Table Of ContentSpringer Series in
Computational
Mathematics
15
Editorial Board
R.L. Graham, Murray Hill (NJ)
J. Stoer, WOrzburg
R. Varga, Kent (Ohio)
Franco Brezzi Michel Fortin
Mixed and Hybrid
Finite Element Methods
With 65 Illustrations
Springer-Verlag
New York Berlin Heidelberg London
Paris Tokyo Hong Kong Barcelona
Franco Brezzi Michel Fortin
University of Pavia Departement de MatMmatiques
Institute of Numerical Analysis et de Statistique
5 Corso Carlo Alberto Universite Laval
1-27100 Pavia Quebec G1 K 7P4
Italy Canada
Mathematics Subject Classification: 73XX, 76XX
library of Congress Cataloging-in-Publication Data
Breul, Franco.
Mixed alld hybrid finite element methods' Franco Breul, Michel
Fortin.
p. cm --(Springer series in computational mathematics: t5)
Includes bibliographical references and index.
ISBN-13: 978-1-46]2-7824-5 ~-ISBN-13: 978-1-4612-3172-1
001: IO.l007f978-]-4612-3172-]
1. Finite element method. I. Breai, F. (Franco),1945-.
II. TItle. III. Series.
TA347.F5F68 1991
620'.00t'51535--dc20 9t-t0909
Printed on acid-free paper.
© 1991 Springer-Verlag New YorK Inc.
Softcover reprint of the hardcover 1st edition 1991
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Photocomposed COpy prepared from the author's TEX file.
9 8 7 6 5 34 21
Preface
When, a few years ago, we began the redaction of this book, we had the naive
thought that the theory of mixed and hybrid finite element methods was ripe
enough for a unified presentation. We soon realized that things were not so
simple and that, if basic facts were known, many obscure zones remained in
many applications. Indeed the literature about nonstandard finite element method
is still evolving rapidly and this book cannot pretend to be complete. We
would rather like to lead the reader through the general framework in which
development is taking place.
We have therefore built our presentation around a few classical examples:
Dirichlet's problem, Stokes problem, linear elasticity, ... They are sketched
in Chapter I and basic methods to approximate them are presented in Chapter
IV, following the general theory of Chapter II and using finite element spaces
of Chapter III. Those four chapters are therefore the essential part of the book.
They are complemented by the following three chapters which present a more
detailed analysis of some problems.
Chapter V comes back to mixed approximations of Dirichlet's problem
and analyses, in particular, the (lambda)-trick that enables to make the link
between mixed methods and more classical non-conforming methods. Chapter
VI deals with Stokes problem and Chapter VII with linear elasticity and the
Mindlin-Reissner plate model.
The reader should not look here for practical implementation tricks. Our
goal was to provide an analysis of the methods in order to understand their
properties as thoroughly as possible. We refer, among others, to the recent
work of BATHE [A] or HUGHES [A] or to the classical and indispensable book
vi Preface
of ZIENKIEWICZ [A] for practical considerations. We are of course strongly
indebted to CIARLET [A] which remains the essential reference for the classical
theory of finite element methods. Finally we also refer to ROBERTS-THOMAS
[A] for another presentation of mixed methods.
This book would never have come to its end without the help, encour
agement and criticisms of our friends and colleagues. We must also thank all
those who took the time reading the first draft of our manuscript and proposed
significant improvements. We hope that the final result will better than what
one might expect, according to the quotations thereafter, of the hybrid resulting
of a collaboration between Pavia and Quebec.
Apris atque sui setosus nascitur hybris. (c. Plinius Caecilius Secondus.)
Mixtumque genus prolesque bi/ormis. (Publius Virgilius Maro.)
Contents
Preface v
Chapter I: Variational Formulations and Finite Element Methods ....... 1
§1 . Classical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§2. Model Problems and Elementary Properties
of Some Functional Spaces. . . . . . . . . . . 4
§3. Duality Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 .1 . . . . . . . . . . . .
3.1. Generalities ...... '.................. .......... ............ 11
3.2. Examples for symmetric problems. . . . . . . . . . . . . . . . . . 1.3 . . . . . .
3.3. Duality methods for non symmetric bilinear forms. . . . . . . . .2 3. .
§4. Domain Decomposition Methods, Hybrid Methods. . . . . . . .. .. . 2.4
§5. Augmented Variational Formulations. . . . . . . . . . . . . . . . ... . . 3. 0. . . .
§6. Transposition Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 3. 3 . . . . . . . . . .
§7. Bibliographical remarks. . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 . . . . . . . . . .
Chapter II: Approximation of Saddle Point Problems. . . . . . . . . . . .. .. . 3.6 . .
§1 . Existence and Uniqueness of Solutions. . . . . . . . . . . . . . . . . .. 3. 6. . . .
1.1. Quadratic problems under linear constraints . . . . . . . . . . . . 3.7 . . .
1.2. Extensions of existence and uniqueness results . . . . . . . . . . .4 .4 .
§2. Approximation of the Problem . . . . . . . . . . . . . . . . . . . . . . 51. . . . . . . . .
2.1. Basic results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . 5.1 . . . . . . . . . . .
2.2. Error estimates for the basic problem. . . . . . . . . . . . . . . . 5. 4. . . . .
2.3. The inf-sup condition: criteria......... .................... 57
2.4. Extensions of error estimates. . . . . . . . . . . . . . . . . . . ... . 6. 1. . . . . .
2.5. Various generalizations of error estimates. . . . . . . . . . . . . . 63. . . .
viii Contents
2.6. Perturbations of the problem, nonconforming methods. . . . . ..6 5
2.7. Dual error estimates. . . . . . . . . . . . . . . . . . . . . . . . ... . . .7 0 . . . . . . . .
§3. Numerical Properties of the Discrete Problem . . . . . . . . . . . . . . 73 . . .
3.1. The matrix form of the discrete problem. . . . . . . . . . . . . . 7. 3. . . .
3.2. Eigenvalue problem associated with the inf-sup condition. .. 75
3.3. Is the inf-sup condition so important? .................... 78
§4. Solution by Penalty Methods, Convergence of
Regularized Problems. . . . . . . . . . . . 8.0 . ..
§5. Iterative Solution Methods. Uzawa's Algorithm. . . . . . . . . . . .8 . 7 . .
5.1. Standard Uzawa's algorithm . . . . . . . . . . . . . . . . . . . . . 8. 7. . . . . . . .
5.2. Augmented Lagrangian algorithm. . . . . . . . . . . . . . . . . . .8 7. . . . . .
§6. Concluding Remarks ............ ... ... ........ . ..... . ....... 88
Chapter III: Function Spaces and Finite Element Approximations .... .. 89
§1. Properties of the spaces HS(0.) and H(div;0.) ................ 89
1.1. Basic results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . 8. 9 . . . . . . . . . . .
1.2. Properties relative to a partition of 0. . . . . . . . . . . . . . . . .9 3. . . . . .
1.3. Properties relative to a change of variables. . . . . . . . . . . . . 9.5 . . .
§2. Finite Element Approximations of Hl(0.) and H2(0.) ..... ..... 99
2.1. Conforming methods. . . . . . . . . . . . . . . . . . . . . . . .. . . . 9. 9. . . . . . . .
2.2. Nonconforming methods. . . . . . . . . . . . . . . . . . . . . . . . 10.7 . . . . . . . . .
2.3. Nonpolynomial approximations: Spaces 'ck (Q;h) . . . . . . .. .. . 11.1
2.4. Scaling arguments ........................................ 111
§3. Approximations of H(div; 0.) ................................ 113
3.1. Simplicial approximations of H (div; K) . . . . . . . . . . . ... . 1.1 .3 . .
3.2. Rectangular approximations of H (div; K) . . . . . . . . . . .. .. .1 1.9 . .
3.3. Interpolation operator and error estimates. . . . . . . . . . . ... . .1 24. .
3.4. Approximation spaces for H( div; 0.) ....................... 131
§4. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . .. .. . 13.2 . . . . . . . . .
Chapter IV: Various Examples. . . . . . . . . . . . . . . . . . . . . . . . .. .. .1 3. 3. . . . . . . . .
§1. Nonstandard Methods for Dirichlet's Problem ............. .... 134
1.1. Description of the problem. . . . . . . . . . . . . . . . . . . . .. .. 1.3 4. . . . . . .
1.2. Mixed finite element methods for Dirichlet's problem . . . .. .. 135
1.3. Primal hybrid methods . . . . . . . . . . . . . . . . . . . . . .. .. .1 .4 0. . . . . . . .
1.4. Dual hybrid methods ................... .................. 149
§2. Stokes Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .1 5.5 . . . . . . . . . . .
§3. Elasticity Problems. . . . . . . . . . . . . . . . . . . . . . . . . . ... . 1.5 9. . . . . . . . . . .
§4. A Mixed Fourth-Order Problem. . . . . . . . . . . . . . . . . . . ... . 1. 64. . . . . .
4.1. The 1/;-\.1) biharmonic problem ............................. 164
§5. Dual Hybrid Methods for Plate Bending Problems. . . . . . . . . .1 .6 .7 .
Chapter V: Complements on Mixed Methods for Elliptic Problems. . . .. . 177
§1. Numerical Solutions ....................... .................. 177
1.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .1 .7 7. . . . . . . . . . .
1.2. Interelement multipliers. . . . . . . . . . . . . . . . . . . . . . ... . 1.7 8. . . . . . . .
Contents ix
§2. A Brief Analysis of the Computational Effort ................. 182
§3. Error Analysis for the Multiplier . . . . . . . . . . . . . . . . . . . . .1 86. . . . . . . .
§4. Error Estimates in Other Norms .............................. 191
§5. Application to an Equation Arising
from Semiconductor Theory ...... .... 193
§6. How Things Can Go Wrong .......................... ....... 195
§7. Augmented Formulations. . . . . . . . . . . . . . . . . . . . . . . . 1. 9.8 . . . . . . . . ..
Chapter VI: Incompressible Materials and Flow Problems. . . . . . . . . ... .2 .0 0
§1 . Introduction................................................. 201
§2. The Stokes Problem as a Mixed Problem ..................... 202
2.1. Mixed Formulation. . . . . . . . . . . . . . . . . . . . . . . . .. . . 2. 0.2 . . . . . . . . .
§3. Examples of Elements for Incompressible Materials ........... . 206
3.1. Simple examples ................... ............... ....... 207
§4. Standard Techniques of Proof for the inf-sup Condition .... .... 219
4.1. General results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 4. . . . . . . . . . . ..
4.2. Higher order methods . . . . . . . . . . . . . . . . . . . . . . .. . . 2. 2.7 . . . . . . . .
§5. Macroelement Techniques and Spurious Pressure Modes ....... 228
5.1. Some remarks about spurious pressure modes .............. 228
5.2. An abstract convergence result ..................... ....... 231
5.3. Macroelement techniques ............... ......... ......... 235
5.4. The bilinear velocity-constant pressure (Ql-PO) element .... 240
5.5. Other stabilization procedures, (Augmented Formulations) ... 246
§6. An Alternative Technique of Proof
and Generalized Taylor-Hood Element. . . . . . . . . . . . . . . .. .. 2. 5. 2. . . .
§7. Nearly Incompressible Elasticity, Reduced Integration
Methods and Relation with Penalty Methods. . . . . . . . . . . . . . 25.8 . . . .
7.1. Variational formulations and admissible discretizations ...... 258
7.2. Reduced integration methods ......................... ..... 259
7.3. Effects of inexact integration ................... ........... 263
§8. Divergence-Free Basis, Discrete Stream Functions ............. 267
§9. Other Mixed and Hybrid Methods for Incompressible Flows .... 272
Chapter VII: Other Applications ....... ........................ ....... 274
§1 . Mixed Methods for Linear Thin Plates ........................ 274
§2. Mixed Methods for Linear Elasticity Problems ................ 282
§3. Moderately Thick Plates ..................................... 295
3.1. Generalities ....................................... ....... 295
3.2. Discretization of the problem. . . . . . . . . . . . . . . . . . ... . . 3. 03 . . . . .
3.3. Continuous Pressure Approximations . . . . . . . . . . . . . . . 3. 2.2 . . . ..
3.4. Discontinuous Pressure Approximations .................... 322
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . .3 .2 .4 . . . . . . . . . . . . . .
Index ......... ... .......... ..................... .. .................. 344
I
Variational Formulations and Finite
Element Methods
Although we shall not define in this chapter mixed and hybrid (or other non
standard) finite element methods in a very precise way, we would like to situate
them in a sufficiently clear setting. As we shall see, boundaries between dif
ferent methods are sometimes rather fuzzy. This will not be a real drawback
if we nevertheless know how to apply correctly the principles underlying their
analysis.
After having briefly recalled some basic facts about classical methods, we
shall present a few model problems. The study of these problems will be the
kernel of this book. We shall thereafter rapidly recall basic principles of duality
theory as this will be our starting point to introduce mixed methods. Domain de
composition methods (allied to duality) will lead us to hybrid methods. Finally,
we shall present a few ideas about transposed formulations as they can help to
understand some of the weak problems generated by the previous methods.
1.1 Classical Methods
We recall here in a very simplified way some results about optimization methods
and the classical finite element method. Such an introduction cannot be complete
and does not want to be. We refer the reader to CIARLET [A) or RAVIART
THOMAS [D), among others, where standard finite element methods are clearly
exposed. We also refer to DAUTRAY-LIONS [A] where an exhaustive analysis
of many of our model problems can be found.
Let us then consider a very common situation where the solution of a
physical problem minimizes some functional (usually an "energy functional"),
2 Variational Formulations and Finite Element Methods §1.1
in a "well chosen" space of admissible functions V that we take for the moment
as a Hilbert space,
(1.1) inf J(v).
vEV
JO
If the functional is differentiable (cf. EKELAND-TEMAM [A] for in
stance), the minimum (whenever it exists) will be characterized by a variational
equation
(1.2) (JI(U),V)V1XV = 0, 'Vv E V,
where (-,.) V v denotes duality between V and its topological dual V', the
I X
derivative J' (u) at point u being considered as a linear form on V.
A classical method, Ritz's method, to approximate the solution of (1.1)
consists in looking for Um E Vm, where Vm is a finite-dimensional subspace of
V, which is solution of the problem
(1.3)
or, differentiating,
(1.4)
Let us consider to fix ideas the quadratic functional
(1.5) J(v) = ~a(v, v) - L(v),
where a(·, .) is a bilinear form on V, which we suppose to be continuous and
symmetric, and LO is a linear form on V. The variational equation (1.4) can
then be written as: Um E Vm and
(1.6)
If a basis W1, W2, ... , Wm of V m is chosen and if one writes
m
(1.7) Um = LCt;W;,
;=1
problem (1.6) is reduced to the solution of the linear system
m
(1.8) La;jCti = bj, 1 $ j $ m,
i=1
where one defines
(1.9)
