ; D) such t hat a(u ,v) = !r.vdx ,

VEHo(div j D) ,

(2.8)

[J

where t he bilinear form a(·,·) is given by a (w, v ) := !(adivwdiVV+BW.V)dx ,

w ,vEH(div ; D).

[J

An energy norm defined by IIIwlll 2 := a(w , w) is associate d with the bilinear form a(·, -). In parti cular , t his energy norm is equivalent to the Hilbert space norm II . Ii div. The equivalence constant s depend on th e coefficient s and can be quit e lar ge or quite small. Before defining t he Raviart- Thomas space of lowest ord er , we consider t he trace of an element in H (div ; D) in more detail. Throughout t he rest of t his sect ion, we will work with scaled Sobolev norm s; the size of the domains is reflected in weight factors. Given a bounded open Lipschitz dom ain D CD, with a boundary aD and a diameter HD, let 1·l s;D denote t he semi norm of the Sobolev space H S(D). Then the scaled norms are defined by 2 2 1 2 11 4>l k D = 14>II ;D + HD211 4>IIO;D'

1

4> E H (D) ,

2.2 Vector Field Discret izati ons

101

and 2 2 1 2 11 ¢11 1'D = 1¢l l. 11 ¢llo·,aD, aD + -H 2 'v 2' D

1

¢ E H 'i (8D ) .

In t he case t hat D = rl, we will drop the reference to the region. In contrast to t he pr evious sections, the H- l/2-norm will from now on reflect t his scaling, and it is defined by Ilw , nll _l2 'aD := '

sup

1 4> E H 2 ( tl D ) 4>'1'0

(w · n, ¢) 11 ¢11 !; aD '

wE H (div ;rl) ,

where (-, .) represe nts t he du alit y pairing between H - 1 / 2( 8D) and H 1 / 2( 8D) . T he following lemm a provides a basic est imate for th e trace of an element in H(div ; rl) . The proof is quite elementary and is based on Green 's formula and a scaling argument . Lemma 2.3. Th ere exists a cons tan t C , which is in depen den t of th e diameter of D but depen d on the shape regularity of D , such that for w E H (div ; rl) Ilw , nll ~ ! ; aD :::; C (1I wI1 6;D + H bll div w!l6;D) . We will use t he well known Raviart- Thomas spaces for th e discreti zation of (2.8). T he finit e element approximation is given on a t riangulat ion Th , t he element s of which are denoted by T . The set of int erior faces and edges of t he trian gulations Th is called :h and [ h , respecti vely. 2.2.1 Raviart-Thomas Finite Elements Our st udy concerns t he lowest order Raviart- Thomas finite elements for t he discrete ap proximation of (2.7); see [BF91]. In Sect . 1.4, t his space was already used to define a mort ar finite element discreti zation. Here, we work in 3D and t he definiti on of t he degrees of freedom is essent ial, and we th erefore consider t his space in more detail in t he rest of th is subsect ion . T he globa l Raviar t-Thomas finite element space RT(rl ; Th) is defined by mean s of t he local ones

RT( rl ; Th) := {w E H(div; rl)1 wI T E RT(T ), T E Th} , where RT(T) stands for t he local Raviart-Thomas space . In the case of a hexahedr al triangulation , where t he elements are cubes , t he local space has dimension six, while in t he case of a simplicial triangulat ion, where t he elements are tetrah edras, t he local space is four dimension al. For a cube with sides parallel to t he coordinate axes, RT(T) is given by

102

2. It erative Solvers Based on Dom ain Decomposition

In t he case of a tetrahedr a, RT(T) is defined by

RT(T) :=

+ f3 X ) + f3y , a3 + f3z

a 1 (

(2.9)

a2

The degrees of freedom of RT( D ; T,.) are given by the averages of th e norm al compo nents over the faces F of th e triangulation:

AF(W) :=

I~I

J

w · n dC7

F

Here, IFI is t he area of th e face F and t he direction of the normal can be fixed arbit ra rily for each face. This formula also defines th e natural int erp olation operator from H(div ; D) onto t he space RT(D ; T,.) . We not e t hat t he norm al component of any Raviar t- Thomas vecto r field is constant and continuous across each face, see Fig. 2.9.

Fig. 2.9 . Local degrees of freedom of a lowest order Raviar t-Thomas vect or field

We define the subspace of vectors with vanishing norm al components on the boundar y of D by

Vi, := RT(D j T,.) n Ho(div ; D) As in t he case of Lagran gian finite elements , t he L 2 -nor m of t hese discrete vect or fields can be bounded from above and below by mean s of t he values of t heir degrees of freedom . For W E RT(T) , we have C

L (H; AF(W)f

F e8T

: :;

Ilw11 6;T :::;

C L (H; AF(W)f Fe8T

'

(2.10)

where H F is the diameter of t he face F . The equivalence constants depend on the aspect ratio of t he elements, bu t do not depend on the diameter. Moreover , t he following inverse estimate holds: 1 IIdivwllo;T :::; C HT II wll o;T,

w

e RT(T)

2.2 Vect or Field Discreti zation s

103

These bounds can easily be shown by using t he affine equivalence of the elements of t he t riangulations and t he finite dimension of th e local spaces . We not e t hat relativ ely few st udies exist of domain decomp osition methods for H(div ; n) and H(curl ; n) in 3D; we refer to [CPRY97, HTOO, TosOO], for two-level overlapping methods, to [AFWOO, BDH+99a, Hip96, Hip97, Hip98] for multilevel and multi grid methods, and to [AV99] for a study of an iterative sub structuring method in H(curl ; n). We refer to [AFW97 , Bre92, EW92, Mat93 a, Mat93b] and to the references therein , for some Schwarz methods for problems in H (div ; n) in 2D. A certain class of multilevel methods for the mixed approximation of th e Laplace equation is discussed in [Sar94] . Here, we const ruct a quasi-opti mal it erative subst ruct ur ing method as well as a hierarchical basis method for t he 3D case. 2 .2.2 An Iterative Substructuring Method

In this subsection, we int rod uce and analyze our iterative subst ruct ur ing met hod, originally described in [WTWOO] . For a quite general introduction to dom ain decomp osition methods for vector fields discretizations , we refer to [Tos99]. We restrict ourselves to the case of a hexahedr al triangulation Tit which is obt ain ed by qu asi-uniform refinement from a coarse macro-tri angulation TIl ' From now on t he generic elements, faces and edges of Tit are denot ed by t, t , and e, and t hose of TIl by T , F , and E , respectively. The set of int erior faces of the macro-tri angulation TIl is called :FII . We use a simplified notation for the ratio of the meshsizes between macro and fine triangulation. In t he context of elementwise est imates, H / h denot es the ratio of t he local meshsizes whereas in globa l bounds, H / h stands for th e maximum of t he local rati os. As we have seen in Sect . 2.1, t he first ste p towards th e introduction of an addit ive Schwarz meth od is to define a set of subspaces; see Lemm a 2.1. As in t he 2D case for standa rd Lagrangian finite elements, we introduce t hree different typ es of subspaces called VII , VF and VT. To obtain scalabl e bounds, we cannot avoid th e use of a globa l space. But in contrast to t he standard Lagr angi an finite element s in 3D, th e low dimensional Raviar t-Thomas space associated with the macro-triangulation, VII := RT(n ;TII) nHo(div ;n) , can be used for this purpose while at the same ti me the constant in t he estimate of the condit ion number does not depend on the ju mp s of the coefficients . The local spaces V'r , T E TIl , are associated with the elements, also called substruct ures, of t he macro-t rian gulation. Each element in VT ,

V'r := {v E Vi, I supp veT} ,

T E TIl ,

has sup port contained in T. Thus, the projections onto VT are orthogonal. The t hird t ype of subspaces is associate d with t he faces F E :FH of the macro-t riangulation. T he supports of its elements are contained in two substructures T 1 and T 2. For each interior face F E :FII , th ere are t wo elements

104

2. Iterative Solvers Based on Domain Decomposition

T 1 and T 2 E TH such t hat F = aT1 n aT2, and we set '1'F := '1'1 U'1'2. The face spaces are defined by VF

:= {v E

Vh

I a (v , w)

= 0, w

s V T1 + V T 2 ,

suppv C '1'F }

We not e t hat an element v E VF is defined uniquely by its valu es v . n on F. Now, we decomp ose 11" into t he coarse space VII , the face spaces VF , F E FH ' and t he int erior spaces V T , T E TH, and we observe that the coarse space VH is contained in t he union of the face and int erior spaces. The decomp osition, Vh=VH

+

L

VT

+

T ETH

L

VF

,

(2.11)

FEFH

on which our iterative sub structuring method is based , is therefore not a direct sum. Fi gur e 2.10 illustrat es t he local decompositi on of Vh into t he t hree different t ypes of subspaces. Each type of subspace is symbolized by t he faces which arc associate d with the degrees of freedom.

/:

/.

/ : / i / : / : :

::v ·('t"-.:+:~·l- - ._;('t-... : //;:--1-- ,::·t:·::f·· ) /-i- .. i7 •.. .. )/:_- -)L .. .. V /',

Fig. 2.10. Decomposition of Vh into t hree types of subspaces For simplicity , we restrict our selves t o t he case that t he exact pr ojections onto the sub spaces arc used to define the additi ve Schwar z method. According to Lemma 2.1, the proof of an upper bound for t he norm of Tad d is elementary. It is based on a coloring argument, and t he coefficient s a and B do not enter int o t he bound. The crucial par t is t o find a decompositi on of v E Vh such that (2.2) holds wit h a Co as small as possible. A bound for t he corresponding multi plicat ive vari ant follows from Lemma 2.2. Ob serving t hat V h = L:TETH V T + L: FEFH V F is a dir ect sum, the decomposit ion v

=VH

+

L

TE TH

VT

+

L

vF ,

F EFH

with V H E V H , V F E V F , and V T E V:r , is unique as soon as VII has been fixed . A first ste p towards t he pr oof of a sharp bound for Co is t o find a suitable v H E VH . In the following subs ubsection, we consider a standard int erpolant PH ont o t he global coarse subspace. By mean s of th e discret e norm equivalence (2.10) , we will establish stability bounds in t he £ 2_ and

2.2 Vector Field Discretizations

105

the H (div ; D)-norm. A detailed discussion of this operator can be found in [WTWOO]. 2.2.2.1 An Interpolation Operator onto V H. An important role in the analysis of our additive Schwarz method is played by the interpolation PH onto the global coarse subspace VH . This interpolation operator is defined in terms of the degrees of freedom of VH by

Ap(PHV):=

I~I /

v· u da,

FE

hi .

F

We note that the stability estimates for the interpolant PH will enter into our estimate of the constant Co. As we will see, PH is not uniformly stable in h. The following lemma can be found in [WTWOO] and gives a bound for the interpolant PH in terms of the ratio H / h. Lernrna 2.4. There exists a constant C, which depends only on the aspect ratios of T E TH and the elements of TI", such that for all v E Vh,

(2.12)

Iidiv (PHv)[16;T ::; Ildivvl16;T , IlpJIv116;T

< C ( (1 + log ~) Ilv116;T + H}lldivvI16;T)

(2.13)

Proof. For a better understanding of the techniques, we review the proof given in [WTWOO]. By a simple computation and the use of Green's formula, we find that (div (pJIv)) IT is constant and

where IIH is the L 2-projection operator onto the space of constants on T E TH; see [BF91, Sect. III.3.4]. Inequality (2.12) follows immediately. The proof of (2.13) uses Green's formula, the norm equivalence (2.10), and a partition of unity very similar to the one given in [DSW94] for the simplicial case. Consider a face F c BT, and note that it is decomposed into non-overlapping faces of the fine triangulation; see Fig. 2.11.

H

Fig. 2.11. Decomposition of F

106

2. It erative Solvers Based on Domain Decompositi on

Number t hese faces so th at h , 1 :S i :S nF , have at least one vertex on an edge of F ; see Fig. 2.11, and let II ,12 ,13 ,14 be t he faces t hat contain a corner point of F . Let t, C T , be t he associated element s. We remark t hat, by ass umption, t he t riangulation restrict ed to t he face is qu asi-uniform, and t hus nF :S C( H l h) . Let 1'JF be a cont inuous , piecewise t rilinear fun cti on defined on T , which vanishes on aT \ F and is equal to one at all int erior mesh point s of F. It equa ls one in the grey shadowed region in Fig. 2.11. The exte nsion of 1'J F t o t he int erior of T has values between zero and one, and t he absolute valu e of it s gradient is bounded by C I max (r , h), where r denotes t he dist an ce to t he wirebas ket of T . The wirebasket of T is t he union of t he twelve edges of T. We refer to [DSW94] for an explicit construction of such a function for a simplex; t his constru ct ion can easily be adapted to the cubic case. The followin g upper bound for t he HI-semi norm and t he £ 2-norm can then be est ablished I1'J Fl tT

:S CHT (1 + log ~),

II1'J F I16;T :S CH'f .

(2.14)

Using (2.10), it is sufficient t o bound t he absolute valu e of AF(PHV) to get an upper bound for t he £ 2-norm of PHV. Appl ying t he definit ion of PH and Green 's formula, we obtain

where f3i = 3/4 for 1 :S i :S 4 and f3i = 1/2 for 5 :S i :S t he £ 2-norm of PHV can be bo unded by H TllpHV I1 6'T

,

:S C 2:= r cer

(fT(

+ C F~T

n.j»:

(1'J F div v + grad 1'J F . v ) 4

Thanks to (2.10),

dX) 2

n

) 2

t i~ IlilA!i (v) + ~ i~ IhlA!i (v)

By mean s of (2.14), we find an upp er bound for t he first sum on t he right han d side (2.15) Applying (2.10) again and keepin g in mind t hat nF is bo unded by C'Hl li , we find an upper bound of t he second te rm

2.2 Vect or Field Discreti zati ons

C

L

FeaT

L 11i1 2(Af i (v)) 2 :::; CHT II vll 6;T

107

np

nF

(2.16)

i= 1

The upp er bound (2.15) and inequality (2.16) finally give

IlpH v ll 6;T :::; CHfildivvll~ ;T + C

(1 +

log

~) IIvll~;T

0

We can obtain a similar est imate for the energy norm on each subst ruc t ure J B

(PHV) ·(PHV)dx:::;C ;~ (1+10g~) J

l'

Bv ·vdx

l'

+ CHf

l TJa divv divvdx aT l'

Here, aT is t he minimum of a( x ) on T, and (31' and IT satisfy

Thus, the const ant in t he corresponding global est imate dep ends on the ratio of the coefficients B and a on individual sub structures, but not on the jumps of t he coefficients between the subst ruc t ures . Remark 2.5 . Th e interpolati on operator PH is logarith mically stable in th e II . IIdiv -norm . Th is resu lt holds fo r the 3D case as well as in the 2D case; see [TWWOO). In contras t, th e nodal in terpolant onto contin uous finit e element spaces has a norm which grows as (H/h)I / 2 in the 3D case; see [DSW94j. Th is is an essentia l difference between the fa ce based R aviart -Thomas finite eleme nts and th e st andard ver tex based Lagrangian finite elements. 2.2.2.2 An Extension Operator onto V F. T he second basic tool in t he analysis of our it er ative subst ructur ing method is an exte nsion op er ator from Wh(F) onto VF. Here, Wh(F) is t he finit e element space of piecewise constants on f C F . The idea is analogous to a result for the st andar d finit e element case , in which the HI-norm of a discret e harmonic function is bounded by t he H 1 / 2-n orm of it s t race on the boundary. The following lemma shows t hat t he exte nsion op erat or is uniquely defined and stable. Lemma 2.6 . Th ere exis ts a unique extension operato r Ji F : Wh(F) ----+ VF satisfyin g (Ji F/-L) . n ip = /-L , /-L E Wh(F) . Furth ermore if JF /-L da = 0, we have th e follo wing sta bility estim ate

where /-L is exte n ded by zero on

aT \ F.

108

2. Iterative Solver s Based on Domain Decomposition

Proof. The degrees of freedom of VF are exactly the normal components on f C F , which are constant on each face f . Thus the trace space of norm al compo nents of VF is exac tl y Wh(F) . Therefore, the existe nce and uniqueness of such an extension operator follows from the definition of VF. The pr oof of t he stability est imate is based on a lemma established in [WTWOO] , where a stable exte nsion to a divergence free vector field is const ructed for f-t E Wh(F) with I F u da = O. Here, we sket ch the ideas of t his proof and refer to [WTWOO] for t he det ails. The starting point is a Neumann boundar y valu e problem for t he Laplace operator on T , where FeaT . We extend f-t by zero on aT \ F . The extension is still denoted by f-t . Then , the Neumann boundar y value problem -Llu = 0,

in T ,

au an =f-t ,

on aT

(2.17)

has a solut ion . Moreover, we obtain a unique solution of (2.17) by imposing the addit ional condi tion u dx = O. An elementary regularity result gives an upper bound of t he H -norm of th e solut ion in te rms of t he boundar y condition, i.e., luh ;T S; C IIf-t II-l!2;8T' The resulting vector field grad u is in H(div ; T) . In a second ste p, we apply the natural int erpolation operator onto t he Raviar t- Thomas vector fields, and denote the result by v /l' The stability, Ilvltlldiv;T S; ClukT , is a consequence of an approximation property and an inverse est imate . By const ruct ion v It is divergence free, and it has the same normal t rac e as 1-lFf-t on D'I', Thus, 1-lFf-t, restricted to T , can be writ t en as 1-lFf-t = v u + VT , where VT E VT . Keeping, t he definiti on of VF in mind , we find t ha t

Ir

where aT(-,' ) is t he restriction of a(', ') on T . 0 As already mentioned , an important step in the pro of of a sha rp bound for Co is t he choice of V H . Defining VH := PHV , we find

and t hus, by means of Lemm a 2.6, we get an upp er bound for t he norm of V F in terms of t he norm al component s of v - PHV restricted to F

where f-tI F = ((v - PHV) . n) IF and f-t = 0 on aT \ F . To apply Lemm a 2.1 , an est imate of t he H -l! 2-norm of f-t in te rms of the norm of v is required . We conclude this par agraph by proving a decomp osition lemm a for piecewise const ant fun ctions on t he boundar y of a subst ructure . We recall that t his space is exactly t he trace space of normal component s of Raviart-Thomas finite elements.

2.2 Vect or Field Discreti zat ions

109

The following lemm a is established in [WTWOO] . Here, we briefly discuss t he main ideas and omit some technical det ails . In the following each element f.L F in Wh(F) is extended by zero onto aT \ F and is st ill denoted by f.LF. The subspace of Wh(F) with zero mean value on F is called WO;h(F) := {f.L E Wh(F ), JF f.L do = O} , and the six dimensional space on aT containing all fun ctions which are constant on each face F is denoted by W H (aT) . Lemma 2.7. Let T be in TH' let f.LF E WO ;h(F) , FeaT , and let f.L := L- FC[)T f.LF · Th en, there exists a constant C, in depen dent 01 f.LH E WH(aT)

and h such that

2 ). IIf.LFII_2!;[)T :s; C (1 + log hH) ((1 + log hH)!If.L + f.LHII_2! ;[)T + 11f.L11_!;[)T (2.18) Proof. By definiti on , t he mean valu e on aT of f.LF is zero. Applying a Poin care-Friedri ch's type inequ ality, we obtain

sup 1 E H'i([)T)

where c is t he mean valu e of¢ on aT. The essent ial idea is now t o replace t he supremum over H 1 / 2( aT) by t he sup remum over a suitable discret e space . We int roduce a space of bubbles on aT by

where Ai , 1 :s; i :s; 4, are t he bar ycentric coordinate funct ions of the face We also use t he space of conforming bilinear finit e elements on aT Sh(aT) := { ¢ E C(aT)1 ¢I! E

QU) , 1 caT}

I·

,

where QU) is t he space of bilinear funct ions on I. Let P : H 1 / 2 (aT ) --+ Bh (aT ) + Sh(aT) be a uniform H 1 / 2 -st able operat or which sat isfies

J

J

F

F

¢ f.LF do =

P ¢ f.LF da,

f.LF E Wh(F) , r c

er .

(2.19 )

For t he existence of such an operator, it is sufficient to construct one example. We set P ¢ := ¢S+¢B , where ¢s := Ps ¢ is given by a st andard locally defined H 1 / 2 -st able quasi-projection opera t or Ps ont o Sh(aT ) j see, e.g., [SZ90], and ¢B E B h(aT) is defined by

J

J

f

f

¢B do =

(¢ - ¢s ) da,

1c

et .

110

2. Iterative Solvers Based on Dom ain Decom positi on

Then , P satisfies (2.19) by construction. Fur th erm ore, t he Hl/2-semi norm of PcP can be bounded by IP¢>l l/2;8T ::; l¢>sI1/2;8T + I¢>BIt/2;8T. If t he quasiproj ection operator Ps satisfies an approximation prope rty, we find for the second term

which yields t he H 1 / 2- st ability of P. Moreover, the spaces Sh(8T ) and B h (8T ) satisfy an orthogona lity relation, and we find ('\l'l/Js , V'tPB)O;8T = 0, 'l/Js E Sh(8T ), 'l/JB E B h(8T). Then , an int erp olation argument and an inverse esti mates yield

These prelimin ar y considerations guarantee the following norm equivalence

Now, the supremum over H 1 / 2 (8T ) can be modified . In particular , we can replace t he cont inuous spa ce H 1 / 2 (8T ) by t he discret e spaces Sh(8T) and B h (8T ), and we find t he following upp er bound

II /1F II -1 '8T < C 2'

-

I I '8T -

sup 4>EBh(8T) 4> =/:0

sup

4>ESh(8T)+Bh(8T)

2'

¢ ;econ st .

(/1F, ¢»

1¢>!I.8T 2

1

(2.21)

(/1F, - ¢» -+ 1¢>I I ;8T 2

sup 4> E Sh(8T)

II;8T 2

The first inequ alit y is a consequence of (2.19), whereas th e second follows from t he st abili t y in t he H 1 / 2- semi norm of P . Finally, t he last inequality follows by using (2.20). We next consider t he two term s on t he right hand side of (2.21) separ at ely, and start with the first te rm. A local inverse est imate and an interp olati on arg ument shows t hat t he rest riction ¢>F of ¢> E B h (8T ) to F is H 1 / 2- st able,

where ¢>F := ¢> on F and ¢>F := 0 on 8T \ F . Using t his bound, we find t hat for ¢> E B h(8T)

1(/1F, ¢»

I = 1(/1, ¢>F) I < 11/111- ~ ; 8T II ¢>FII ~ ; 8T I¢>I ~ ;8T I¢>I ~ ; 8T I¢>I ~;8T 11/111-1'8T I¢>F!I.8T ::; C < C I I /1 I I-~;8T 1¢II '8T 2

2'

2'

(2.22)

2.2 Vector Field Discretizations

111

Unfortunately, the second term in (2.21) cannot be bounded as easily. We start by defining for each ¢ E Sh(aT) a weighted average c¢ C¢./ {}F dO":= F

.I

h({}F¢) do .

F

Here, '1') F is given in the proof of Lemma 2.4, and h is the nodal interpolation operator onto Sh(aT). Then, the supremum in the second term on the right hand side in (2.21) can be replaced by sup ¢ ESh (8T ) 1'¥:const.

(MF, ¢) 1¢ll'BT 2'

sup

(2.23)

¢ E S h (8 T ) ¢#const.

i.e., we need only consider functions ¢ which have a zero weighted mean value on F. In the last step, we have used the following norm equivalence

This is a Poincare-Friedrich's type inequality, and can be easily proved by contradiction. We can now apply techniques well established for the standard Lagrangian finite elements in 3D; see [DSW94]. In general, ¢ E Sh(aT) cannot be decomposed in a uniformly stable way into face and edge contributions. Therefore, we start by decomposing ¢ into a sum of contributions ¢F supported on individual faces FeaT, and ¢w supported in a neighborhood of the wirebasket which is one clement wide; see Fig. 2.12.

Fig. 2.12. Neighborhood of the wirebasket

By means of this decomposition, we can rewrite ¢ as

¢=

L

¢F +¢w .

(2.24)

FeBT

The H 1 / 2 -semi norm on aT of ¢w can be bounded by means of an inverse estimate in term of its one dimensional L 2-norm over the wirebasket W

112

2. It er at ive Solvers Based on Dom ain Decom posit ion

Upper boun ds for Il cPwll& ;w and Il cPFll i/2;8T are established for t he simplicial case in [DSW94]. Following [DSW94, Lemm a 4.3 and Lemma 4.5], and observing t hat t he H I-norm on T of a discrete harmonic function can be bounded by t he H I / 2 -norm of its t race on aT, it can be shown t hat 2 IlcPwllo ;w

H) IlcPll2!;8T

~ C 1 + log h (

(2.25)

'

Il cPFII! ;8T ~ C(1 + log ~ )21IcPll ! ;8T .

(2.26)

The proofs in [DSW94] are for t he simplicial case, but they can be carr ied out in exactly t he same way for the hexahedral case and det ails are t herefore omitted. We find, by using t he splitting (2.24) , t hat

(/-LF, cP) =

L

(/LF, cPp) + (/-LF,cPw ) = (/-L,cPF ) + (/-LF, cPw ) .

(2.27)

Pe8T

Since h (7'J FcP) = cPF = t hat cq, = 0, we obtain

h(7'JFcPF ) on aT,

and since we can always assume

and Cq,F = O. The first term on t he right side of (2.27) can be bounded by mean s of (2.26):

For each

cPw , t here

cPB E B h (aT)

is a uniqu e

JcPw do JcPB da, =

f

f

such that

cPB =

E r.;

F .

f c

0 on aT \ F and

f

Moreover , t his mapping is locally uniformly stable in th e £ 2-norm on aT; Il cPBIIO;j ~ CllcPw llo;j. An inverse estimate toge t her with the definition of cPB easily yield

Il cPBII! ;8T ~

C ~ llcPB I I&;8T ~ C~ l lcPw ll&;8T ~ C Il cPwll&;w

By mean s of t his bo und and (2.25), we finally obtain

I(/-LF, cPw )1= I(/-LF, cPB )1= I(/-L, cPB )1~ CII/-LII-!;8TllcPw llo;w ~ C (l

H ! + log h) 211/-L11- ! ;8TllcPll! ;8T .

(2.29)

The pro of is complete d by combining (2.21), (2.22), (2.23), (2.24), (2.28), and (2.29). 0

2.2 Vector Fi eld Discreti zations

113

2.2.2.3 Quasi-optimal Bounds. We ar e now able to formul at e the cent ral resul t of t his section. In t he following theorem , we specify a suitable decomposit ion of v E Vh and give an upp er bound of t he constant Co in Lemma 2.1. This result was originally given in [WTWOO].

Theorem 2.8. For each v E Vh, there exis ts a decomposition v = v II +

L: VT + L: TE TH

vp ,

P EFH

corresponding to (2.11) , such that a(VH,VH ) +

L: a(VT ,VT) + L: a(vp ,VP) :::;C(1+1 0g~ra(v ,v) ,

TETH

PEFH

with a constant C, independent of h, H, and v . Proof. The choice VH := PHV and Lemm a 2.4 yield

where aTC· ) is the restriction of a (' , ') on one substructure T . Then , Vp is uniquely given as t he discret e harmonic exte nsion of J.Lp := (v - VH ) . n lF' The minimization property of 1-l p gives

see Lemm a 2.6. In a final step, we apply Lemm a 2.7 and bound IIJ.Lpl l- 1/ 2;8T in te rms of aT(v, V)1/ 2. Setting J.L H := VH . n , we obtain

Lemm a 2.3, the trian gle inequ ality, and Lemm a 2.4 yield

and t hus by applying Lemm a 2.3 to Ilv . n ll-l / 2;8T' we find aT(Vp,

H

vr ) :::; C (l + log h)

2

aT (v , v ) .

An upper bound for aT(vT , VT) is now an easy consequence of the trian gle inequ ality. Finally, t he global upp er esti mate is obtained by summing t he local ones over the elements of t he macro-trian gulati on. 0

114

2. Iterative Solvers Based on Domain Decomposit ion

Remark 2.9. Th e cons tant C in Th eorem 2.8 depends on the coeffici ents but not on the m eshsize. Since all estim ates are done locally on ind ividual su bstructures, it can be shown by a more careful analy sis that C is in dependent on the jumps of th e coeffi cients. However, it depends lin early on

Using exac t orth ogonal projections onto t he sub spaces, t he addit ive Schwar z operat or Ta d d , defined by t he decomposition (2.11), has a condition number which is bounded logari thmically in t erms of the ratio H / h K: (Tad d )

::;

C (1

+ log ~) 2

,

where th e constant C does not depend on the jumps of th e coefficients. This is an easy consequence of Lemm a 2.1, Theorem 2.8, and a coloring argument. 2.2.3 A Hierarchical Basis Method In t his subsect ion, we introdu ce a hierarchical basis method for Ravi artThomas finit e elements in 3D. We obt ain th e same qualit ative upp er bound for t he condition number of t he additive Schwar z meth od as for standa rd Lagr angian finit e elements in 2D; see, e.g., [Ba n96, Yse86, Yse93]. We recall that t his result does not hold in 3D for th e Lagran gian finit e elements. The starting poin t for the definition of our hierar chical bas is method is a nest ed sequence of ad ap tively generate d simplici al t riangulat ions , 70,' " Tj. We can use some standard refinement rules gua ra nteeing t hat t he t riangulations form a famil y of shape regular and locally quasi-uniform triangulations; see, e.g. , [Bey95]. The sets of int erior faces are denoted by Fi , o ::; I ::; j , and the associated Ravi art- Thomas finite element spaces are called VI , 0 ::; I ::; j. They form a sequence of nest ed spaces satisfying Vo C VI C ... C 10 C Ho(div ; rl). The local Ravi ar t-Thomas spaces with four degrees of freedom per element ar e defined by (2.9), and th e global space has one degree of freedom per int erior face. To define our Schwarz method, we consider two different types of decompositions - a horizontal and a vertical. The horizontal decomposition is based on the hierarchy of finit e element spa ces whereas th e vertical one reflects a Helmh olt z-typ e decomposition of the vector fields. This ty pe of split t ing has already been used in [HW97, Woh95] for the 2D case . We also note that a hierar chical basis pr econdi tioner for the saddle point pr oblem ari sing from t he mixed finite element discretiz ation of an ellipt ic second order operat or has been introduced in [Woh95]. Similar decomp ositi on techniques are used in [Hip96 , Hip97] t o construct an optima l multigrid method for Raviar t- Thomas finite elements in 3D based on a uniformly refined sequence of t riangulat ions .

2.2 Vector Field Discretizations

115

Efficient preconditioners and multigrid techniques are discussed and analyzed in [AFW97, AFW98, AFWOO]. The natural interpolation operator PI : Vj ---+ Vi, 0 ::; l ::; i . will play an essential role in the analysis of our hierarchical basis method. For the convenience of the reader, we recall the definition already given in the previous subsection AF(PI(V)) :=

I~I

J

v· nda,

FE:FI ,

F

wher e AF( ') are the degrees offreedom as given in Subsect. 2.2.1. We remark that Pj = Id, and we define P-l := O. 2.2.3.1 Horizontal Decomposition. We define a family of subspaces by

o::; l ::; i , of Vj

Then, obviously Vo = Vo and Vi C Vi and Vi n Vk Furthermore, each element v E Vj can be written as ~

~

j

0, 0


k . We consider one eleme nt T E T,. at a ti me , and observe that v klT E R/(T). In a first st ep , we show that t he divergence of Vk and of VI are orthogonal with resp ect t o the £ 2-scalar product

J

div v. div vs d»

=

T

J J

div(Plv-PI_ lv)divvkdx

T

(III (div v) -

=

tu., (div v)) div v» dx

= 0

T

Her e, III and lIl- 1 are the £ 2-projecti ons onto t he space s of piecewise constants associate d with the t riangulat ions 11 and 11-1 . To get an upper bound for (VI , V k) d iv ;T , we have to consider in a second step t he cont ribut ions of t he £ 2-term. Observing t hat v klT can be written as a gradient of a qu ad ratic function wit h mean value zero, VkIT = grad ¢, ¢ E P2 (T) , we find by applying Gr een 's formul a

J

VI ' v» dx

J

=-

div v, ¢ dx

T

+

T

By t he definitions of VI and

J

¢ VI . u da .

8T

Ti-1, we get the following orthogonality relations

J

div v,

fil - 1¢ dx

= 0 ,

T

where fi l - 1 is t he £ 2-proj ecti on onto the space of piecewise constant s assoand ciate d with t he t rian gulat ion

Ti-1 ,

Jfi

l- 1 ¢ VI . ti da

=0

8T

By mean s of these orthogonality relations and t he discret e norm equivalence (2.10), we obtain the followin g upper bound

J

J

T

T

VI ' v» dx = -

divvI(¢ - n -1 ¢) dx

:::; C2 -(l- 1- k)

+

J

(¢ -

8T

fil- 1 ¢ )VI . u da 1

(hTlldivvdlo;TllvkIIO;T + hj,llvl' nllo;8Tllv kllo;T)

:::; C2- (I - k)/ 2(\Idiv vdlo;T + \Ivdlo;T)llv k\lo;T .

o

118

2. It erative Solvers Based on Dom ain Decomposition

An upper bound for p(£) can be now derived by using the equivalence between t he energy norm and the Hilb ert space norm and Lemm a 2.11. It is bounded by a constant ind ep end ent of t he number of refinement levels

p(£)

~

C .

Combinin g t he est imate for p(£) and Lemma 2.10 with Lemma 2.1, we find that the condit ion number of this addit ive Schwarz method with exac t projections onto t he subspaces is bounded quadrati cally in j

The computat ional cost for t he applicat ion of t he exac t pr ojection is in pr act ice t oo high . Thus, the exac t pr oj ecti on onto the subspaces will, very oft en , be repl aced by qu asi-projecti on operat ors Ti onto Vi.:- We define our qu asipr oj ecti ons Tl in terms of a vertical decomp osition of Vi , 1 ~ 1 ~ j , and show t hat t he condit ion number is st ill bounded qu adrati cally in terms of j . In particular , we prove that (2.3) is satisfied with w ind epend ent of j . 2.2.3.2 Vertical Decomposition. We decompose t he global space Vi int o local low dim ension al spaces which are assoc iate d with t he faces and elements of the triangulati on on level 1- 1. This split t ing is motiv at ed by a Helmholt ztyp e decompositi on . In case of a Helmholt z decompositi on , t he space would be writ t en as a dir ect sum of a divergence free subspace and it s or thogonal complement. Here, we decomp ose Vi int o a sum of divergence free subspaces plu s a kind of qu asi-orthogonal complement which is called, in the following, a surplus space. We not e t hat t he divergence free Raviart-Thomas vect or fields can be obtain ed from Ned elec finit e elements; see [Ned82]. The family of Nedelec finit e elements form a subspace of t he Hilb ert space H( curl ; n )

We work wit h the Ned elec finit e elements of lowest order which are defined by

N V(n ; Tt) := {q E H(curl ; n)1

qlT E N V (T ), T E Tt} ,

0 ~ 1~ j ,

wher e N V(T) stands for t he local Nedelec space. In t he case of a simplicial triang ulation, t he local space has dim ension six

NV (T) :=

(~~: ~~~) (Y3

+ (J3 Z

.

The degrees of freedom of t he global space are given in t erms of t he t an genti al components on t he edges

Ae(q)

:=

I~I

J

q . tds ,

e

2.2 Vector Fi eld Discret izati ons

119

where e is an edge of the triangulation Ti, and th e directi on of t depends on t he dir ection of n. We now int roduce, for each face F E F I-I which is not an element of FI , a subspace of at most three degrees of freedom

NVI ;F := {q E N V (D;Ti)1 Ae(q) = O, e

et F}

see Fig. 2.14. Moreover , we have curlNV (D;Ti) C lIi. F

Fig. 2 .14. Different refinement techniques on F E FI- 1

We define our divergence free Raviar t- Thomas subspaces associated with t he faces in te rms of t he Nedelec spaces N V I;F by

Vi ;F := curl NVI ;F . It can be easily seen t hat Vi ;F is a subspace of Vi supp orte d in two elements of Ti- I ' An element VI E lIi is also in Vi if and only if VI . n do = 0, for F E F I-I . Let V = cu rl q with q E N V I;F, th en

IF

f

v . n do

=

f

q . t ds

= 0,

F E FI-I

of

F

The sur plus spaces Vi ;T , T E Ti-I \

Vi ;T := {v E

Ti , will be defined

Vii

locally by

supp veT} .

T he dimension of Vi ;]" is given by the number of faces F E FI which are in th e interior of T ; it is bounded by eight . We point out that Vi ;T can also contain a divergence free element. It has such a one dimensional divergence free subspace, in t he case t hat T E Ti- I is refined in such a way t hat it has an interior edge; see Fig. 2.15. We can now define our vertical decomp osition in te rms of t he local divergence free face spaces Vi ;F and t he element spaces Vi;T . We find t he following direct sum repr esent ation for Vi

lIi =

L

FEFl _l\F,

Vi ;F (fJ

L

Vi ;T '

(2.32)

TE 'Ti-l \ 'Ti

To define our quasi-projection operators, we have to specify a new bilinear form 0,(". ) on Vi x Vi , It is given by means of t he dir ect splitting (2.32) and t he origina l bilinear form a(' , .)

120

2. Iterative Solvers Based on Domain Decomposition

Fig. 2.15. Adaptive refinement such that ~;T contains a divergence free element

a(v[, v.) :=

L

aiv »; v»)

L

+

a(vT' VT) ,

TETi-l \ t.

PCF'-l \:F,

where according to (2.32) v: := 2:PE:F,_,\:F, Vp + 2:TETi_l\Ti VT· The following lemma shows the equivalence of the bilinear forms and a (', .) when restricted to Vz x Vz.

aC')

Lemma 2.12. There exist constants c and C such that

Proof. We start by proving the lower bound. It can be obtained easily by a coloring argument. We define vp := 2:PE:F'_l\:F, v tr, VT:= 2:TETi_l\Ti VT, and find

a(v[,vz) = a(vp+VT,VP+VT) ::; ~a(vF,vF)+5a(vT,vT)

::;5(

2:

FE:F'_l\:F,

a(vp,vp)

+

2:

TETi-l\Ti

a(vT,vT)) =

5a(v[,vz)

The proof of the upper bound relies on the norm equivalence (2.10). Let v r be an element in Vi;p, then a(vp, v r )1/2 is equivalent to its L 2-norm : 2:

FE:F'_l \:F,

a(vp, vr-)

0 since Gil and Al are symmet ric and positive definit e. Let us first assume XR =J O. Then since Gi l Al is a lower blo ck trian gular matrix , A is an eigenvalue of Gil Al with the eigenvect or Xl only if XR is an eigenvect or of the lower dim ensional problem

By usin g t hat x~GRxR get

2: x~ ARRx R follows from (2.39) and

A NN > 0, we

and t hus A < 1. Let us n-;w consider t he second case x T = (0, x];;. ), i.e., XR = O. Then , t he eigenvalue pr oblem Gi l AI XI = A Xl redu ces t o t he following eigenvalue problem on a smaller space

To prove A ~ 1, we use t hat Al is posit ive definit e. We set wT ~ 1 T ~ 1 T T~ ((AFlRMYN) , ( - A ]:m YN) ) , then w I AI WI 2: 0 yields T

T

~

l

T~l

yNM AFlRMYN ~ yNA]:mYN .

Using t he ass umpt ion (2.39) on G R and the definition of a, a straightforward calculation shows T

~-1

1

T

T

-1

1

T

T

~-1

T

A yNANNYN = 2YN M G R M YN + aYNYN ~ 2YNM A RRMYN

1

T

~-1

+ 2YNANNYN
is Young's modulus and v E (0,1/2) is the Poisson ratio . Then , t he components of t he st ress tensor can be writ ten in terms of E , v and the infinit esimal st rain € te nsor as

Using the Lam e constants f-l and A, th e st ress tensor satisfies u(u) = 2 f-l €(u)

+ A tr(€(u))

0 ,

where the trace of t he strain tensor is given by tr(€(u)) := Ekk (U), and the Lam e constants ar e defined by

E

f-l := 2(1 + v) '

A:=

E v (1 + v)(l - 2v)

Here, in an abuse of not ati on we use the same symbol for one of the Lame constants as for the Lagr ange multiplier in the mortar setting. In the following, we use Poisson's ratio and Youn g's modulus to specify a material.

144

2. It er ative Solvers Based on Domain Decomposition

T he boundar y aD = TD U TF is decompos ed into two complementary parts, a Dirichlet par t T D with non zero measure and a Neumann part TN , TN n TD = 0. On TD t he displacement is set to zero and on TN the sur face t raction is given. Then, t he space of admissible displacement s H;(D) is a subspace of H 1(D) := (H1(D))d and is defined by

H ;(D) := {v E H 1(D) I vl r D = o} St arting with (2.45) , int egration by part s yields t he weak formulation. Let u E H; (D) be the solut ion of the following vari ational pr oblem

a(u , v) = f(v) , where f(v) := (v ,f)o;!t by

a(w , v)

:=

+ (v ,p)o;rF'

J

v E H;(D) ,

(2.46)

and the bilinear form a(·,·) is defined

E i jl m W i ,j Vl ,m

dx ,

W,

v E H 1(D) .

a

Associated with a(' , ') is t he energy norm I I · I I , IIIvll1 2 := a(v , v). Appl ying the Lemm a of Lax-Milgram , the well-posedness of (2.46) follows from the cont inuity of the bilinear form a(·, ·) and the linear form fO and the second Korn inequality ; see, e.g., [Bra97, BS94]. We refer to [Cia88, Gur81 , MH94, K0 88] for a general introdu ction to continuum mechanics and elasticity. The decomposition of t he domain into subdomains is chosen according to t he different materials or bodies. We consider two different sit ua tions. In t he first sit uation, we assume that th e different materials are glued to gether such t hat t he jump of the displacements in t angenti al and normal dir ection vani shes. In Sub subsect . 2.3.6.1, t he question of uniform ellipticity is add ressed, and in Sub sub sect. 2.3.6.2, num erical result s illustrate t he deformation of a body. We solve a simplified lineari zed form of a contact problem in Sub sub sect. 2.3.6.3. In t hat sit uation, the weak solution of t he conti nuous vari ati onal problem does not have to be in H; (D) . At th e int erface, arbit ra ry displacement s in tangential dir ection are admissible, bu t t he jumps of the displacement s in norm al direction have t o be zero across the int erface, i.e., [u-.n] =0. For th e discreti zation , we use t he same finite element spaces of ord er one as before, see (1.7) . The unconstrained vector valued finite element space Xh is defined as pro du ct space X h := (Xh )d. In the case of the coupling in t ang enti al and norm al direction , t he Lagrange multiplier space M j, := (Mh) d is also vector valued . Then , t he saddle point formul ati on of the mortar method can be defined in te rms of th e bilinear form b(·, .) M

b(v, p,) =

L ([vd, m =l

ft i) , = ,

v E X j, , P, E Mj, .

2.3 A Multigrid Method for the Mortar Product Space Formulation

a(Uh , v)

+ b(v , Ah) =

b(Uh ' JL)

f(v) ,

v E Xh , JL E M h .

= 0,

145

(2.47)

Here, the bilinear form a(·, ·) is extended to th e nonconforming space X h by replacin g the int egral over il by it s broken form ~~=l Jfl k' The second equation of t he saddl e point problem gua ra ntees the weak continuity of t he solution us , In ana logy to Sect . 1.1, the nonconforming space V h is defined as the kern el of the operator B T : Xh - 7 Mj, associate d with th e bilinear form bt-, ') ,

and we find V h = (Vh)d . Under the assumption t hat a(',') is uniformly elliptic on V h x V h , i.e., K

«(v .vi

> cllvlli := Lllvlli;flk'

VEVh ,

k=l

th e following variational pro blem has a unique solution: Find Uh E V h such t hat

a(Uh 'v)

= f(v) ,

v E

Vh .

(2.48)

In the next subsubsection, we address th e question of ellipticity. A uniform discrete inf-sup condition yields in combina tion with the ellipt icity of a(' , ') on the kernel of the operator B T th e uniqu e solvability of (2.47) . We refer to Sub sect . 1.2.3 for t he pro of of the inf-sup condition in the scalar case. Since Mj, and X h are product spaces , the inf-sup condition follows from the scalar case. Moreover , the positive definite syst em (2.48) is equivalent to the saddle poin t problem (2.47) . 2.3.6 .1 Uniform Ellipticity. Here, we consider the uniform ellipt icity of th e bilinear form a(·, .) on the constrained spa ce V h X V h . We start with the special case that Bilk n TD has a non zero measur e for ali I :::; k :::; K . Then , Korn's inequa lit y can be applied to each subdomain , and we find

a(v, v )

=

K

L ak(V,V) ;::: k=l

K

CLllvlli;flk = Cllvlli , k=l

where akh ·) stands for th e restriction of a(' , ') to t he subdomain ilk; see, e.g., [Bra97, BS94]. Thus a(·, ·) is elliptic on H ;(il) x H ;(il). We remark that C does not depend on the numb er of subdomains. Unfortunately, many interesting cases do not satisfy this assumpt ion . However for the uniqu e solvability of (2.48), it is sufficient to have the uniform ellipticity of a(·,·) on v ; x v».

146

2. Iterative Solvers Ba sed on Domain Decomposition

In t he scalar ellipt ic case, the kernel of th e corr esponding bilinear form is th e subspa ce of piecewise constant functions. It s dimension is given by the number of subdomains fh such that a[h n To is empty. In th e elasticity setting, the kern el is of higher dimension . The rigid body motions per sub domain define a three dimension al space in 2D and a six dimension al space in 3D. To get a bet ter feeling for the kernel of a(·, '), we consider the case of two unit squa res nl and nz in 2D with homogeneous Dirichlet boundar y condition on one side of anI nan and homogeneous Neumann boundary condition elsewhere. We set vl n! := and vl», := (3 (y - Ye, Xc - x)T, (3 :f. 0, where (x c, Ye)T denotes th e center of gravity of 'Y := anI n anz . Then, v E (Y) Z but a(v , v) = 0, and thus a(· ,·) is not ellipt ic on (Y) Z x (Y) Z, where the nonconforming space Y is defined by (1.6) . Moreover to obtain ellipt icity on V h X V h, the dimension of the Lagr ang e multiplier space has to be lar ger t ha n d. In t he following, we assume th at N m :::: 2 in 2D, and that in 3D the t riangulation at the int erface is a tensor product mesh with N m :::: 4. We recall that N m is t he dimension of Mh b m). Based on this observation, we define the nonconforming space

°

V H :=

{VEH~(n)1

![V] 'J.LdS=O 'J.LEMH b m) ,I"'5.m "'5.M} ,

,=

where MH b m) := (MH b m))d is a suitable te st space. If M Hb m) C Mh b m) t hen V h C V H, and to obtain t he uniform ellipt icity on V h X V h it is sufficient to show the ellipt icity on V H xV H. A natural choice for MH b m) is Pl b m) . Unfor tunat ely, none of t he considere d Lagrange multiplier spaces satisfy Pr b m) C Mh b m). We start with t he const ruction of a new macro Lagrange multiplier space MH b m) C M hb m) in 2D. Let t E [0,1] be a par ametrization of t he ID int erface "[m» i.e., x E ;:Ym if and only if x = Xo + t x(XN=+l - xo ), t x E [0,1]' where Xo and XN= +l are the two endpoints of "[ra- Then , we decompose t he set of int erior vertices {Xi 11 "'5. i "'5. N m} int o two disjoint subsets . Introducing "'m:= max{ I "'5. i "'5. N ml t X i "'5. 0.5}, we define a test functi on PH by

where 'ljJi is eit her t he nod al st andard basis function 'ljJt or the du al one 'IjJ[; see Subsubsect . 1.2.4.1. Figur e 2.28 shows the shap e of the test function PH. In t he left , PH is given for t he st andard Lagran ge multiplier space and in t he right for the du al Lagrange multiplier space . It is easy to see t hat t he mean value of PH is equa l zero for both choices. Now, we define in t erms of PH the macro space MH bm ) := span{ipH E P H} , where PH := {I , PH} in t he 2D case. In 3D, we work wit h t he simplified dua l basis fun ctions given in Sub subsect. 1.2.4.2. We use the tensor product

2.3 A Mult igrid Method for the Mort ar Product Space Formulation

147

st ruct ure to define PH := {l,tLk-,tL1I ,tLk-tL1I} , where tLk- and tL1I are th e one dimensional t est functi ons with respect to the two different dir ections. Thus for the 2D and 3D case , we have M Hbm) C M hbm) .

5/4

5/4

~H

\I,~ o

0.5

1/3

o

1

5/9

_

·1

_

----"'-,· 1

Fig. 2.28. Test fun cti on J..tH for standar d (left) and du al (right) Lagrang e multiplier

Lemma 2.30. Let v E V H body m otion , th en v = O.

an d v restric ted on

,

fh , 1 :s;

k

:s;

K , be a rigid

Proof. We start with a subdomain rho such that afh o n r D has a non zero measur e. Due to t he Dirichlet boundary condition, v restricted to t his subdomain is zero. On each adjace nt subdomain nl , we can write vl n, al + W I , where a l E ]Rd and

WI

y - Yc ) = (XI ( Xc - X

,

in 2D and 3D, respectively. The center of gravity of I'mo := anko n ani is denot ed by (xc, Yc) in 2D and by (xc,Yc, zc) in 3D. Then , the int erface condition yields that for all adjace nt subdomains, we find in 2D

J

(al

+ W I)

. ei

da = 0,

1 :S; i

:s; d

I'~o

where e, denot es the i-t h unit vector, and t hus al = O. Introducing in 2D VII E P1b mo) , VII( XO) = -VH(XN~+d = 1, see Fig. 2.28, a st raightforward computation shows t hat

J

J..tH vHdO'

I'~o

= a u bmo l

148

2. It erative Solver s Based on Domain Decomposition

We find the lower bound aH 2: 1/3 if Mh(')'mo) is the piecewise linear dual Lagr ang e multiplier space as discussed in Subsubsect . 1.2.4.1 , and an 2: 2/9 if Mh (')'mo) is the st andard Lagrange multiplier space and thus WI = O. In 3D , WI = 0 follows from the 2D situation and the tensor product st ruct ur e. We not e that the lower bound for an does not depend on the mesh on "'(mo' St arting from rh o, we can move to each other subdomain by crossing interfaces. 0 The following lemm a yields the uniform ellipt icity of a(' , .) on V h X V h . Lemma 2.31. Th e bilin ear form a(· , ·) is uniformly elliptic on V H xV H. Proof. The proof follows from th e definition of V H and th e following upp er bound for t he broken HI-norm M

Ilvll i :S C1a(v, v)

+ C2 L

L

J

'PII [V] dO'1I 2,

II

E

V

m=l

l .

(2.55)

An int eresting alte rnative formulation can be obtained by eliminating t he int erior variables. Introducing a sca led Lagran ge mul tiplier '\h defined on t he non-m or t ar side, we can work on t he interface. We recall that the mortar side is defined on il2 and t he non-mort ar side on ill . For simplicity, we restrict ourse lves to t he special case WD = 1. Now, we define our pr econdi t ioned Richardson method for '\h as follows: (2.56) wher e the Schur complement S, := A~r - A~I(A}I )- 1A }r , i = 1,2, is t he discret e St eklov-Poincar e operato r on ili , and q := 5 11ql - M T 5:;1 q2 with qi := f} - A~I(A}I)- l f} . We not e that the discr et e St eklov-Poincare operato rs are symmet ric and positive definit e. In the conforming set t ing, the convergence rate of the Richard son it eration (2.56) depend s only on the cont inuity and coerciveness constants of 51 and 52; see [QV97]. Here, the norm of M which do es not dep end on the meshsize also enters. Und er the assumpt ion t hat t he damping param eter WN is small enough, t he convergence rate of t he preco ndit ione d Richardson it eration (2.56) is boun ded ind ep end ently of t he meshsize, i.e.,

158

2. Iterative Solvers Based on Domain Decomposition

II'\h- '\hll :::; pnll'\~ - '\hll with 0:::; p < 1 independent ofthe meshsize and '\ h := (51 1 +M T S 21 M) -l q. The following lemm a shows th e relation between th e iterates u 2 and '\h'

ur,

Lemma 2.34. Un der the assumpti ons th at A~ = WN( O Id)(1I - A }yu~) and A rvu g = Jz , th e N eumann-Dirichlet algorithm with WD = 1 and the precond ition ed R ichardson ite rati on (2.56) are equivalent. Th e fini te elem ent it erat es can be obtain ed from '\h by

U2 =

(A rv)-I( Jz

ur = (Ab)- I(Jl

+ S T(O (,\~ -I) T) T) + Su 2)

n >l

Moreover, th e scaled Lagrange multiplier satisfie s

'\h =

(1 - WN ),\~-1

= WN(O Id)

n

+ WN(O Id)(1I

I: (1 -

- A }yul)

WN )n-I (II - A }yuD '

(2.57)

1= 0

The assertio ns can be obtained by indu ction and are based on (2.55) , (2.56) and t he definition of t he Schur complement s. Compa ring (2.57) with (1.40) , we find that '\h is the scaled Lagran ge multiplier of the mortar set ting, i.e., '\h = D Ah where D is a diagonal matrix, th e ent ries of which are proportional to the local meshsize. The assumption Arvug = Jz in Lemm a 2.34 can be weakened . For t he equivalence, it s is sufficient if there exists a Yl such t ha t A rvug = h + S T us. A three term recur sion for '\h can be obtained for more general damping par ameters. Remark 2.35. Th is type of algorith m can be us ed to construct an efficient itera tive solver fo r nonlinear problem s. It can be applied to solve multi body cont act problems. Th en in each ite rati on step, a lin ear N eumann problem and a S igno rini problem have to be solved. Th e N eumann data on the mortar side are obtain ed in terms of ST an d the Lagrang e multiplier on th e non-mortar side. Applying S to th e Dirichlet values on th e mo rta r side gives th e onesided obstacl e fo r th e Signorin i problem on the non-morta r side. W e refer to [KWOOc] for th e n onlinear Dirichlet-Neumann algorithm an d numerical resu lts illus tra ti ng th e deformation of the bodies and th e bounda ry stresses at th e contact zone . An extension of th e algorithm to 3D an d Coulomb fr iction can be f ound in [KW01]. Th e nonlinear Signorini problem can be solved effic iently by m on otone m ultigrid methods; see [Kor91, KK99, KKOO, Kratll], 2.4.2 Numerical R esults

In t his subs ection , we pr esent num erical results for the pr oposed NeumannDirichlet algorit hm. We consider t he ext remely simple case of two unit squares Di , i = 1, 2, and -L\u = f on D, D := D 1 U D 2 . The right hand side f is

2.4 A Dirichlet-Neumann Type Method

159

chosen to be const ant on each subdomain, fi n! = -1 and fl n2 = 1. Inhomogeneous Dirichlet boundary condit ions ar e imposed on the sides of the unit squar es parallel to the int erface, and homogeneous Neumann boundary condition ar e applied elsewhere . We work with non-matching triangulations at the int erface and use uniform refinement t echniques. The coarse triangulation has 4 element s on the non-mortar sub domain D 1 and 9 element s on the mor tar subdomain D2 . In the Neumann-Dirichlet algorit hm, the start it erate u~ is given by a random vector on each level, the valu es of the component s are cont ained in [-5,5] . Furthermore, we set pI = ST(J1 - AJvu~) which corresponds t o the choice ug = (A'7v)-l(h + ST(J1 - AJvu~)). Replacing t he upper block triangl e matrix in (2.54) by the lower block triangle matrix yields a Diri chlet -Neum ann algorit hm . In t hat case, we have to initialize gl. We define l = Sug , where ug is a random vector. This choice corresponds t o u~ = (Ab)-1(j1 + Sug). In the Dirichlet -Neum ann algorit hm, we start with a Dirichlet problem on t he non-mortar subdomain D1 followed by a Neum ann problem on the mortar subdomain D 2 . We compare the proposed Neum ann-Dirichlet algorit hm, WN E {0.7,0.75,0.8,0.85,0.9} and WD = 1, with the Dirichlet-Neum ann algorit hm , W D E {O. 7,0.75,0.8,0.85, 0.9} and W N = 1. In Tabl es 2.5 and 2.6, the number of required it er ation st eps to obtain an error redu ction of 10- 6 is given for different damping par am et ers. The numbers in parenthesis show the numeric al results for a different coarse trian gulation. In that case , we st art with 25 element s on D1 and 4 element s on D 2 . Table 2.5. Numbe r of it eration ste ps, (t race norm) level 1 2 3 4 5 6 7

Dirichlet-Neumann WD 0.85 0.75 0.8 7 (9 6 7 6 6 10 11 7 9 5 8 6 6 10 7 9 5 7 6 6 10 7 9 6 7 6 6 11 7 9 5 8 6 6 10) 7 9 6 8 6 6 10 7 9 6 7 6 6

0.7 9 9 9 9 9 9 9

0.9 8 6 8 7 8 7 8 7 8 7 8 7 8 7

Neumann-Dirichlet W N 0.7 0.75 0.8 0.85 9 10) 7 8 6 7 6 5 9 9 7 (8 5 7 6 6 7 8 6 7 5 6 9 9 9 9 7 8 6 7 5 6 9 10 ) 7 8 6 7 5 6 9 (9 7 8 6 7 6 5 9 9 7 8 6 7 5 6)

0.9 7 7) 8 7) 7 7 7 7 7 7 7 7) 7 7)

In Tabl e 2.5, t he error is measur ed in a weighted L 2-norm for the trace. The weighting factor is given by 1/h2 where h2 is the local meshsiz e on the mortar side, i.e., (2.58) Her e, Sh2 is the set of edges at the interface on the mor t ar side , and U 2 is t he exac t discrete solution on the mortar side. We rem ark that this weighted

160

2. It erative Solvers Based on Domain Decomposition

L 2 -nor m is equivalent to the Euclidean vector norm of t he interface nod es on t he mor tar side.

U2 -

u~ restricted to

Table 2.6. Number of it eration steps, (Lagr ange multiplier norm)

level 1 2 3 4 5

6 7

0.7 9 11 9 11 8 10 8 11 8 11 8 (11 8 11

Dirichlet-Neumann WD 0.8 0.75 0.85 7 9 5 8 6 6 7 9 5 8 6 7 7 9 5 7 6 6 7 9 5 8 6 6 7 (9 5 8 6 6 7 (9) 5 (8) 6 (6) 7 (9 5 8 6 (6

0.9 8 6 8 6 8 6 8 6 8 6 8 6 8 6

Neumann-Dirichlet WN 0.7 0.75 0.8 0.85 9 10 7 9 5 7 6 6 9 10 7 9 5 8 6 6 9 11 7 9 6 8 6 6 9 10 7 9 5 8 6 6 9 11 7 9 5 8 6 6 9 (10) 7 (9) 5 (8) 6 (6) 9 (11 7 8 5 8 6 (6)

0.9 8 7 8 7 8 7 8 7 8 7 8 6 8 7

In Table 2.6, th e err or in the Lagrange multiplier defines th e stopping crite ria for t he iterat ive solver. It is measur ed in a weight ed L 2 -nor m . The weighting factor is given by the local meshsize on t he non-mortar side. We note t hat the Lagrange multiplier is an approximation of th e flux. It can be obtained from t he residuum of a Neum ann problem on the non-mortar side by a diagonal scaling, i.e., Ah = D-I(OId)(h - Atvud, where UI is the exac t discret e soluti on on th e non-mort ar sub domain D I. Moreover if we set A~ = D-I(OId)(h - Atvul) , we have the following norm equivalence C

2

II(Atv(UI - ur ))r I1

::;

L

hellAh - A~II~; e ::; C II (Atv(UI - ur))rI1

2

e ESh ,

Here, Sh, is the set of edges at the int erface on th e non-mortar side . This equivalence motivates t he stopping crite ria (2.59) We note th at th e choice of our st ar t vectors gua ra ntee t hat Atv(UI - ur) = 0 for all interior nod es on DI , i.e., II(Atv(UI - ur))rll = IIAtv(UI - ur)ll · The number of requir ed iteration st eps is independent of the refinement level for all damping parameters . But it depends highly on th e damping pa ra mete r, see also Figs. 2.37 and 2.38. Both algorit hms, Dirichlet-Neum ann and Neum ann-Dirichlet , require approximate ly the same number of iteration ste ps to obtain the requir ed error reducti on. The choice of th e norm for th e error plays only a minor role, and almost the same results are obtained for th e stopping crite rias (2.58) and (2.59). Figur es 2.37 and 2.38 show the computed errors and t he best least-square fit for different damping par ameters on level 5 and level 7. The compute d err ors ar e marked by different symbols whereas the corresponding best fit is given by a solid line. In Fig . 2.37, the error reductio n is measure d in the

2.4 A Dirichlet -Neumann Typ e Method

161

• damp ing: 0.7

• damping:0.75 e damping: 0.8

D

o

da mping: 0.85 da m in : 0.9 2

4

6

8

Number of iterations, level 7

10

Fig. 2.37. Error reduct ion (trace norm) on level 5 (left) and level 7 (righ t) for the Diri chlet -Neumann algorit hm

sp ecified £2- norm for t he trace on the mortar side . The results for the err or in t he Lagrange multiplier on the non-mortar side are given in Fig. 2.38.

Fig. 2.38. Error reduct ion (Lagrange multiplier norm) on level 5 (left) and level 7 (right) for t he Diri chlet-Neumann algorit hm

Finally, Tabl es 2.7 and 2.8 give t he asympt ot ic convergence rates. The lines of the two tables corres pond t o the levels 1-7. To define the convergence rate q, we use a first order logari thmic polynomial least- squ ar e fit based on 10 it er ation ste ps. Then , q := lOa where 10

10

2)ek-ak-b) 2= min 2:)ek- ak- (J) 2 , k =l

a,!3 EIRk = l

and ek is the logarithm of the it er at ion err or in one of the sp ecified norms. Tabl e 2.7 shows the convergence rates for different damping factors in the t race norm on the mortar side, and Tabl e 2.8 gives the corresponding results for t he Lagrange multiplier norm on the non-mortar side . The convergence rates dep end highly on t he choice of t he damping par am et er. In pr actic e, t he choice of a good damping par am et er is a delicat e point. T he influence of the damping param et er can be considerably weakened by using the block GauBSeidel method as a pr econditioner for a Krylov subspace method. Another

162

2. Iterative Solvers Based on Dom ain Decomp osit ion

possibility might be to use ada ptive st rategies for the damping par am et er. We not e t hat in general Ul and U2 are unknown. Then , u 2 - u~- l an d uf - U ?-l can be used t o define a stopping criteria. T able 2.7. Asym pt ot ic convergence rat es q , (trace) 0.7 0.151 0.142 0.145 0.147 0.146 0.149 0.148

Dir ichlet-Neumann WD 0.75 0.8 0.85 0.094 0.038 0.052 0.084 0.034 0.048 0.088 0.036 0.051 0.091 0.037 0.053 0.089 0.036 0.053 0.092 0.037 0.051 0.091 0.037 0.052

0.9 0.109 0.108 0.109 0.113 0.113 0.109 0.111

0.7 0.160 0.150 0.159 0.160 0.160 0.158 0.160

Neumann- Dirichlet WN 0.75 0.8 0.85 0.100 0.040 0.046 0.089 0.028 0.050 0.099 0.039 0.042 0.100 0.040 0.044 0.100 0.040 0.040 0.099 0.039 0.047 0.100 0.040 0.042

0.9 0.101 0.107 0.087 0.090 0.084 0.100 0.088

Table 2.8. Asymptot ic convergence rat es q, (Lagra nge multiplier norm) 0.7 0.139 0.140 0.139 0.138 0.138 0.138 0.138

Dirichlet- Neumann WD 0.8 0.85 0.75 0.081 0.035 0.049 0.079 0.026 0.046 0.078 0.027 0.050 0.078 0.028 0.051 0.078 0.028 0.053 0.078 0.027 0.051 0.078 0.028 0.052

0.9 0.109 0.106 0.110 0.111 0.113 0.111 0.112

0.7 0.146 0.142 0.154 0.141 0.145 0.140 0.142

Neumann-Dirichlet WN 0.75 0.8 0.85 0.088 0.036 0.045 0.083 0.027 0.050 0.095 0.037 0.050 0.085 0.035 0.051 0.089 0.036 0.053 0.081 0.032 0.050 0.085 0.034 0.052

0.9 0.105 0.110 0.106 0.112 0.113 0.111 0.111

R emark 2.36. We rema rk that the application of the operators S an d S T

is extrem ely cheap. Th is is due to the use of dual basis fun ctions for the Lagrange m ultiplier.

2.5 A Multigrid Method for the Mortar Saddle Point Formulation In Sect . 2.3 and in Sect. 2.4, two different it erati ve solvers for mortar discret ization t echniques have been discussed . We have considere d the case of du al Lagrange multipliers. T he t heoret ical results also hold if standa rd Lagrange mu ltipliers are used . But in t hat case , the computational cost for one it erati on step is considerably higher. This is due to t he fact , t hat t he scaling of t he mass matrix on t he mor t ar side involves the inverse of a mass matrix on t he non-mortar side. In this section, we analyze a multigrid method based on t he mortar saddle point formulation wit h standard Lagran ge mul t iplier

2.5 A Mult igrid Method for the Mortar Saddle Point Formulati on

163

spaces . In cont ras t to t he two previous secti ons, th e scaled mass matrix does not ente r in t he computation. There are different approaches to th e efficient solution of t he indefinite saddle point problem (1.9). One possibility is to use a good preconditioner for the exac t Schur complement as ana lyzed in [Kuz95a, Kuz95b] and further exploite d in [EHI+98 , EHI+OO, HIK+98]. A different technique is based on an idea of Br aess and Sar azin which is present ed for the Stokes problem in [BS97] . It has been successfully adapte d to mortar situations in [BD98, BDW99, WW98]. In t ha t approac h, a modified Schur complement syste m has to be solved exac t ly in each smoot hing step . More recently, a simplified version of this idea has been studied in [WW99]. The modification is based on ideas introduced in [ZulOl]' and a more genera l approximation property ana lyzed in [WohOOb] . Here, we st udy a multigrid meth od for t he saddle point problem (1.9) . T wo different ty pes of smoot hers are discussed ; a block diagonal one and one reflecting the sad dle point st ruc t ure . There is no need to solve a Schur complement system exact ly. The first smoot her works on the squa red syste m. In t he second case , t he exac t solution of the modified Schur complement syste m is replaced by an iteration, resulting in an inner and outer iterati on cycle. This multi grid method is given for th e standa rd mortar formulation as pr esent ed in Sect. 1.1. In cont rast to [BD98, BDW99 , WW98], we are not workin g in the subspace on which t he saddle point problem is positive definite; th e iterates do not have to satisfy the const ra ints at the int erfaces, and we do not have to solve a Schur complement system exactly . To obtain convergence results, it is therefore necessar y to establish an appro priate approximation property for both vari ables Uh and A h . This issue was originally addressed in [WohOOb] ; a weaker approximation property has also been studied in [BD98, BDW 99]. For bot h smoot hers, we obtain level independent convergence rates for the W- cycle provided t hat the numb er of smoothing ste ps is large enough. We assume full H 2 -regularity. In this section, the multigrid analysis is carr ied out in t he operator setting. We work with t he standa rd Lagrange multiplier spaces M l , 0 ~ l ~ i . as defined by (1.2) , and t he unconstrained product spaces Xl which are asso ciated with a nested famil y of quasi-uniform trian gulations Ti, hi = 2h1H . The t riangulations on the non-mort ar sides are called S m ;l ' We then find for the finit e element spaces t hat

The spaces X; x M; are Hilbert spaces equipped with the mesh dependent bilinear forms

((v, f.1 ), (w, V)) hl ;S? XS := (v,w)o + (f.1 , v) _ ~ , hi

;S

(v, f.1) , (w, v) E X;

X

M, ,

where t he mesh dependent bilinear form (' , ·)h- 3/ 2.S on M, x M, is given by I

'

164

2. Iter ative Solvers Based on Domain Decomposit ion

The corres ponding norm s are denoted by 11 ·llhl;!lXS and 11 ·lIh-s /2.s. 1 , Before focusing on t he approx imation property, we consider suit able t ransfer operators. Let ILl : X l- l --t X, and JL l : M l- l --t M, be th e natural injecti ons. We t hen define If-l and Jf- l by Wl- l E Xl- I , Vl- l E M l- l

Since th e spaces are nest ed , we have Wl- l = IL l Wl - l, Wl-l E X l- I , and Vl-l = JL lvl- l , Vl-l E Mi.:«. A st ra ightforward calculat ion now shows that if (713[ , Vl ) E X, X M, satisfies the saddle point probl em a(u5[ ,Vl)

+ b(Vl' Vl) =

= (~ ,J-Ll)

b(u5[ ,J-Ll)

and if (Wl- l ,Vl - l ) E X l- l a(uh- l,Vl - d

X

a(Wl' vd b(un, J-LI)

_.3.

h i 2 ;S

,

J-LlEMl ,

M l- l satisfies

+ b(Vl- l ,Vl-l)

b({ih- l ,J-Ll-l)

Vl EXl ,

(d; ,Vl )O,

= (If- ld; ,Vl- l) O, 1 = (Jf- 8l ,J-Ll-l) _ .3. , hl_; ;S

+ b(vl ' Vl ) =

_ .3.

hi 2 ;S

J-Ll-l E Ml- l ,

VI E XI ,

(dl ,VI)O,

= (01 , J-Ll)

Vl-l E Xl- I ,

,

J-LlEMI ,

with ((dl , Ol) , (VI- I , J-Ll-d)hl;!l XS = 0, for all (VI- I , J-Ll-d E X l- l x M I - l . After t hese prelimin ar y remarks, we will establish a genera l approximat ion property and introduce two different kind s of smoot hing operators. 2.5.1 An Approximation Property

The following lemm a can be found in an abst rac t form , as well as in th e special case of mixed finit e element approximations of the St oke's equation, in [Ver84 , Lemm a 4.2]. Here, we ada pt it to mortar finite elements . It shows t hat t he coarse grid correction in the multigrid framework yields a good approximation of the solut ion of the defect equation on t he fine level. Lemma 2.37. (Approximation property) Let (d l , 01) E X, x M I be orthogo nal to X l - l x M I -

l ,

i .e.,

2.5 A Multigrid Method for the Mortar Saddle Point Formulation

165

and let (WI , VI ) E XI x MI be the solution of: Find (WI , VI) E XI x M I such that

(2.60) Then, there exists a constant C satisf ying

Proof. The qu asi-uniform ity of t he triangulation s yields 3

c (1lwtllo + h? Ilvtll o;s)

~

lI(wI , vI)llh /;S2 XS

~

3

C (1lwtllo + h? IIvtllo;s)

The estimate for the Lagran ge multipli er term will be based on the discrete inf-sup condition (1.10) whereas the bound for Ilwtllo is based on du ality techniqu es. We start with an estimate for the upp er bound of h; /21Ivtllo ;s . The continuity of a(·, '), th e discrete inf-sup condit ion (1.10) , and the orthogonality of dl on XI-I , i.e., (dl , VI- d o = 0, VI- l E X I- I , yield an upp er bound for Ilvtll h;-1/2;S in terms of Ilwllh and IIdtllo: (dl ' VI -

VI- l)OIlvtlll

a(wI ,VI )

(2.61) The second equation in (2.60) yields b(wI ,J.LI-d = 0, JLI-l E M I- 1 , and thus WI-l E Y . Recallin g t ha t the bilinear form a(· , ·) is ellipt ic on Y x Y , see (1.5) , and using (2.61) , we find for WI-l E XI-l C

Ilwtlli

WI-l )O- (151, VI)h

~ a(wI ,WI) = (dl ,WI )O- b(wI , VI) = (dl ,WI -

~

3

Chi

7

(lIdtilo + h? IIJtllo;s) Ilwtlh + Chl 1ldtllollJtllo;s . 2

To obtain an upp er bound for IIwtlh in terms of sufficient to consider t he quadrati c polynomial g(8) :=

8

2

-

a(al

+ a2)8 -

h;/21IJtllo;s

aa la2,

8

and

_3 I

'1·s '

Ildtl lo, it

is

E IR ,

where al := h~/21I Jtllo;s 2: 0, a2 := htlldtllo 2: 0, and a is a positive constant . In the case that al = a2 = 0, t he uniqu e solvability of (2.60) gives Ilwtlh = O. For al + a2 > 0, an easy calculatio n shows that g(8) > 0 for 82: (1 + a )(a l + a2)' Thus, t he following upp er bound for Ilwtlll holds in t erms of IIJtllo;s and

Ildlllo:

166

2. Iterative Solvers Based on Dom ain Decomposition

(2.62) Combining (2.62) with the upp er bound (2.61) for

Ilvtllh-l/2.S' we find I

'

(2.63) In our next step, we focus on an estimate for Ilwtllo. To obtain an upp er bound for Ilwtllo, we use Aubin - Nit sche typ e arguments. Let w E H6(D) be the solution of the cont inuous vari ational problem: Find w E HJ (D) such t ha t a(w , v) = (WI , v)o, v E HJ(D) . Taking into account t ha t, in genera l, WI E XI is not contained in get IIwtlI6= a(w ,WI ) + b(WI , v) ,

HJ (D) , we

where v .- a'Vw . n is t he flux of w across the interfaces. Using t ha t a(wI' VI-I ) + b(VI_l , VI) = 0 , VI- l E XI- I , b(WI , /-LI-l) = 0 , /-LI-l E MI-l , and observing t ha t b(w , VI) = 0, VI E M I , we find for VI- l E X I- l and /-LI-l E M I - l th at

IIwtlI6 = a(w - VI- I, WI) + b(WI , v- /-LI-d + b(w - VI-I,VI) :::; C(11w - vl- l lll Ilwtll l + II[wdllo;s Ilv - /-LI- lllo;s + It[w- vl- dllo;s Ilvtllo;s) . Here, we have used t ha t th e spaces are nested . We choose VI- l E X I -

local quasi-projection of w such that

111V- vi-l ih :::; Chl- l llwl12 , II[w- vI-l]llo;s :::; Ch?_11IwI12 111-1 E M I -

l

l

as a

(2.65)

3

see, e.g., [SZ90] , and

(2.64)

satisfies 1

1

llv - /-LI-lllo;s :::; Ch?_lllvil k;s :::; Ch?-111w112

(2.66)

Remark 1.13 shows t hat M I - l is also a suit able Lagrange multipli er space for X I. Thus, we can apply Lemm a 1.7, for ~ := { v E X I , b(v ,/-L) = 0, /-L E MI-d , and obtain 1

II[wdllo;s :::; Ch?llwtlll . In a last ste p , we use t he H2-regularity, IIwl12:::; Cllwtll, and a t race theorem. Combing t he upp er bounds (2.64)-(2.66) yields

IIwtlI6:::; Chi (1lwtlh + Ilvtllh;k;s) Ilwtllo , which, t ogether with (2.62) and (2.63) , proves th e assertion.

0

2.5 A Multigrid Method for the Mortar Saddle Po int Formulat ion

167

Before introducing our smoot hing op erators, we consider t he operator K, associated with the saddle point problem (2.60) on levell: Let Al : X, ---7 Xi , Bi : M, ---7 Xl , B] : X, ---7 M, be t he operators defined by (A1Vl ,Wl)O := a(vl ,w l) , (Bl/..Ll ,Wl)O:= b(Wl ,f.Ll) , (Bt wl ,f.Ll) _ ~ h/

;S

b(Wl ,f.Ll )

:=

Then , th e self-adjoint non-singular operator K, : X, X M, associate d with t he saddle point problem (2.60) , is given by Kl(Vl ,f.Ll) := (A1 VI

+ Blf.Ll ,Btvl) ,

(Vl ,f.Ll) E Xi

X

---7

Mi

X,

X

Ml ,

(2.67)

The solution (WI , VI) of t he saddle point problem (2.60) satisfies K; (WI , VI) = (dl ,81) ,

and t hus IIK1- 1(dl ,81)llh/ ;!l XS ::; Ch rll(dl ,81)llh,;!l XS for those (dl ,81) E X, X M, which are orthogonal onto X l - 1 x M l - 1 with respect to (., ·)h/;!l xS. Equivalently, we find (2.68) for (WI , VI) E X, x M, satisfying (Kl(Wl ,Vl),(Vl- l ,f.Ll-l))h/ ;!l XS = 0, for all (Vl - l ,f.Ll-d E X l- 1 x M l- 1 . 2.5 .2 Smoothing and Stability Properties

The second basic tool to establish convergence within t he multi grid framework is the smoothing property. In t his subsection, we introduce two typ es of smoot hers. We show that t hey are st able and sati sfy a suitable smoothing property. Before we give the definition of the smoo th ers, we consider th e opera to r Bl in mor e det ail. The st ability of the block diagonal smoother can be easily established by mean s of the properties of th e operator Bi , The following lemm a shows that th e condition numb er of Bi is uniformly bounded. Lemma 2.38. There exist constants such that

(2.69) P roof. The upper bound is obtained by using an inverse estimate and t he definiti on of Bi :

IIBlf.Llllo = sup

W IE X , w /;cO

168

2. It er ative Solvers Based on Domain Decomposit ion

To establish the lower bound, we observe t hat

Each WI E X, is uniquely defined by it s valu es at the vertices of the t riangulat ion . We define Wl (P) := J-ll(P) for an int erior vert ex P of a non-m ort ar side "(m , 1 :::; m :::; M . For all other vertices q of Ti , we set Wl (q) := O. This special choice yields b(Wl, J-ll) ~ cll[wdllo;sIIJ-ltllo;s; we refer to [Woh99c] for det ails. The lower bound in (2.69) now follows from II[wdllo;s ~ ch~ I/2 1 Iwtl lo . 0 Remark 2.39 . Using a sim ilar construction as in the proof of Lemma 2.38, we fin d by a straightforward computati on and by m eans of (2.69)

WI EXI .

inf

Vt EXl

(2.70)

Bi Vl = B jw l

Using t he definition (2.67) of th e operat or Ki , we can rewrite the saddle poin t problem (1.9) on X, x M, as an operator equation: Find zt := (Ul' AI) E X, X M, such t hat Kl zt

= fl

,

where li > (1;,0) E X, X M, is defined by (j;,Vl )O:= (f,Vl )O, VI E Xl . To establish appropriate smoot hing pr operties, we will use suitable operator norms . We recall t hat if 5 is a linear cont inuous operat or 5 : H I --+ H 2 , where H I and H 2 are Hilb ert spaces with norm s II . 111ft and II .IIII2' respect ively. T hen the standa rd operator norm is given by

11511:=

sup

xE H l

x#O

115xllII2 II x II n, .

In the next two subsubsections, we establish smoot hing and st ability pr operties for two different ty pes of it erations. In par ticular , we do not requ ire that t he iterates satisfy th e weak cont inuity conditions at th e int erfaces exactly. As a consequence, we neither need a good pr econdi tioner for the exact Schur complement nor an exact solver for a modified Schur comp lement. 2.5.2.1 A Block Diagonal Smoother. Following the ideas of [Ver84], we introduce a smoother for t he squa red positiv e definite syste m. The operato r Kl : X, x M, --+ X, X M, is defined by means of the symmet ric posit ive definit e bilin ear forms a (·, ·) and d(·, ·) on X, x X, and M, x Mi , respectively,

(Kl (Vl, J-lt) , (WI , Vl ))h/;S!XS := a(vI' WI ) + d(VI' J-lI) , It has a block diagonal st ructure

K l(vt,J-lI ) = (AIVl, DI J-lI) ,

(WI , vt) E X,

X

M, .

(2.71)

2.5 A Multigrid Method for the Mortar Saddle Point Formulati on

169

where the op erators A l : X I --+ XI , and D I : M I --+ M I are asso ciated with the bilin ear forms a( ·, ·) and d(·, '), resp ectively. One smoothing it er at ion on level l is given by m

m -l

ZI := zi

K IZIm - l ) , + K~I-IKI K~-l(d I 1-

(2.72)

where dl represents the right hand side of t he syst em Ki z, = dl , which has to be solved, ZI is t he exact solution, zl' denotes the it er at e in the mthst ep , and zp is the initial guess. The blo ck diagonal smoother works on the squared syst em which is positive definit e. Each smoot hing ste p can be easily performed provid ed that the applicat ions of Al l and DI I are cheap. The following lemma gives t he smoot hing rat e and can be found in [WohOOb].

Lemma 2.40. (Smoothing property) Let KI be defin ed as in (2. 71), where A I, IIAdl ::; Clh;, an d D I , IIDdl ::; are self- adj oint positive definite operators. If th ere exists for each WI E XI an aWl' 0 < aWl < 1 such that

cn«,

I

(BID I BtwI , wI)o ::; (1- a wl)(Alwl , wl)o ,

(AIWL,WI)O ::; awl (Alw l,wl )o,

then th e follo wing sm oothin g property holds for the it eration (2.72) : IIKl el'II''1 ;!7 Xs ::;

h;~lle?llhl ;!7 XS ' m ~ 1

.

(2.73)

Here, ei := zl' - ZI, m ~ 0, is the iterati on error in the mth-smoothing step, and th e const an t C does not depend on the aWl' Proof. The it er ation err or ei sat isfies m

el

= (Id -

K ~-IK K ~-IK) m 0 I

I

I

l

ei '

Since KI is a self-adjoint positive definit e oper ator and KI is self-adjoint , t here exists a complete set of orthogonal eigenfunctions zj satisfying

.

.

z;, we find for the eigenvalues Si =j:. 0, that w; =j:. 0, .

- _ 1.

Setting (Wi, pi) := K I and t hat

2

T hen , t he ass umpt ions on

Al

and DI yield

170

2. Iterative Solvers Base d on Dom ain Decomp ositi on

and t he norm of K1 ei is bounded by m

IIKl el Ilh/;!2XS

- ! - -! - -! -- -! - -! 2 m- ! 0 :S II K I tc, KIKI (Id - ttc, K IKI )) «, el llh,;!2xS