الرئيسية
LeastSquares Finite Element Methods
LeastSquares Finite Element Methods
Pavel B. Bochev, Max D. Gunzburger
Categories:
Mathematics\\Computational Mathematics
السنة:
2009
الناشر:
Springer
اللغة:
english
الصفحات:
664
ISBN 10:
0387308881
ISBN 13:
9780387689227
Series:
Applied mathematical sciences 166
File:
PDF, 12.52 MB
تحميل (pdf, 12.52 MB)
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Applied Mathematical Sciences Volume 166 Editors S.S. Antman J.E. Marsden L. Sirovich Advisors J.K. Hale P. Holmes J. Keener J. Keller B.J. Matkowsky A. Mielke C.S. Peskin K.R. Sreenivasan For further volumes: http://www.springer.com/series/34 Pavel B. Bochev • Max D. Gunzburger LeastSquares Finite Element Methods 123 Pavel B. Bochev Sandia National Laboratories Applied Mathematics and Applications MS 1320, P.O. Box 5800 Albuquerque NM 871851320 USA pbboche@sandia.gov Max D. Gunzburger Florida State University Department of Scientific Computing 400 Dirac Science Library Tallahassee FL 323064120 USA gunzburg@fsu.edu Editors: S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park MD 207424015 USA ssa@math.umd.edu J.E. Marsden Control and Dynamical Systems, 10781 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu ISBN 9780387308883 DOI 10.1007/9780387689227 L. Sirovich Laboratory of Applied Mathematics Department of Biomathematical Sciences Mount Sinai School of Medicine New York, NY 100296574 USA Lawrence.Sirovich@mssm.edu eISBN 9780387689227 Library of Congress Control Number: 2008943966 Mathematics Subject Classification (2000): 65N30, 65N35, 65N12, 65N21, 65M60, 65M70, 65M12 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acidfree paper springer.com To my mother and Biliana To Janet v Preface Since their emergence in the early 1950s, finite element methods have become one of the most versatile and powerful methodologies for the approximate numerical solution of partial differential equations. At the time of their inception, finite element methods were viewed primarily as a tool for solving problems in structural analysis. However, it did not take long to discover that finite element methods could be applied with equal success to problems in other engineering and scientific fields. Today, finite element methods are also in common use, and indeed are often the method of choice, for incompressible fluid flow, heat transfer, electromagnetics, and advectiondiffusionreaction problems, just to name a few. Given the early connection between finite element methods and problems engendered by energy minimization principles, it is not surprising that the first mathematical analyses of finite element methods were given in the environment of the classical Rayleigh–Ritz setting. Yet again, using the fertile soil provided by functional analysis in Hilbert spaces, it did not take long for the rigorous analysis of finite element methods to be extended to many other settings. Today, finite element methods are unsurpassed with respect to their level of theoretical maturity. A finite element method is first and foremost a quasiprojection scheme.1 This truly fundamental property establishes a link with sophisticated mathematical structures and has a tremendous impact on the algebraic problems generated by the method. The paradigm that describes and defines the key properties of what we refer to as a quasiprojection is the marriage of two ingredients: a variational principle and a closed subspace. Approximate solutions are then characterized as quasiprojections of the exact (weak) solutions onto the closed subspace. Finite element approximations are an example of this rule. Indeed, a finite element method and its properties are completely determined by specifying the variational principle and 1 A projection in a Hilbert space is, of course, defined with respect to an inner product and a closed subspace. Finite element methods do not always involve a true projection because they are not always based on inner products; however, as we show, finite element approximations do, in general, possess many of the important properties of projections. vii viii Preface the approximation subspace.2 The cornerstone of the great success of finite element methods is the remarkable fact that the combination of these two ingredients prove to be exceptionally well suited for the numerical solution of partial differential equations. From a mathematical viewpoint, this success is rooted in the existence of strong intrinsic links between differential equations and variational principles; in fact, it is often the case that the latter serve as the primary mathematical model. On the other hand, the practical appeal of finite element methods and their wide acceptance in the engineering community results from the choice of approximating subspaces. These spaces are spanned by locally supported, piecewise polynomial functions defined over a subdivision of a domain into simple geometrical subdomains referred to as finite elements. Such spaces are not only simple to use, but allow for the almost automatic generation of highorder methods3 with respect to arbitrary unstructured subdivisions, a trait that, if not outright impossible, is not easy to accomplish with other schemes. The popularity of finite element methods is also largely due to the small support of standard finite element basis functions. This property implies that the resulting algebraic problems involve banded and sparse matrices. Although the choice of approximation space has, especially from a practical point of view, a tremendous influence on the attributes of the resulting discretized systems, all fundamental properties of finite element methods are ultimately governed by the variational principles from which they are defined. It is, therefore, a fortuitous coincidence that the variational foundations of the first finite element methods were provided by unconstrained, quadratic energy minimization principles, a setting that turned out to be, by far, the most attractive one for finite element methods. Such principles search for a point (in a suitable function space) that is the unconstrained minimizer of a convex quadratic functional and give rise to true innerproduct projections in Hilbert spaces. This property, exemplified by the classical Rayleigh–Ritz principle, results in distinct, unique, and very desirable computational and analytic advantages for the resulting finite element methods. Most notably, true inner product projections allow for a wide and nonrestrictive choice for the approximating spaces and lead to symmetric and positive definite algebraic systems. The connection between finite element methods and Rayleigh–Ritz principles was not immediately recognized as the fundamental cause for the remarkable success of the first finite element methods. As a result, some of the early attempts directed at extending finite element methods beyond the class of problems whose solutions can be characterized as global minimizers of convex quadratic functionals 2 There exists a fundamental philosophical difference between finite difference and finite element methods. The former substitutes difference operators for differential operators and leads to approximations defined with respect to a discrete lattice, i.e., the fundamental discretization step is to approximate operators. Finite element methods, on the other hand, do not deal directly with the differential operators; instead, discretization is effected using alternative weak formulations of these equations posed over finitedimensional function spaces. This leads to functional approximation as opposed to operator approximation. Finite volume methods occupy a middle ground as they exhibit features shared by both finite element and finite difference methods. 3 This is due to the possibility of using internal degrees of freedom that enrich the finite element space inside each element. Preface ix led to disappointing results and a reluctance (by some) to apply the methods outside the field of structural analysis. However, with the rapid development of the mathematical theory of finite element methods and the rapid accumulation of a wealth of practical experience in applying the methods, there came the understanding for what were the reasons behind these early setbacks. It was realized that differential equations not associated with unconstrained minimization principles lead to two other classes of variational principles with strikingly different properties. In one of them, the quasiprojection is induced by a search for a stationary point of an indefinite functional; constrained minimization problems provide an example of this class. The other type offers even less mathematical structure and defines the quasiprojection by a formal residual orthogonalization process. In both cases, the variational principle does not lead to a true innerproduct projection; this causes the associated finite element methods to operate in a much less favorable setting as compared to those based on Rayleigh–Ritz principles. The computation of stationary points that is the paradigm of mixedGalerkin methods demands strict compatibility conditions on the discrete spaces, if stable and accurate approximations are desired. Likewise, the formal residual orthogonalization process that provides the template for Galerkin methods may require onerous stability conditions on the approximation spaces. In both cases, one is confronted by the problem of solving indefinite and/or nonsymmetric algebraic equations. Not surprisingly, there have been many efforts aimed at developing finite element methods that, for problems not connected to unconstrained energy principles, share some, if not all, of the attractive mathematical and algorithmic properties of the Rayleigh–Ritz setting. Broadly speaking, there are two different ways to approach the construction of better projections or quasiprojections than those afforded by the original problem. The first one improves the quasiprojection by penalization or regularization of the original variational principle. In this approach, the principal role of the naturally occurring variational principle4 is retained, but the principle itself is modified so as to behave more like a true innerproduct projection.5 Galerkin leastsquares, stabilized Galerkin, penalized Lagrangian methods, artificial diffusion, and upwind Petrov–Galerkin methods are all examples of finite element methods resulting from this approach. A second approach that leads to better quasiprojections, indeed to true projections, is to abandon completely the naturally occurring variational principle and to devise a Rayleigh–Ritzlike environment by formulating an artificial, externally defined energytype principle. This energy principle most often takes the form of residual minimization in a suitable Hilbert space; thus, there arises the commonly used adjective leastsquares to denote the resulting finite element methods. Residual 4 “Naturally occurring” variational principles include Galerkin principles that are based on residual orthogonalization and may or may not have any physical interpretation. 5 Often the same effect can be achieved by posing the original variational principle on a modified finite element space. One example is the SUPG method which can be viewed both as resulting from the use of a modified test space or a modified variational principle. Another example is finite element spaces “enriched” by bubble functions. In many cases, enriched spaces lead to exactly the same formulations as modified Galerkin principles posed on conventional finite element spaces. x Preface minimization is as universal as residual orthogonalization so that, in principle, leastsquares finite element methods have the same wide range of applicability as do Galerkin finite element methods, i.e., they are both applicable to virtually any partial differential equation problem. However, residual minimization differs fundamentally from other variational settings, including modified and formal Galerkin principles. Unlike the case for other settings, leastsquares finite element methods are consistently capable of recovering almost all of the advantages of the Rayleigh– Ritz setting over a wide range of problems and, with some additional effort, they can often create a completely analogous variational environment for finite element methods. This is what makes leastsquares finite element methods stand apart from the rest of the methods in the finite element universe. Mostly over the last ten to fifteen years, efforts focused on leastsquares finite element methods have achieved tremendous success in making the methods viable alternatives to existing schemes. There is a wealth of theoretical and practical experience with the methods and they are steadily gaining a solid reputation and popularity among researchers and practitioners for providing robust, efficient, and practical computational tools. Nevertheless, compared with established finite element techniques, such as mixedGalerkin methods, the theory for leastsquares finite element methods remains much less unified and is not always well understood. Because such methods are based on innerproduct projections, they tend to be exceptionally robust and stable. As a result, one is often tempted to forego analyses and proceed with the seemingly most natural and simple leastsquares formulations. This sometimes leads to unsatisfactory results and methods that cannot fully exploit the advantages of the leastsquares approach. This book is motivated by the premise that there exists a real need to put leastsquares finite element methods on a common, mathematically sound foundation and also to discuss important implementation issues that are critical to their success in practice. It is intended to give both the researcher and the practitioner a guide to the theory and practice of leastsquares finite element methods, their strengths and weaknesses, caveats to be followed, and established successes. The factors that set leastsquares finite element methods apart from other finite element methods also call for a different approach in the algorithmic development of these methods, if truly reliable, efficient, and accurate schemes are desired. Most notably, reliance of a leastsquares method on externally defined variational principles makes the choice of this principle the single most important step in the design of a method. Unlike traditional finite element methods for which the variational principle is almost always dictated by the given problem, leastsquares finite element methods dictate the variational principle and then fit the problem into the principle. Thus, the flexibility afforded by the freedom to choose the variational principle places the principal responsibility for the success of the method on the “fitting” process. There are two, often opposing, forces that affect this process. One is the desire to keep the method as simple as possible so as to develop a scheme that is easy to implement. The other is the need to adhere to the basic premises of a Rayleigh–Ritzlike framework, namely, to work with projections defined by inner products that are equivalent to the natural inner products in suitable Hilbert spaces. The interaction Preface xi between these two competing forces is, in our opinion, the key to understanding leastsquares finite element methods; we make it the central theme of this book. Indeed, we show over and over again that the choices made in reconciling ease of implementation with adherence to the Rayleigh–Ritz framework affect all aspects of leastsquares finite element methods, from condition numbers of the algebraic problems and their efficient preconditioning to the existence of quasioptimal asymptotic error estimates. An overview of the book Throughout the book, careful attention is paid to not only the rigorous analysis of leastsquares finite element methods, but also to practical issues that arise in their implementation. For those who wish to gain a full understanding of the mathematical analyses connected with leastsquares finite element methods, space limitations necessitate the requirement of certain prerequisites. In particular, a basic familiarity with functional analysis in Hilbert spaces and with the theory and implementation of standard finite element methods is assumed at the level of, e.g., [250] and [76, 123, 188], respectively. More advanced background material on functional analysis, partial differential equations, and finite element methods is contained within the book, especially in the appendices. Those who merely wish to learn about leastsquares finite element algorithms and their implementation can still use the book by focusing on those sections that consider these topics. In this case, space limitations necessitate familiarity with the algorithmic and implementation aspects of standard finite element methods. The book is organized into parts, each containing several chapters. In Part I of the book, we provide a necessarily brief review of the finite element universe. It is not meant to give a comprehensive treatment, but rather to provide a context for understanding where leastsquares finite element methods fit into that universe. In Chapter 1, “classical” finite element methods, i.e., those based on Rayleigh–Ritz, mixedGalerkin, and Galerkin variational principles, are discussed. Then, in Chapter 2, a discussion is provided about several of the attempts that have been made to try to recover some of the advantageous features of finite element methods in the Rayleigh–Ritz setting through the definition of modifications to the “classical” principles. Along the way, the strengths and weaknesses of the specific finite element methods encountered are pointed out. At that point, one is ready for the discussion provided in the latter part of Chapter 2 about why and how leastsquares finite element methods are meant to improve on the three “classical” approaches and their modifications. In Part II, we provide the theoretical core of the book that is repeatedly referred to in the rest of the book. Chapter 3 is devoted to an abstract theory of leastsquares finite element methods for systems of elliptic partial differential equations. The abstract framework can itself be divided into three classes of leastsquares finite element methods that are characterized by discrete leastsquares principles having xii Preface different derivations and properties. The first is defined by simply restricting the minimization of a continuous leastsquares functional for the partial differential equations to a finite element space. The other two classes additionally involve a discretization of the leastsquares functional itself and differ from each other by the type of equivalence relations that exist (or do not exist) between leastsquares functionals and the norms used to measure the size of the solution and the data for the problem. All the possible scenarios emanating from this classification of leastsquares finite element methods are analyzed. For a specific problem, one can usually define several different realizations of leastsquares finite element methods. Most of these can be viewed as particular applications of one or another of the different scenarios covered by the abstract framework. In addition, several additional basic ingredients needed to define and analyze leastsquares finite element methods but which transcend the specific context to which those methods are applied are also discussed. In Chapter 4, the Agmon–Douglis–Nirenberg theory for elliptic partial differential equations is exploited to provide “automated” mechanisms for using solution and data spaces and norms for the definition of leastsquares variational principles. The abstract frameworks developed and analyzed in Part II can be used to define and analyze many leastsquares finite element methods for a number of specific settings. In these cases, one need only show how the setting fits into the abstract frameworks and that the hypotheses invoked in those frameworks hold. Thus, there is no need to repeat over and over again proofs specialized to each setting. Using this approach, Part III of the book is devoted to the application of the abstract framework developed in Part II to concrete elliptic partial differential equations.6 In Chapters 5 and 6, leastsquares finite element methods for scalar and vector elliptic partial differential equations are considered, respectively, and, in Chapter 7, leastsquares finite element methods for the Stokes equations are considered. In Part IV, leastsquares finite element methods in contexts that do not fit into the abstract theory of Part II, but nevertheless rely on some aspects of that theory, are examined. In Chapter 8, the nonlinear stationary Navier–Stokes equations are considered. In addition to the intrinsic interest engendered by the Navier–Stokes equations, this setting provides an example of how leastsquares finite element methods can be used for nonlinear problems. In Chapters 9 and 10, parabolic and hyperbolic (timedependent) partial differential equations are considered, respectively. The application of leastsquares finite element methods to optimization and control problems for systems governed by elliptic partial differential equations is treated in Chapter 11. Then, in Chapter 12, other settings to which leastsquares finite element methods have been applied, but that are not discussed in Chapters 5 through 11, are briefly considered.7 In that chapter, some variations on the leastsquares finite element 6 There are a few settings considered in Part III that do not fit into the abstract frameworks of Part II; these are treated directly on a casebycase basis. 7 Providing a full treatment of all the topics discussed in Chapter 12 would have easily doubled the length of the book. However, we believe these topics should be included in the book, even in a somewhat cursory manner, not only because they are important, but because they serve to further Preface xiii methods are also briefly discussed; these are methods that have a leastsquares character or that use leastsquares principles in different ways or for different purposes than those discussed in the rest of the book. Included in Chapter 12 are discussions of boundary condition treatments, LL∗ leastsquares methods, mimetic reformulations of leastsquares methods, leastsquares collocation methods, restricted leastsquares methods, optimizationbased leastsquares methods, advection–diffusion–reaction problems, higherorder problems, div–grad–curl systems, domain decomposition leastsquares methods, multiphysics problems, problems with singular solution, Treffetz leastsquares methods, a posteriori error estimation and mesh refinement, leastsquares wavelet methods, and meshless leastsquares methods. In the four Appendices, results from functional analysis, partial differential equations, and finite element theory that are used in various places in the rest of the book are provided. In particular, we define a consistent and unified notational system that does not vary from setting to setting, even though much of the source material used for the book is not consistent in this regard. This not only greatly reduces the notational definitions introduced, but, more important, facilitates making connections between the different settings treated in the book. We also include, in addition to the expected index, a list of often used acronyms and a glossary of often used notations. Acknowledgements Achi Dosanjh, our editor at Springer, has shown great patience and support throughout the process of getting this book to print. It has been a pleasure to work with her. Much of this book and much of our own work on leastsquares finite element methods would not have come to pass without the help of our collaborators who have worked with us on this subject, especially students and many colleagues with whom we had pleasant and productive interactions. Thus, a great deal of thanks is owed to Dana Bedivan, John Burkardt, Zhiqiang Cai, Yanzhao Cao, ChingLung Chang, Jungmin Choi, Christopher Cox, Jennifer Deang, Raytcho Lazarov, HyungChun Lee, Hugh MacMillan, Tom Manteuffel, Steve McCormick, Roy Nicolaides, and Panayot Vassilevski. Both authors especially acknowledge George Fix who was instrumental to their progress in the early stages of their careers and who, directly in the case of Max Gunzburger and indirectly in the case of Pavel Bochev, introduced them to the world of leastsquares finite element methods. This book is dedicated, by Pavel Bochev, to his wife, Biliana, for her unyielding support, kindness and exceptional patience, and for nurturing his resolve in those moments when the sheer magnitude of this endeavor made its success seem distant and unattainable. A special debt of gratitude is owed by Pavel to his mother Dora for believing in his dreams and making the many sacrifices that helped turn these dreams into reality. This book is also dedicated, by Max Gunzburger, to his wife, Janet Peterson, not so much for the real support she provided throughout the long and arduous path he illustrate the very significant progress that has recently been made in algorithmic, theoretical, and application aspects of leastsquares finite element methods. xiv Preface took to the completion of the book, but for the constant and enduring patience she has shown, support and help she has provided, sacrifices she has made, and love she has given that have been instrumental to his success, not only in his professional life, but, more important, in his personal life. Pavel Bochev and Max Gunzburger Albuquerque and Tallahassee November, 2008 Contents Part I Survey of Variational Principles and Associated Finite Element Methods 1 2 Classical Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Variational Methods for Operator Equations . . . . . . . . . . . . . . . . . . . 1.2 A Taxonomy of Classical Variational Formulations . . . . . . . . . . . . . 1.2.1 Weakly Coercive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Strongly Coercive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mixed Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Relations Between Variational Problems and Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Approximation of Solutions of Variational Problems . . . . . . . . . . . . 1.3.1 Weakly and Strongly Coercive Variational Problems . . . . . . 1.3.2 Mixed Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The Equations of Linear Elasticity . . . . . . . . . . . . . . . . . . . . . 1.4.3 The Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 A Scalar Linear AdvectionDiffusionReaction Equation . . 1.4.6 The Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 A Comparative Summary of Classical Finite Element Methods . . . 3 4 8 8 9 10 Alternative Variational Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Modified Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Enhanced and Stabilized Methods for Weakly Coercive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Stabilized Methods for Strongly Coercive Problems . . . . . . 2.2 LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 A Straightforward LeastSquares Finite Element Method . . 2.2.2 Practical LeastSquares Finite Element Methods . . . . . . . . . 35 36 12 15 15 18 22 22 25 26 28 30 30 31 36 46 49 51 53 xv xvi Contents 2.3 2.2.3 NormEquivalence Versus Practicality . . . . . . . . . . . . . . . . . . 58 2.2.4 Some Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . . . 60 Putting Things in Perspective and What to Expect from the Book . 62 Part II Abstract Theory of LeastSquares Finite Element Methods 3 Mathematical Foundations of LeastSquares Finite Element Methods 69 3.1 LeastSquares Principles for Linear Operator Equations in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.1.1 Problems with Zero Nullity . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1.2 Problems with Positive Nullity . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Application to Partial Differential Equations . . . . . . . . . . . . . . . . . . . 75 3.2.1 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2.2 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . 77 3.3 General Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . 80 3.3.1 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.2 The Need for Continuous LeastSquares Principles . . . . . . . 84 3.4 Binding Discrete LeastSquares Principles to Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4.1 Transformations from Continuous to Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.5 Taxonomy of Conforming Discrete LeastSquares Principles and their Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.1 Compliant Discrete LeastSquares Principles . . . . . . . . . . . . 92 3.5.2 NormEquivalent Discrete LeastSquares Principles . . . . . . 94 3.5.3 QuasiNormEquivalent Discrete LeastSquares Principles 96 3.5.4 Summary Review of Discrete LeastSquares Principles . . . 100 4 The Agmon–Douglis–Nirenberg Setting for LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1 Transformations to FirstOrder Systems . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.1 Homogeneous Elliptic Systems . . . . . . . . . . . . . . . . . . . . . . . 107 4.2.2 NonHomogeneous Elliptic Systems . . . . . . . . . . . . . . . . . . . 107 4.3 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.1 Homogeneous Elliptic Systems . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.2 NonHomogeneous Elliptic Systems . . . . . . . . . . . . . . . . . . . 110 4.4 LeastSquares Finite Element Methods for Homogeneous Elliptic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.5 LeastSquares Finite Element Methods for NonHomogeneous Elliptic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.1 QuasiNormEquivalent Discrete LeastSquares Principles 114 4.5.2 NormEquivalent Discrete LeastSquares Principles . . . . . . 124 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Contents xvii Part III LeastSquares Finite Element Methods for Elliptic Problems 5 Scalar Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.1 Applications of Scalar Poisson Equations . . . . . . . . . . . . . . . . . . . . . 135 5.2 LeastSquares Finite Element Methods for the SecondOrder Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.2.1 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . 138 5.2.2 Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . 139 5.3 First–Order System Reformulations . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3.1 The Div–Grad System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.3.2 The Extended Div–Grad System . . . . . . . . . . . . . . . . . . . . . . 145 5.3.3 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.4 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.4.1 Energy Balances in the Agmon–Douglis–Nirenberg Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.4.2 Energy Balances in the VectorOperator Setting . . . . . . . . . . 152 5.5 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.6 Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.6.1 The Div–Grad System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.6.2 The Extended Div–Grad System . . . . . . . . . . . . . . . . . . . . . . 169 5.7 Error Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.7.1 Error Estimates in Solution Space Norms . . . . . . . . . . . . . . . 171 5.7.2 L2 (Ω ) Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.8 Connections Between Compatible LeastSquares and Standard Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . 176 5.8.1 The Compatible LeastSquares Finite Element Method with a Reaction Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.8.2 The Compatible LeastSquares Finite Element Method Without a Reaction Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.9 Practicality Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.9.1 Practical Rewards of Compatibility . . . . . . . . . . . . . . . . . . . . 184 5.9.2 Compatible LeastSquares Finite Element Methods on NonAffine Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.9.3 Advantages and Disadvantages of Extended Systems . . . . . 192 5.10 A Summary of Conclusions and Recommendations . . . . . . . . . . . . . 194 6 Vector Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.1 Applications of Vector Elliptic Equations . . . . . . . . . . . . . . . . . . . . . 200 6.2 Reformulation of Vector Elliptic Problems . . . . . . . . . . . . . . . . . . . . 201 6.2.1 Div–Curl Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2.2 Curl–Curl Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.3 LeastSquares Finite Element Methods for Div–Curl Systems . . . . 206 6.3.1 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.3.2 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . 209 6.3.3 Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . 211 xviii Contents 6.3.4 6.4 6.5 6.6 7 Analysis of Conforming LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.3.5 Analysis of NonConforming LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 LeastSquares Finite Element Methods for Curl–Curl Systems . . . . 221 6.4.1 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.4.2 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . 224 6.4.3 Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . 225 6.4.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Practicality Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.5.1 Solution of Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . 232 6.5.2 Implementation of NonConforming Methods . . . . . . . . . . . 234 A Summary of Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 The Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.1 FirstOrder System Formulations of the Stokes Equations . . . . . . . . 238 7.1.1 The Velocity–Vorticity–Pressure System . . . . . . . . . . . . . . . . 239 7.1.2 The Velocity–Stress–Pressure System . . . . . . . . . . . . . . . . . . 242 7.1.3 The Velocity Gradient–Velocity–Pressure System . . . . . . . . 243 7.2 Energy Balances in the Agmon–Douglis–Nirenberg Setting . . . . . . 246 7.2.1 The Velocity–Vorticity–Pressure System . . . . . . . . . . . . . . . . 247 7.2.2 The Velocity–Stress–Pressure System . . . . . . . . . . . . . . . . . . 250 7.2.3 The Velocity Gradient–Velocity–Pressure System . . . . . . . . 251 7.3 Continuous LeastSquares Principles in the Agmon–Douglis–Nirenberg Setting . . . . . . . . . . . . . . . . . . . . . 253 7.3.1 The Velocity–Vorticity–Pressure System . . . . . . . . . . . . . . . . 253 7.3.2 The Velocity–Stress–Pressure System . . . . . . . . . . . . . . . . . . 256 7.3.3 The Velocity Gradient–Velocity–Pressure System . . . . . . . . 256 7.4 Discrete LeastSquares Principles in the Agmon–Douglis–Nirenberg Setting . . . . . . . . . . . . . . . . . . . . . 257 7.4.1 The Velocity–Vorticity–Pressure System . . . . . . . . . . . . . . . . 258 7.4.2 The Velocity–Stress–Pressure System . . . . . . . . . . . . . . . . . . 260 7.4.3 The Velocity Gradient–Velocity–Pressure System . . . . . . . . 260 7.5 Error Estimates in the Agmon–Douglis–Nirenberg Setting . . . . . . . 261 7.5.1 The Velocity–Vorticity–Pressure System . . . . . . . . . . . . . . . . 261 7.5.2 The Velocity–Stress–Pressure System . . . . . . . . . . . . . . . . . . 263 7.5.3 The Velocity Gradient–Velocity–Pressure System . . . . . . . . 264 7.6 Practicality Issues in the Agmon–Douglis–Nirenberg Setting . . . . . 264 7.6.1 Solution of the Discrete Equations . . . . . . . . . . . . . . . . . . . . . 265 7.6.2 Issues Related to NonHomogeneous Elliptic Systems . . . . 266 7.6.3 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.6.4 The Zero Mean Pressure Constraint . . . . . . . . . . . . . . . . . . . . 274 7.7 LeastSquares Finite Element Methods in the VectorOperator Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.7.1 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Contents xix 7.7.2 7.7.3 7.7.4 7.7.5 7.7.6 7.7.7 7.8 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . 281 Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . 281 Stability of Discrete LeastSquares Principles . . . . . . . . . . . 284 Conservation of Mass and Strong Compatibility . . . . . . . . . 287 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Connection Between Discrete LeastSquares Principles and MixedGalerkin Methods . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.7.8 Practicality Issues in the Vector Operator Setting . . . . . . . . . 304 A Summary of Conclusions and Recommendations . . . . . . . . . . . . . 306 Part IV LeastSquares Finite Element Methods for Other Settings 8 The Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 8.1 FirstOrder System Formulations of the Navier–Stokes Equations . 313 8.2 LeastSquares Principles for the Navier–Stokes Equations . . . . . . . 314 8.2.1 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . 315 8.2.2 Discrete LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . 316 8.3 Analysis of LeastSquares Finite Element Methods . . . . . . . . . . . . . 317 8.3.1 Quotation of Background Results . . . . . . . . . . . . . . . . . . . . . . 318 8.3.2 Compliant Discrete LeastSquares Principles for the Velocity–Vorticity–Pressure System . . . . . . . . . . . . . 321 8.3.3 NormEquivalent Discrete LeastSquares Principles for the Velocity–Vorticity–Pressure System . . . . . . . . . . . . . 329 8.3.4 Compliant Discrete LeastSquares Principles for the Velocity Gradient–Velocity–Pressure System . . . . . . 340 8.3.5 A NormEquivalent Discrete LeastSquares Principle for the Velocity Gradient–Velocity–Pressure System . . . . . . 344 8.4 Practicality Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.4.1 Solution of the Nonlinear Equations . . . . . . . . . . . . . . . . . . . 348 8.4.2 Implementation of NormEquivalent Methods . . . . . . . . . . . 351 8.4.3 The Utility of Discrete Negative Norm LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.4.4 Advantages and Disadvantages of Extended Systems . . . . . 359 8.5 A Summary of Conclusions and Recommendations . . . . . . . . . . . . . 364 9 Parabolic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 367 9.1 The Generalized Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 9.1.1 BackwardEuler LeastSquares Finite Element Methods . . 369 9.1.2 SecondOrder Time Accurate LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 9.1.3 Comparison of FiniteDifference LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 9.1.4 Space–Time LeastSquares Principles . . . . . . . . . . . . . . . . . . 391 9.1.5 Practical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 9.2 The TimeDependent Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . 396 xx Contents 10 Hyperbolic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 403 10.1 Model Conservation Law Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 404 10.2 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 10.2.1 Energy Balances in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 407 10.2.2 Energy Balances in Banach Spaces . . . . . . . . . . . . . . . . . . . . 409 10.3 Continuous LeastSquares Principles . . . . . . . . . . . . . . . . . . . . . . . . . 410 10.3.1 Extension to TimeDependent Conservation Laws . . . . . . . . 412 10.4 LeastSquares Finite Element Methods in a Hilbert Space Setting . 413 10.4.1 Conforming Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 10.4.2 NonConforming Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 10.5 Residual Minimization Methods in a Banach Space Setting . . . . . . 416 10.5.1 An L1 (Ω ) Minimization Method . . . . . . . . . . . . . . . . . . . . . . 416 10.5.2 Regularized L1 (Ω ) Minimization Method . . . . . . . . . . . . . . 418 10.6 LeastSquares Finite Element Methods Based on Adaptively Weighted L2 (Ω ) Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 10.6.1 An Iteratively ReWeighted LeastSquares Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 10.6.2 A Feedback LeastSquares Finite Element Method . . . . . . . 420 10.7 Practicality Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 10.7.1 Approximation of Smooth Solutions . . . . . . . . . . . . . . . . . . . 422 10.7.2 Approximation of Discontinuous Solutions . . . . . . . . . . . . . 423 10.8 A Summary of Conclusions and Recommendations . . . . . . . . . . . . . 427 11 Control and Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 11.1 Quadratic Optimization and Control Problems in Hilbert Spaces with Linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 11.1.1 Existence of Optimal States and Controls . . . . . . . . . . . . . . . 432 11.1.2 LeastSquares Formulation of the Constraint Equation . . . . 435 11.2 Solution via Lagrange Multipliers of the Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 11.2.1 Galerkin Finite Element Methods for the Optimality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 11.2.2 LeastSquares Finite Element Methods for the Optimality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 11.3 Methods Based on Direct Penalization by the LeastSquares Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 11.3.1 Discretization of the Perturbed Optimality System . . . . . . . 450 11.3.2 Discretization of the Eliminated System . . . . . . . . . . . . . . . . 453 11.4 Methods Based on Constraining by the LeastSquares Functional . 455 11.4.1 Discretization of the Optimality System . . . . . . . . . . . . . . . . 457 11.4.2 DiscretizeThenEliminate Approach for the Perturbed Optimality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 11.4.3 EliminateThenDiscretize Approach for the Perturbed Optimality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 11.5 Relative Merits of the Different Approaches . . . . . . . . . . . . . . . . . . . 460 11.6 Example: Optimization Problems for the Stokes Equations . . . . . . . 461 Contents xxi 11.6.1 The Optimization Problems and Galerkin Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 11.6.2 LeastSquares Finite Element Methods for the Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 11.6.3 LeastSquares Finite Element Methods for the Optimality Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 11.6.4 Constraining by the LeastSquares Functional for the Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 471 12 Variations on LeastSquares Finite Element Methods . . . . . . . . . . . . . . 475 12.1 Weak Enforcement of Boundary Conditions . . . . . . . . . . . . . . . . . . . 475 12.2 LL* Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 12.3 Mimetic Reformulation of LeastSquares Finite Element Methods . 483 12.4 Collocation LeastSquares Finite Element Methods . . . . . . . . . . . . . 488 12.5 Restricted LeastSquares Finite Element Methods . . . . . . . . . . . . . . 490 12.6 OptimizationBased LeastSquares Finite Element Methods . . . . . . 492 12.7 LeastSquares Finite Element Methods for Advection–Diffusion–Reaction Problems . . . . . . . . . . . . . . . . . . 494 12.8 LeastSquares Finite Element Methods for HigherOrder Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 12.9 LeastSquares Finite Element Methods for Div–Grad–Curl Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 12.10 Domain Decomposition LeastSquares Finite Element Methods . . . 507 12.11 LeastSquares Finite Element Methods for MultiPhysics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 12.12 LeastSquares Finite Element Methods for Problems with Singular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 12.13 Treffetz LeastSquares Finite Element Methods . . . . . . . . . . . . . . . . 521 12.14 A Posteriori Error Estimation and Adaptive Mesh Refinement . . . . 523 12.15 LeastSquares Wavelet Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 12.16 Meshless LeastSquares Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Part V Supplementary Material A Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 A.1 General Notations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 A.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 A.2.1 The Sobolev Spaces H s (Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . 536 A.2.2 Spaces Related to the Gradient, Curl, and Divergence Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 A.3 Properties of Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 A.3.1 Embeddings of C(Ω ) ∩ D(Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . 547 A.3.2 Poincaré–Friedrichs Inequalities . . . . . . . . . . . . . . . . . . . . . . 548 A.3.3 Hodge Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 A.3.4 Trace Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 xxii Contents B Compatible Finite Element Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 B.1 Formal Definition and Properties of Finite Element Spaces . . . . . . . 554 B.2 Finite Element Approximation of the De Rham Complex . . . . . . . . 557 B.2.1 Examples of Compatible Finite Element Spaces . . . . . . . . . 559 B.2.2 Approximation of C(Ω ) ∩ D(Ω ) . . . . . . . . . . . . . . . . . . . . . . 567 B.2.3 Exact Sequences of Finite Element Spaces . . . . . . . . . . . . . . 569 B.3 Properties of Compatible Finite Element Spaces . . . . . . . . . . . . . . . . 571 B.3.1 Discrete Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 B.3.2 Discrete Poincaré–Friedrichs Inequalities . . . . . . . . . . . . . . . 576 B.3.3 Discrete Hodge Decompositions . . . . . . . . . . . . . . . . . . . . . . 577 B.3.4 Inverse Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 B.4 Norm Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 B.4.1 QuasiNormEquivalent Approximations . . . . . . . . . . . . . . . 581 B.4.2 NormEquivalent Approximations . . . . . . . . . . . . . . . . . . . . . 582 C Linear Operator Equations in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 585 C.1 Auxiliary Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 C.2 Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 D The Agmon–Douglis–Nirenberg Theory and Verifying its Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 D.1 The Agmon–Douglis–Nirenberg Theory . . . . . . . . . . . . . . . . . . . . . . 593 D.2 Verifying the Assumptions of the AgmonDouglisNirenberg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 D.2.1 Div–Grad Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 D.2.2 Div–Grad–Curl Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 D.2.3 Div–Curl Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 D.2.4 The Velocity–Vorticity–Pressure Formulation of the Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 D.2.5 The Velocity–Stress–Pressure Formulation of the Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 Chapter 1 Classical Variational Methods In order to provide a context for leastsquares finite element methods (LSFEMs) and a means for judging their effectiveness, we briefly review, in this chapter and the next, other finite element approaches. In this chapter, we focus on what we refer to as “classical” finite element methods, a designation that is admittedly somewhat arbitrary, but that is nonetheless generally accepted for the methods we include in the sonamed class. Finite element methods belong to the class of variational methods. As such, they are quasiprojections1 whose properties are intrinsically connected to an underlying variational formulation. Thus, it is instructive to establish, in an abstract setting, taxonomies of variational formulations that provide the basis for defining variational methods, including finite element methods. We begin the chapter with a review of variational formulations for abstract linear operator equations. We distinguish between weakly and strongly coercive problems and also subdivide the formulations into those that are related to an optimization problem and those that are not. The taxonomy of variational formulations induces a like taxonomy of the associated variational methods, i.e., the quasiprojections that serve to define approximations of solutions of the variational equations. We explain how the properties of the variational formulations affect those of the associated variational methods. We apply our taxonomy to examine finite element methods for several canonical examples of partial differential equation (PDE) problems. We use these examples to show the influence that the underlying variational setting has on each discretization method and to explain the reasons that led to interest in the use of alternative variational formulations, including leastsquares principles. Because the aim of this chapter is to explain and fix important notions that surface regularly throughout the book, we keep the discussion at a fairly informal level. 1 We explain what we mean by a quasiprojection in Remark 1.1. P.B. Bochev, M.D. Gunzburger, LeastSquares Finite Element Methods, Applied Mathematical Sciences 166, DOI: 10.1007/b13382 1, c Springer Science+Business Media LLC 2009 3 4 1 Classical Variational Methods 1.1 Variational Methods for Operator Equations Given Banach spaces X and Y , a bounded linear operator2 Q from X into Y , and an element f ∈ Y , consider the abstract, linear operator equation find u ∈ X such that Qu = f in Y . (1.1) Typically, computational solutions of (1.1) rely on a process, called discretization, that converts this problem into a parameterized family3 Qh~uh = ~f h (1.2) of systems of linear algebraic equations for the unknown vector ~uh .4 The main goal in the design of a discretization process is to obtain parametrized families (1.2) of discrete problems whose solution sequence {~uh }h>0 converges, in some sense as h → 0, to an exact solution u ∈ X of (1.1). For variational methods, the conversion of (1.1) into a discrete problem (1.2) begins with the reformulation of (1.1) into a suitable variational equation. An abstract variational equation or variational formulation is defined in terms of two Hilbert spaces U and W , a continuous bilinear form Q(·, ·) : U × W 7→ R, and a bounded linear functional F(·) : W 7→ R, and is given by find u ∈ U such that Q(u, w) = F(w) ∀w∈W . (1.3) The space U in which we seek the solution u is referred to as the trial space and the space W is referred to as the test space. Variational equations such as (1.3) are often called weak formulations or generalized formulations5 of operator equations such as (1.1). An operator equation such as (1.1) may be reformulated into several different variational equations of the type (1.3); the specifics of which particular variational equation one uses, i.e., the relationships between the spaces X,Y and U,W and between the operator Q and the bilinear form Q(·, ·), usually depend on the nature of the operator equation and practicality issues. In Section 1.4, we provide examples of the reformulation process. 2 For the class of problems of interest to us, i.e., PDEs, Q is a partial differential operator. Usually, h is related to the number of degrees of freedom used in the definition of an approximate solution. In the finite element context, it is some measure of the size of the grid used to define the finite element spaces. 4 The connection between the components of the vector ~ uh and the solution u of (1.1) depends on the specific discretization process used. For finite difference methods, the components of ~uh are associated with approximations of u on a set of grid points. For finite element methods, those components are the coefficients of an expansion of an approximate solution uh in terms of a basis for a finitedimensional approximating subspace. For finite volume methods, these components are usually some averaged quantities related to u and associated with the edges, faces, or cells in a grid. 5 The origin of these terminologies is that, in the PDE setting, variational equations often have solutions that are not smooth enough to be pointwise solutions of the corresponding PDE. 3 1.1 Variational Methods for Operator Equations 5 On the other hand, the continuous bilinear form Q(·; ·) implies the existence of e : U 7→ W ∗ through the relation an operator6,7 Q e wiW ∗ ,W Q(u, w) = hQu, ∀ u ∈ U, w ∈ W . Similarly, the bounded linear functional F(·) implies the existence of an element fe ∈ W ∗ through the relation F(w) = h fe, wiW ∗ ,W ∀w ∈ W . Then, (1.3) is equivalent to the problem find u ∈ U e = fe in W ∗ . such that Qu (1.4) Because the pairs of spaces X,Y and U,W ∗ are not necessarily the same, it is clear e 6= Q and fe 6= f .8 that, in general, Q One of the key reasons for the popularity of variational methods in general and finite element methods in particular is the ease with which variational equations such as (1.3) can be used to obtain parameterized discrete systems such as (1.2). The discretization process is simple to describe. A discrete variational equation consists of the same bilinear form and functional9 as in (1.3), but restricted to two families of finitedimensional subspaces10 U h ⊂ U and W h ⊂ W . Thus, we obtain the family of discrete weak formulations find uh ∈ U h such that Q(uh , wh ) = F(wh ) ∀ wh ∈ W h . (1.5) To connect (1.5) with the parameterized linear algebraic system (1.2), we choose h h h bases {φ jh }Nj=1 and {ψih }M i=1 for U and W , respectively. We then write u = ∑Nj=1 ~uhj φ jh and set Qhij = Q(φ jh , ψih ) and ~fih = F(ψih ). Then, ~uh ∈ RN , Qh ∈ RM×N , ~f h ∈ RM , and, for every h > 0, (1.5) is equivalent to the linear system (1.2). Throughout, W ∗ denotes the dual space of a Hilbert space W , i.e., W ∗ is the space of all bounded linear functionals on W . Then, W ∗ itself is a Hilbert space. Also, h·, ·iW ∗ ,W denotes the duality pairing between elements of W and its dual space W ∗ . The inner product on W is denoted by (·, ·)W . 7 We adopt the notation that, e.g., Q e denotes the operator induced by the bilinear form appearing in a weak formulation of a PDE that is defined in terms of the operator Q. 8 For example, (1.1) could represent a PDE for u ∈ X that holds pointwise in a domain so that e could involve generalized derivatives and Q involves classical derivatives. On the other hand, Q then u ∈ U is viewed as a generalized solution of (1.1), i.e., one that does not satisfy (1.1) in a e and Q pointwise manner. Thus, although formally (1.1) and (1.4) may look alike, the operators Q have different domains and ranges. 9 It is possible to define discretizations based on bilinear forms and functionals that are different from those in (1.3), e.g., meshdependent bilinear forms arising from the use of quadrature rules to approximate integrals. However, for brevity, we do not discuss such discretizations here. 10 Whenever the inclusion U h ⊂ U holds, the resulting discrete approximations are referred to as being conforming. Nonconforming approximations for which U h 6⊂ U are also used in practice. Unless explicitly noted, in this book, we confine our presentation to conforming approximations. 6 6 1 Classical Variational Methods Remark 1.1 Because W h ⊂ W , (1.3) implies that Q(u, wh ) = F(wh ) ∀ wh ∈ W h ⊂ W . Together with (1.5), this relation implies that Q(uh , wh ) = Q(u, wh ) ∀ wh ∈ W h . (1.6) Thus, if u ∈ U and uh ∈ U h are solutions of (1.3) and (1.5), respectively, then (1.6) has the appearance of defining a projection uh ∈ U h ⊂ U of u ∈ U. This is what we mean when we say that finite element methods define quasiprojections. In fact, in general, (1.6) does not define a true projection; indeed that is the case only if the bilinear form Q(·, ·) defines an inner product, a situation for which a necessary but not sufficient condition is W = U. Note that (1.6) can be rewritten as Q(u − uh , wh ) = 0 ∀ wh ∈ W h . (1.7) This relation is often referred to as a Galerkin orthogonality relation for the error u − uh ; of course, it defines a true orthogonality relation only if the bilinear form Q(·, ·) defines an inner product. 2 Finite element methods are distinguished from other variational methods by the manner in which one defines the approximation spaces U h and W h . Construction of a finite element space begins with partitioning the problem domain into finite elements.11 A locally supported basis is then obtained by mapping polynomial functions defined on a standard reference element to every element in the domain partition. For further discussion of finite element spaces, see Appendix B. An important special class of variational problems is provided by mixed variational problems. In this setting, we have two Hilbert spaces V and S, two continuous bilinear forms a(·, ·) : V ×V 7→ R and b(·, ·) : V × S 7→ R , (1.8) two bounded linear functionals D(·) : V 7→ R and G(·) : S → 7 R, and the mixed variational problem ( find {v, p} ∈ V × S a(v, r) + b(r, p) = D(r) ∀r ∈ V (1.9) such that b(v, s) = G(s) ∀s ∈ S. 11 In two dimensions, triangles or quadrilaterals are used; in three dimensions, tetrahedrons, rectangular parallelepipeds, or hexahedrons are among the finite elements commonly employed. Although the explicit use of finite elements as building blocks for the approximation spaces is still the prevailing way of developing and using finite element methods, finite element methods have a much broader scope and encompass any quasiprojection scheme that uses approximation spaces with locally supported bases. This notion includes, e.g., meshless methods for which a traditional subdivision of a domain into finite elements is not explicitly used in the definition of the approximation space. 1.1 Variational Methods for Operator Equations 7 Using the duality pairings between the spaces V and S and their dual spaces V ∗ and S∗ , respectively, the continuous bilinear forms in (1.8) give rise to the bounded linear operators Ae : V 7→ V ∗ , Be : V 7→ S∗ , and Be∗ : S 7→ V ∗ defined by e riV ∗ ,V a(v, r) = hAv, ∀ v, r ∈ V and e siS∗ ,S = hBe∗ s, viV ∗ ,V b(v, s) = hBv, ∀ v ∈ V, s ∈ S , respectively. The bounded linear functionals D(·) and G(·) respectively give rise to the functions de∈ V ∗ and ge ∈ S∗ defined by e riV ∗ ,V D(r) = hd, ∀v ∈ V Then, (1.9) is equivalent to ( and G(s) = he g, siS∗ ,S e + Be∗ p = de Av in V ∗ e Bv in S∗ . = ge ∀s ∈ S. (1.10) Remark 1.2 Analogous to the relation between (1.1) and (1.4), (1.10) or, equivalently, (1.9), can be viewed as a weak or generalized formulation of a system of PDEs of the form ( Av + B ∗ p = d (1.11) Bv = g, where these equations are thought of as holding pointwise. 2 By setting U = W = V × S, u = {v, p}, w = {r, s}, F({r, s}) = D(r) + G(s), and Q({v, p}, {r, s}) = a(v, r) + b(r, p) + b(v, s) , (1.9) assumes the form (1.3) and, by setting ! e Be∗ A e= Q and Be 0 fe = de , ge (1.12) (1.13) (1.10) takes the form (1.4). The parameterized linear system (1.2) also has a special structure in the case of mixed variational problems. To define this system, we choose conforming subspaces V h ⊂ V and Sh ⊂ S and restrict (1.9) to these subspaces, i.e., we solve the problem: ( find {vh , ph } ∈ a(vh , rh ) + b(rh , ph ) = D(rh ) ∀ rh ∈ V h (1.14) V h × Sh such that b(vh , sh ) = G(sh ) ∀ sh ∈ Sh . h h If {ξ jh }Nj=1 and {η hj }M j=1 denote bases for V and S , respectively, then (1.14) is equivalent to the linear system 8 1 Classical Variational Methods Ah Bh Bh 0 T ! ~vh ~ph = d~h ~gh , (1.15) where vh = ∑Nj=1 ~vhj ξ jh and ph = ∑M phj η hj and (·)T denotes the transpose matrix. j=1 ~ In (1.15), we have that Ahij = a(ξ jh , ξih ) for i, j = 1, . . . , N , Bhij = b(ξ jh , ηih ) for i = 1, . . . , M, j = 1, . . . , N, d~ih = D(ξih ) for i = 1, . . . , N, and ~ghi = G(ηih ) for i = 1, . . . , M . Thus, the coefficient matrix Qh in the parameterized linear system (1.2) is given by ! h Bh T A h Q = . Bh 0 1.2 A Taxonomy of Classical Variational Formulations We now discuss the basic classes of variational formulations upon which classical finite element methods are based and the conditions that ensure the wellposedness of the weak problems (1.3) and (1.9). We then explore the connections between the variational formulations of Section 1.1 and optimization problems. 1.2.1 Weakly Coercive Problems A bilinear form Q(·, ·) : U ×W 7→ R is called continuous if Q(u, w) ≤ αkukU kwkW ∀ u ∈ U, w ∈ W , (1.16) where α < ∞, and is called weakly coercive if12 inf sup u∈U w∈W Q(u, w) ≥β kukU kwkW (1.17) and 12 We should more precisely write (1.17) and similar expressions as inf sup u∈U, u6=0 w∈W, w6=0 Q(u, w) ≥β. kukU kwkW For the sake of economy of notation, we omit the obvious necessity to have u 6= 0, and so on. 1.2 A Taxonomy of Classical Variational Formulations inf sup w∈W u∈U 9 Q(u, w) > 0, kukU kwkW (1.18) where β > 0.13 The following theorem, often referred to as the Nečas theorem (see [280,281]), shows that (1.16)–(1.18) are sufficient14 to guarantee that the variational problem (1.3) is well posed. Theorem 1.3 Let F(·) denote a bounded linear functional on W and let Q(·, ·) denote a continuous and weakly coercive bilinear form on U × W , i.e., (1.16)–(1.18) hold. Then, the abstract variational problem (1.3) has a unique solution u ∈ U. Moreover, that solution satisfies kukU ≤ 1 kFkW ∗ β 2 so that it depends continuously on the data.15 1.2.2 Strongly Coercive Problems If W 6= U, the most one can expect from a bilinear form is weak coercivity. However, if W = U, i.e., if the test space W and the trial space U are the same, a stronger notion of coercivity can be defined. Note that if W = U, then (1.16) becomes Q(u; v) ≤ αkukU kvkU ∀ u, v ∈ U . (1.19) 13 It is clear why (1.17) and (1.18) are referred to as inf–sup conditions. An equivalent way to express, e.g., (1.17) is Q(u, w) sup ≥ β kukU ∀u ∈ U . w∈W kwkW Because of its equivalence to (1.17), we also refer to conditions of this type as inf–sup conditions. 14 The necessity of these conditions can also be demonstrated. 15 It is instructive to consider an example in finitedimensional spaces for which U = RN , W = e ∈ RM×N , ~f ∈ RM , Q(~u; ~w) = ~wT Q~ e u, and F(~w) = ~wT ~f . With these identifications, W ∗ = RM , Q e u = ~f . In one can then verify that the abstract problem (1.3) is equivalent to the linear system Q~ e this case, the weak coercivity conditions (1.17) and (1.18) imply that Q has full row and column ranks, i.e., it is a square invertible matrix. Note the distinction between this example, where the problemdefining operator is finite dimensional, and a parameterized linear system such as (1.2). In the former case, the given problem is a linear system with a coefficient matrix having fixed dimensions. In the case of (1.2), the dimensions of the coefficient matrix grow as the parameter h tends to zero; we have emphasized this point by attaching the superscript h to the matrix Qh in (1.2). This distinction between the coefficient matrices is important because, in the case of (1.2), it is not enough to know that the coefficient matrix is invertible for any fixed value of h; one also needs to know about the uniformity of the invertibility, e.g., the uniformity of a bound on the inverse matrix as h → 0. Of course, this last point does not apply to the example discussed in the previous paragraph. 10 1 Classical Variational Methods A bilinear form Q(·, ·) : U × U 7→ R is called strongly coercive or Uelliptic if, for some β > 0, Q(u; u) ≥ β kukU2 ∀u ∈ U . (1.20) It is easy to verify that (1.20) implies the weak coercivity conditions (1.17) and (1.18), i.e., strong coercivity implies weak coercivity; the converse is not true. The classical Lax–Milgram lemma shows that strongly coercive problems are well posed.16 Corollary 1.4 Let F(·) denote a bounded linear functional on U and Q(·, ·) denote a continuous, strongly coercive bilinear form on U ×U, i.e., (1.19) and (1.20) hold. Then, the abstract variational problem (1.3) has a unique solution. Moreover, that solution satisfies 1 kukU ≤ kFkU ∗ β so that it depends continuously on the data.17 2 1.2.3 Mixed Variational Problems Mixed variational problems provide an example of weakly coercive problems and therefore they can be analyzed using Theorem 1.3.18 However, due to their special structure, it is advantageous to state the hypotheses needed to guarantee the wellposedness of problems such as (1.9) in terms of the constituent bilinear forms and linear functionals. To this end, we need to use the subspace Z = {v ∈ V  b(v, s) = 0 ∀ s ∈ S} ⊂ V (1.21) e consisting of those elements v ∈ V that belong to the null space of the operator B. The following generalization of the Brezzi theorem (see [83,87,183]) establishes hypotheses for which the mixed variational problem (1.9) is well posed. Theorem 1.5 Given the Hilbert spaces V and S, the bounded linear functionals D(·) and G(·) on V and S, respectively, and the bilinear forms a(·, ·) and b(·, ·) on 16 Obviously, because strong coercivity implies weak coercivity, Corollary 1.4 is simply a special case of Theorem 1.3. Here, we state it as a separate result because of its historical importance. 17 We again visit the finitedimensional setting. Let U = W = W ∗ = RN , Q e ∈ RN×N , ~f ∈ RN , e u, and F(~w) = ~wT ~f . A simple argument shows that the strong coercivity of Q(·, ·) is Q(~u; ~w) = ~wT Q~ e being a real matrix with a positive definite symmetric part. A real, positive definite equivalent to Q matrix is always invertible. This is the algebraic version of the statement that strong coercivity implies weak coercivity. On the other hand, because a matrix does not have to be positive definite to be invertible, it is clear that strong coercivity is a sufficient but not necessary condition for guaranteeing the results of Theorem 1.3. 18 Indeed, if a(·, ·) and b(·, ·) satisfy the hypotheses of Theorem 1.5 below, then it can be shown that the compound form Q(·, ·) defined in (1.12) satisfies (1.16)–(1.18). 1.2 A Taxonomy of Classical Variational Formulations 11 V × V and V × S, respectively. Assume that the bilinear forms a(·, ·) and b(·, ·) are continuous, i.e., for some constants αa < ∞ and αb < ∞, a(v, r) ≤ αa kvkV krkV ∀ v, r ∈ V (1.22) ∀ v ∈ V, s ∈ S . (1.23) and b(v, s) ≤ αb kvkV kskS e i.e., Assume further that a(·, ·) is weakly coercive on the null space of B, inf sup v∈Z r∈Z a(v, r) ≥ βa kvkV krkV a(v, r) inf sup > 0, r∈Z v∈Z kvkV krkV (1.24) where βa > 0, and that b(·, ·) satisfies the inf–sup condition inf sup s∈S v∈V b(v, s) ≥ βb , kvkV kskS (1.25) where βb > 0. Then, the mixed variational problem (1.9) has a unique solution {v, p} ∈ V × S and that solution satisfies kvkV + kpkS ≤ C(kDkV ∗ + kGkS∗ ) so that it depends continuously on the data.19 2 Null space methods The problem (1.9) can, in principle, be “simplified” by restricting it to an affine space on which the second equation is satisfied. For the sake of simplicity, we assume that G(·) = 0 in (1.9)20 so that now we have that b(v, s) = 0 for all s ∈ S, i.e., v ∈ Z. We then use, in the first equation in (1.9), test functions r ∈ Z ⊂ V so that b(r, p) = 0. Then, (1.9) is reduced to the problem Considering again the finitedimensional setting, let V = V ∗ = RN , S = S∗ = RM , Ae ∈ RN×N , e ev, b(~v,~s) = ~sT B~ ev, D(~r) =~rT d, ~ and G(~s) = ~sT~g. It is B ∈ RM×N , ~v ∈ RN , ~p ∈ RM , a(~v,~r) =~rT A~ e not difficult to show that (1.22)–(1.25) imply that the matrix B has full row rank M and that the e i.e., that the problem Z T AZ~ e q = Z T d~ has a unique matrix Ae is invertible on the null space of B, solution ~q ∈ RN−M for any d~ ∈ RN , where Z ∈ RN×(N−M) is a matrix whose columns form a basis e where N(·) denotes the null space. These conditions on the matrices Ae and Be in turn for N(B), e ∈ R(M+N)×(M+N) defined in (1.13) is invertible. Because clearly that guarantee that the matrix Q e satisfies the weak coercivity conditions matrix is not positive definite, we see that in this case Q (1.17) and (1.18). 20 The more general case of G(·) 6= 0 can, in principle, be reduced to this case by subtracting from v any particular solution vb ∈ V satisfying the equation b(b v, s) = G(s) for all s ∈ S. 19 12 1 Classical Variational Methods find v ∈ Z ⊂ V such that a(v, r) = D(r) ∀ r ∈ Z . (1.26) Note that (1.26) is of the form (1.3) with U = W = Z, Q(·, ·) = a(·, ·), and F(·) = D(·). Under the same hypotheses as those of Theorem 1.5, the problem (1.26) is a weakly coercive variational problem; indeed, (1.22) implies that the bilinear form a(·, ·) is continuous and (1.24) is exactly the definition of weak coercivity whenever that form is restricted to act on Z × Z. Then, by Theorem 1.3, we have that (1.26) has a unique solution v ∈ Z, and, because Z ⊂ V and V ∗ ⊂ Z ∗ , that solution satisfies kvkV ≤ β1a kDkV ∗ . If one wishes to also determine the variable p in (1.9), one need only solve, after determining v ∈ Z from (1.26), the problem: find p ∈ S such that b(r, p) = D(r) − a(v, r) ∀ r ∈ Z ⊥ ⊂ V . (1.27) The continuity condition (1.23) and the inf–sup condition (1.25) not only guarantee that Z ⊥ is well defined, but also that the problem (1.27) has a unique solution p ∈ S and that solution satisfies kpkS ≤ CkDkV ∗ ; see, e.g., [183], for details.21 1.2.4 Relations Between Variational Problems and Optimization Problems If we make further assumptions on the bilinear form Q(·, ·) of the problem (1.3) and the bilinear form a(·, ·) of the problem (1.9), we can relate those two problems to optimization problems. Unconstrained optimization problems (the Rayleigh–Ritz setting) We first consider the classical Rayleigh–Ritz setting in which strongly coercive variational problems can be associated with the minimization of energy functionals. Proposition 1.6 Assume that the hypotheses of Corollary 1.4 hold. Assume further that the bilinear form Q(·, ·) is symmetric, i.e., that Q(u, w) = Q(w, u) for all u, w ∈ U. Then, the problem (1.3) is equivalent22 to the unconstrained optimization problem 21 With G(·) = 0, in the finitedimensional setting we have been following, we have that (1.26) has e q = Z T d~ for ~q ∈ RN−M . The hypotheses of Theorem 1.5 imply that the matrix the form Z T AZ~ e ∈ R(N−M)×(N−M) is invertible. Then, after solving for ~q, ~v ∈ RN satisfying (1.9) is found Z T AZ by setting ~v = Z~q. If the matrix Z⊥ ∈ RN×M has columns that form a basis for Z ⊥ , then the TB eT~p = Z T d~ − Z T A~ e problem (1.27) for ~p ∈ RM takes the form Z⊥ ⊥ ⊥ v; the hypotheses of Theorem T T M×M e 1.5 imply that the matrix Z⊥ B ∈ R is invertible. 22 Equivalence of two problems means that solutions of one problem solve the other and conversely. 1.2 A Taxonomy of Classical Variational Formulations find u ∈ U 13 1 that minimizes J(u; F) = Q(u; u) − F(u) . 2 2 (1.28) In fact, (1.3) is the firstorder necessary condition that solutions u ∈ U of the unconstrained minimization problem (1.28) are required to satisfy. In the RayleighRitz e associated with the bilinear form Q(·, ·) is symmetric and setting, the operator Q positive definite. Note also that, in this setting, the discrete problem (1.5) is equivalent to the discrete optimization problem find uh ∈ U h 1 that minimizes J(uh ; F) = Q(uh ; uh ) − F(uh ) . 2 It is important to note that in the RayleighRitz setting, the bilinear form Q(·, ·) defines an equivalent inner product23 on the Hilbert space U and the functional J(·; 0) defines an equivalent norm24 on U. These observations follow from the symmetry of Q(·, ·) and the relation β kukU2 ≤ 2J(u; 0) = Q(u; u) ≤ αkukU2 which in turn follows easily from (1.19) and (1.20). Constrained optimization problems We now consider the relation between mixed variational principles and constrained optimization problems. Proposition 1.7 Assume that the hypotheses of Theorem 1.5 hold. Assume further that the bilinear form a(·, ·) is symmetric, nonnegative, and strongly coercive on the e i.e., that null space of B, a(v, r) = a(r, v) and a(v, v) ≥ 0 for all v, r ∈ V (1.29) and a(v, v) ≥ βa kvkV2 for all v ∈ Z , (1.30) where βa > 0. Then, the problem (1.9) is equivalent to the constrained optimization problem find v ∈ V that minimizes subject to 23 1 J(v; D) = a(v, v) − D(v) 2 b(v, s) = G(s) ∀ s ∈ S . (1.31) 2 Thus, in the RayleighRitz setting, (1.6) implies that the approximate solution uh ∈ U h is a true projection of u ∈ U with respect to the inner product Q(·, ·). Similarly, in that case, (1.7) implies that the error u − uh is truly orthogonal to all elements of the approximating space W h = U h , again with respect to the inner product Q(·, ·). In the more general setting of Section 1.1, (1.6) does not define a true projection and (1.7) does not define a true orthogonality relation. 24 The norm induced by the inner product Q(·, ·) is often referred to as the energy norm. 14 1 Classical Variational Methods In fact, if we enforce the constraint in (1.31) by introducing the Lagrange multiplier p ∈ S and the Lagrangian functional L(v, p; D, G) = J(v; D) − b(v, p) + G(p) , then the equations in (1.9) are the firstorder necessary conditions that saddle points {v, p} ∈ V × S of L(·, ·; ·, ·) are required to satisfy. In the setting of Proposition 1.7, the operator Ae associated with the bilinear form a(·, ·) is symmetric and positive semidefinite; it is also positive definite with respect to the null space of the operator Be associated with the bilinear form b(·, ·). However, even in this case, the composite e defined in (1.13), although symmetric, is indefinite so that the associated operator Q bilinear form Q(·, ·) remains weakly coercive. This, of course, is the nature of saddle point problems. Note also that in the setting of constrained optimization problems, the discrete problem (1.14) is equivalent to the discrete constrained optimization problem find vh ∈ V h that minimizes subject to 1 J(vh ; D) = a(vh , vh ) − D(vh ) 2 b(vh , sh ) = G(sh ) ∀ sh ∈ Sh . A second approach for handling the constraint in the problem (1.31) is to restrict the minimization problem to an affine space on which the constraint is satisfied, i.e, to use the null space method described in Section 1.2.3. With G(·) = 0, the hypotheses of Theorem 1.7 imply that (1.26) are the firstorder necessary conditions for v ∈ Z to solve the reduced, unconstrained optimization problem find v ∈ Z 1 that minimizes J(v; D) = a(v, v) − D(v) 2 (1.32) that is equivalent to the constrained optimization problem (1.31). Under the same hypotheses as those of Theorem 1.7, the problem (1.26) is now a strongly coercive variational problem; this should be contrasted to the situation for Lagrange multiplier methods for which the same hypotheses lead to the weakly coercive variational problem involving the bilinear form (1.12). The unconstrained optimization problem (1.32) is of the RayleighRitz form (1.28) and therefore the former may be discretized in the same manner as the latter. One chooses a conforming subspace Zeh ⊂ Z and then restricts the problem (1.32) to that subspace, i.e., one solves the discrete unconstrained optimization problem ) find vh ∈ Zeh ⊂ Z ⊂ V 1 J(vh ; D) = a(vh , vh ) − D(vh ) . 2 that minimizes The firstorder necessary conditions corresponding to this problem are given by (1.45). 1.3 Approximation of Solutions of Variational Problems 15 1.3 Approximation of Solutions of Variational Problems Problems such as (1.3) are generally referred to as Galerkin25 formulations of the operator equation (1.1); finite element methods based on (1.5) or, equivalently, (1.2), are referred to as Galerkin finite element methods. As seen in Sections 1.2.1 and 1.2.2, Galerkin formulations can involve either a strongly or weakly coercive bilinear form Q(·, ·). The Rayleigh–Ritz setting refers to the special case for which this bilinear form is not only strongly coercive, but is also symmetric. MixedGalerkin formulations of the operator equation (1.11) refer to the system (1.9); finite element methods based on (1.14) or, equivalently, (1.15), are therefore known as mixedGalerkin finite element methods or, more simply, as mixed finite element methods. There is less need to distinguish between problems of the form (1.9) that are related to a constrained optimization problem and those that are not because in both cases one is led to weakly coercive problems. Recall that discretized variational problems are restrictions of variational problems to finitedimensional subspaces so that their wellposedness is subject to similar conditions as were imposed for the parent problems. In this section, we apply this observation to state sufficient stability conditions, first for discrete problems spawned from weakly and strongly coercive variational problems, and then for those spawned from mixed variational problems. It is important to note that, in some cases, a condition imposed on the discretized problem is automatically inherited from the corresponding condition imposed on the parent problem and, in other cases, it is not. 1.3.1 Weakly and Strongly Coercive Variational Problems For the discretization of weakly coercive problems, we have the following theorem that is sometimes referred to as Babuška’s theorem [19, 20]. Theorem 1.8 Let the spaces U and W , the bilinear form Q(·, ·), and the functional F(·) satisfy the hypotheses of Theorem 1.3. Let U h ⊂ U and W h ⊂ W be closed subspaces.26 Then, Q(·, ·) is continuous on U h ×W h . Assume also that the bilinear form Q(·, ·) satisfies the discrete inf–sup conditions Q(uh , wh ) ≥ βh h h uh ∈U h wh ∈W h ku kU kw kW (1.33) Q(uh , wh ) > 0, h h wh ∈W h uh ∈U h ku kU kw kW (1.34) inf and inf 25 sup sup Strictly speaking, the term “Galerkin formulation” refers to variational problems for which the test and the trial spaces coincide. In the general case when U 6= W , one speaks of “Petrov–Galerkin variational formulations.” For brevity, we use the former terminology in all cases. 26 Note that it is not required for the subspaces to be finite dimensional. 16 1 Classical Variational Methods where β h ≥ βb > 0. Then, for every h > 0, the discretized problem (1.5) has a unique solution uh ∈ U h . Moreover, that solution satisfies the stability estimate kuh kU ≤ 1 kFkW ∗ βb and, if u ∈ U denotes the unique solution of (1.3), the optimal error estimate27,28 α ku − uh kU ≤ 1 + inf ku − vh kU . 2 (1.35) h ∈U h b v β In Section 1.1 it is seen that the discretized problem (1.5) is equivalent to a parameterized family of linear systems (1.2). From this theorem, it follows that for every h > 0, the matrix Qh in (1.2) is square and nonsingular. This is basically all that can be inferred about the discretized equations that result from weakly coercive variational formulations. For the discretization of strongly coercive problems, we have the following result that is sometimes referred to as Céa’s lemma [82, 108, 123]. Theorem 1.9 Let the space U, the bilinear form Q(·, ·), and the linear functional F(·) satisfy the hypotheses of Corollary 1.4. Let U h ⊂ U be a closed subspace. Then, Q(·, ·) is continuous on U h ×U h and satisfies Q(uh , uh ) ≥ β kuh kU2 ∀ uh ∈ U h (1.36) and, for every h > 0, the discretized problem (1.5) has a unique solution uh ∈ U h . Moreover, that solution satisfies the stability estimate kuh kU ≤ 1 kFkU ∗ β and, if u ∈ U denotes the unique solution of (1.3), the optimal error estimate ku − uh kU ≤ α β inf ku − wh kU . wh ∈U h 2 (1.37) Remark 1.10 Remarkably, if, in addition, the bilinear form Q(·, ·) is symmetric, e.g., in the RayleighRitz setting of unconstrained optimization problems, the error estimate (1.37) can be improved to 27 An optimal error estimate is one for which the error in a finitedimensional approximation is bounded from above by a constant multiple of the error of the best approximation out of the same finitedimensional subspace. An approximate solution that satisfies an optimal error estimate is referred to as being optimally accurate. 28 It is important to keep in mind that optimality of an error estimate of a numerical method is not sufficient to guarantee convergence of the numerical solution uh to the exact solution u. The latter requires an additional approximability property of the discrete spaces. See Appendix B for further details. 1.3 Approximation of Solutions of Variational Problems ku − uh kU ≤ r α inf ku − vh kU ; β vh ∈U h 17 (1.38) see, e.g., [123]. That this is indeed an improvement over (1.37) follows from the fact that necessarily α ≥ β . 2 This theorem also asserts that, for every h > 0, the matrix Qh in (1.2) is square and nonsingular. However, the discrete coercivity (1.36) implies that, in addition to these properties, strongly coercive variational formulations give rise to positive definite linear systems. Furthermore, in the Rayleigh–Ritz setting, these systems are also symmetric. A cursory inspection of Theorems 1.8 and 1.9 points out an important difference between the conditions that guarantee the stability of discretizations of weakly and strongly coercive variational problems. In both cases, the continuity property of the bilinear form Q(·, ·) with respect to conforming discrete subspaces is inherited from the continuity property that holds on the parent spaces. Likewise, the strong coercivity property (1.36) with respect to U h ⊂ U automatically follows from the strong coercivity property (1.20) with respect to U. Thus, for strongly coercive variational problems, the conformity of the approximating subspace U h ⊂ U is all that is needed to guarantee the stability of the discrete variational problem and the optimal accuracy of the approximate solution, i.e., conformity of the spaces is necessary and sufficient for their compatibility. However, this is not true for weakly coercive variational problems; the inclusions U h ⊂ U and W h ⊂ W are not by themselves sufficient to guarantee that the discrete weak coercivity conditions (1.33) and (1.34) hold, even if the conditions (1.17) and (1.18) are known to hold, i.e., conformity of the spaces is necessary but not sufficient for their compatibility. Thus, in Theorem 1.8, the discrete weak coercivity conditions must be stated as additional hypotheses on the bilinear form Q(·, ·) and the discrete subspaces U h and W h . Let us examine these issues in more detail. Assume first that Q(·, ·) is continuous on U × W . Let uh and wh denote arbitrary elements belonging to the conforming discrete subspaces U h ⊂ U and W h ⊂ W , respectively. Obviously, uh ∈ U and wh ∈ W as well and (1.16) holds for uh and wh considered as elements of the parent spaces. Because uh and wh are arbitrary, we conclude that (1.16) holds on U h ×W h . The same argument can be used to show that if Q(·, ·) is strongly coercive on U ×U, then (1.36) holds on any conforming subspace U h ⊂ U. To understand why the weak coercivity properties with respect to the discrete spaces are not inherited from the analogous property on the parent spaces, note that an equivalent way to express (1.17) is that, given u ∈ U, there exists wu ∈ W such that Q(u, wu ) ≥ β kukU kwu kW . (1.39) 18 1 Classical Variational Methods Thus, given uh ∈ U h ⊂ U, we obviously have that uh ∈ U and so (1.39) implies that there exists a wuh ∈ W such that Q(uh , wuh ) ≥ β kuh kU kwuh kW . However, the function wuh is not guaranteed to belong to the discrete subspace W h . Unfortunately, it is this last property that is required for (1.33) to hold; see Figure 1.1. Fig. 1.1 The strong coercivity property is inherited on the subspace U h (left). The weak coercivity property is not inherited by the subspaces U h and W h (right) because the function wuh whose existence follows from (1.17) is not guaranteed to belong to W h . 1.3.2 Mixed Variational Problems In the case of mixed variational problems, the wellposedness of (1.14) is subject to the conditions (1.24) and (1.25) but restricted to the subspaces V h ⊂ V and Sh ⊂ S. As is the case in Section 1.3.1 for weakly coercive variational problems, the conformity of the subspaces V h and Sh is not sufficient for the discrete versions of (1.24) and (1.25) to follow directly from the corresponding conditions on the parent spaces. For this reason, the discrete versions of these conditions have to be stated as assumptions in the following generalization of a result found in, e.g., [83, 87, 183]. Theorem 1.11 Let the spaces V and S, the linear functionals D(·) and G(·), and the bilinear forms a(·, ·) and b(·, ·) satisfy the hypotheses of Theorem 1.5. Let V h ⊂ V and Sh ⊂ S denote conforming subspaces. Then, a(·, ·) and b(·, ·) are continuous on V h ×V h and V h × Sh , respectively. Define Z h ⊂ V h by Z h = {vh ∈ V h  b(vh , sh ) = 0 and assume that a(·, ·) is weakly coercive on Z h , i.e., ∀ sh ∈ Sh } (1.40) 1.3 Approximation of Solutions of Variational Problems 19 a(vh , rh ) ≥ βah h h vh ∈Z h rh ∈Z h kv kV kr kV inf sup a(vh , rh ) inf sup > 0, h h rh ∈Z h vh ∈Z h kv kV kr kV (1.41) where βah ≥ βba > 0. Assume that b(·, ·) verifies the LBB29 or discrete inf–sup condition b(vh , sh ) inf sup ≥ βbh , (1.42) h h sh ∈Sh vh ∈V h kv kV ks kS where βbh ≥ βbb > 0. Then, for every h > 0, the discrete problem (1.14) has a unique solution {vh , ph } ∈ V h × Sh . Moreover, for some C > 0 whose value is independent of h, that solution satisfies the stability estimate kvh kV + kph kS ≤ C(kDkV ∗ + kGkS∗ ) and, if {v, p} ∈ V × S denotes the solution of (1.9), the optimal error estimate kv − vh kV + kp − ph kS ≤ C inf kv − rh kV + inf kp − sh kS . 2 (1.43) rh ∈V h sh ∈Sh The mixed discrete problem (1.14) is equivalent to the parameterized family of linear systems (1.15). Similarly to Theorems 1.8 and 1.9, Theorem 1.11 also asserts that, for every h > 0, the 2 × 2 block matrix in this system is square and invertible. If, in addition, the parent mixed variational formulation is related to a constrained minimization problem, this matrix is also symmetric. However, in contrast with the strongly coercive setting, the matrix in (1.15) is always indefinite. It is instructive to examine (1.42) more closely. Just as is the case for (1.17)– (1.18) and (1.33)–(1.34), the conformity of the subspaces V h ⊂ V and Sh ⊂ S is not sufficient to guarantee that the inf–sup condition (1.25) automatically implies that the discrete inf–sup condition (1.42) holds; thus, the latter must be explicitly assumed in Theorem 1.11; see Figure 1.2. Clearly, the weak coercivity conditions in (1.24) for the bilinear form a(·, ·) with respect to the space Z do not immediately imply the like conditions in (1.41) with respect to the space Z h . But there is more to this particular issue. Suppose we replace the weak coercivity assumptions in (1.24) with the strong coercivity assumption (1.30) and then replace, in Theorem 1.11, the discrete weak coercivity assumptions in (1.41) by the discrete strong coercivity assumption a(vh , vh ) ≥ βah kvh kV2 29 ∀ vh ∈ Z h , (1.44) The terminology “LBB” originates from the facts that this condition was first explicitly discussed in the finite element setting for saddle point problems by Brezzi [83] and that it is a special case of the general weakcoercivity condition (1.17) first discussed for finite element methods by Babuška [19] and that, in the continuous setting of the Stokes equation, this condition was first proved to hold by Ladyzhenskaya; see [253]. 20 1 Classical Variational Methods Fig. 1.2 Conditions for the wellposedness of discrete saddlepoint problems. Coercivity on the null space (left) and the discrete inf–sup condition (right) must be imposed independently on the discrete spaces because 1) in general Z h 6⊂ Z and 2) the function w ph whose existence follows from the discrete inf–sup condition is not guaranteed to belong to V h . where βah ≥ βba > 0. Strong coercivity is inherited on subspaces; thus, the inclusion of (1.44) as a hypothesis for Theorem 1.11 might seem redundant in light of (1.30). However, although the latter condition does imply that the bilinear form a(·, ·) is coercive on any subspace of Z, it is generally the case that Z h 6⊂ Z. As a result, even for the case for which the bilinear form a(·, ·) is strongly coercive on the null space Z, e.g., for constrained optimization problems, (1.44) must also be imposed as a hypothesis on the discrete space Z h in Theorem 1.11; see Figure 1.2. Null space methods We are now in a position to discuss two approaches for using null space methods to “simplify” the discretization of (1.9). One can first reduce to the null space, i.e., derive the reduced problem (1.26), then choose a subspace Zeh ⊂ Z, and then restrict (1.26) to that subspace to obtain the discrete problem find vh ∈ Zeh ⊂ Z ⊂ V such that a(vh , rh ) = D(rh ) ∀ rh ∈ Zeh . (1.45) Alternately, one can discretize (1.9) using the subspaces V h ⊂ V and Sh ⊂ S to obtain (1.14) and then reduce the latter (discrete) problem using the discrete null space Z h ⊂ V h defined in (1.40). The discrete problems resulting from the two approaches are not the same. For one thing, we cannot choose Zeh = Z h because we want Zeh ⊂ Z but, in general, Z h 6⊂ Z. For another, in (1.45), one directly chooses a subspace Zeh of the null space Z without any need to introduce the intermediate spaces V h and Sh . The distinction between the reducethendiscretize and discretizethenreduce approaches becomes even more evident when we consider the corresponding pae rameterized linear systems. If {ξeh }Nz denotes a basis for Zeh ⊂ Z, then the solution j j=1 z vh ∈ Zeh of (1.45) can be expressed as vh = ∑Nj=1 ~zhj ξejh and then (1.45) is equivalent e 1.3 Approximation of Solutions of Variational Problems 21 to the linear system e h~zh = deh , A (1.46) e e h ∈ RNez ×Nez and deh ∈ RNez are respectively given by where~zh ∈ RNz and where A e h = a(ξeh ; ξeh ) A ij j i and dei h = D(ξeih ) ez . for i, j = 1, . . . , N Now, let Zh denote a matrix whose columns form a basis for N(Bh ). For the sake of simplicity, we assume that (1.42) is satisfied so that Bh has full row rank, dim(Z h ) = N − M, and Zh ∈ RN×(N−M) . Then, from (1.15), we easily obtain that T T Zh Ah~vh = Zh d~h . (1.47) Furthermore, the vector of coefficients ~vh ∈ RN (with respect to a chosen basis for V h ) corresponding to any function vh ∈ Z h can be expressed in the form ~vh = Zh~qh for some ~qh ∈ RN−M . Then, it follows from (1.47) that b h~qh = dbh , A where b h = Zh T Ah Zh A and (1.48) T dbh = Zh d~h . Clearly, the systems (1.46) and (1.48) are not the same. It would seem that, in the case for which the bilinear form a(·, ·) is strongly coercive on Z, finding approximate solutions of constrained optimization problems by solving the reduced problem (1.45) is preferable to solving (1.14). The application of the Lagrange multiplier rule results in the discrete mixedGalerkin formulation (1.14) that is necessarily weakly coercive.30 However, because, in general, the discrete null space Z h 6⊂ Z, it is clear that it cannot be used in (1.45). Thus, Zeh must be constructed separately, even if a stable pair {V h , Sh } for (1.14) is already available. On the other hand, if (1.30) holds and if one has available a subspace Zeh ⊂ Z, we have that the bilinear form a(·, ·) is strongly coercive with respect to Zeh so that the problem (1.45) has a unique solution, that solution depends continuously on the data, and satisfies the error estimate kv − vh kV ≤ C inf kv − rh kV , rh ∈Zeh where v denotes the solution of (1.26). Thus, the reduced formulation has the advantage of offering a strongly coercive setting for discrete variational methods for the constrained optimization problem (1.31). Unfortunately, in practice, it is often not an easy matter to construct subspaces of the space Z defined by (1.21). 30 An additional advantage of (1.45) over (1.14) is that the former does not involve ph , i.e., it involves fewer degrees of freedom. 22 1 Classical Variational Methods 1.4 Examples We now provide examples of variational problems that form the basis for classical finite element methods. We begin with the classification of these problems according to the taxonomy given in Section 1.2, and then infer the properties of the associated finite element methods from the discussions given in Section 1.3. We first consider some PDEs related to optimization problems, starting from the optimization problems themselves and working our way to the equivalent PDE formulations. The optimization problems are turned into weak formulations of the related PDEs by the application of standard techniques from the calculus of variations. For PDEs such as conservation laws and advectiondiffusionreaction problems that are not related to optimization problems, we cannot use the calculus of variations to obtain variational formulations. Instead, variational methods can be based on Galerkin principles that are derived directly from the PDEs. The paradigm of a Galerkin principle is formal residual “orthogonalization.” This process can, in principle, be applied to any PDE, even if there is no underlying optimization problem. Because of their universality, Galerkin principles have provided a natural and popular choice for extending finite element methods beyond PDE problems associated with optimization problems. On the other hand, if such an optimization problem exists, the associated optimality system can be recovered through the use of the Galerkin paradigm. In Sections 1.4.1 and 1.4.2 we respectively consider the Poisson equation and the equations of linear elasticity; both of these problems can be associated with unconstrained optimization problems. In Section 1.4.3, we consider the Stokes equations that can be associated with a constrained optimization problem. In Section 1.4.4 we consider the Helmholtz equation and in Section 1.4.5 a linear advectiondiffusionreaction equation; neither of these problems can be related to an optimization problem. Finally, in Section 1.4.6, we consider the nonlinear example of the Navier– Stokes equations which also cannot be associated with an optimization problem. 1.4.1 The Poisson Equation We consider the Dirichlet and Kelvin principles which give rise to two different variational formulations for the Poisson equation. The Dirichlet and Kelvin principles provide examples of unconstrained and constrained, respectively, optimization problems; the former leads to a symmetric, strongly coercive, Rayleigh–Ritz setting for the Poisson problem and the latter leads to a symmetric, weakly coercive, mixedGalerkin formulation for the same problem. We assume that the domain Ω ⊂ Rd has boundary ∂ Ω which is made up of two disjoint, nonempty parts Γ and Γ ∗ . 1.4 Examples 23 The Dirichlet principle Given31 f ∈ (HΓ1 (Ω ))∗ , consider the functional 1 JD (φ ; f ) = 2 Z 2 ∇φ  dΩ − Z (1.49) f φ dΩ Ω Ω and the Dirichlet principle min JD (φ ; f ) . (1.50) φ ∈HΓ1 (Ω ) Clearly, (1.50) defines an unconstrained optimization setting. The Euler–Lagrange condition corresponding to this principle is given by Z ∇φ · ∇ψ dΩ = Z Ω ∀ ψ ∈ HΓ1 (Ω ) . f ψ dΩ (1.51) Ω Letting U = W = HΓ1 (Ω ) and Z Q(φ , ψ) = ∇φ · ∇ψ dΩ Z F(ψ) = and Ω f ψ dΩ ∀ φ , ψ ∈ HΓ1 (Ω ) , Ω we see that (1.51) is of the form (1.3). Furthermore, it is well known [82, 123] that, in this case, the bilinear form Q(·, ·) is symmetric and strongly coercive on HΓ1 (Ω )× HΓ1 (Ω ) so that we find ourselves in the Rayleigh–Ritz unconstrained optimization setting discussed in Section 1.2.4. Furthermore, the error estimate (1.38) applies to solutions of the finite element discrete problem (1.5) corresponding to (1.51) for any conforming finite element approximation space U h ⊂ HΓ1 (Ω ). Equivalent PDE. Formally, by integration by parts, (1.51) is equivalent to the Poisson problem −∆ φ = f in Ω , (1.52) ∂φ φ = 0 on Γ , and = 0 on Γ ∗ . ∂n Thus, (1.51) provides a symmetric, strongly coercive weak formulation for the Poisson problem (1.52). The Kelvin principle Consider the functional JK (v) = 31 1 2 Z v2 dΩ Ω HΓ1 (Ω ) is the space {ψ ∈ H 1 (Ω )  ψ = 0 on Γ }. When Γ = ∂ Ω this space is denoted