Scientific Computation Editorial Board J.-J Chattot, Davis, CA, USA P Colella, Berkeley, CA, USA W E, Princeton, NJ, USA R Glowinski, Houston, TX, USA M Holt, Berkeley, CA, USA Y Hussaini, Tallahassee, FL, USA P Joly, Le Chesnay, France H.B Keller, Pasadena, CA, USA J.E Marsden, Pasadena, CA, USA D.I Meiron, Pasadena, CA, USA O Pironneau, Paris, France A Quarteroni, Lausanne, Switzerland and Politecnico of Milan, Italy J Rappaz, Lausanne, Switzerland R Rosner, Chicago, IL, USA P Sagaut, Paris, France J.H Seinfeld, Pasadena, CA, USA A Szepessy, Stockholm, Sweden M.F Wheeler, Austin, TX, USA Roland Glowinski Numerical Methods for Nonlinear Variational Problems With 82 Illustrations 123 Roland Glowinski University of Houston Dept Mathematics 4800 Calhoun Road Houston, TX 77004-3008, USA Reprint of the Hard cover edition published in 1984 ISBN 978-3-540-77506-5 e-ISBN 978-3-540-77801-1 DOI 10.1007/978-3-540-77801-1 Scientific Computation ISSN 1434-8322 Library of Congress Control Number: 2007942575 © 2008, 1984 Springer-Verlag Berlin Heidelberg This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for usemust always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: supplied by the author Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig, Germany Cover design: eStudioCalamar S.L., F Steinen-Broo, Girona, Spain Printed on acid-free paper 987654321 springer.com To my wife Angela and to Mrs Madeleine Botineau Preface When Herb Keller suggested, more than two years ago, that we update our lectures held at the Tata Institute of Fundamental Research in 1977, and then have it published in the collection Springer Series in Computational Physics, we thought, at first, that it would be an easy task Actually, we realized very quickly that it would be more complicated than what it seemed at first glance, for several reasons: The first version of Numerical Methods for Nonlinear Variational Problems was, in fact, part of a set of monographs on numerical mathematics published, in a short span of time, by the Tata Institute of Fundamental Research in its well-known series Lectures on Mathematics and Physics; as might be expected, the first version systematically used the material of the above monographs, this being particularly true for Lectures on the Finite Element Method by P G Ciarlet and Lectures on Optimization—Theory and Algorithms by J Cea This second version had to be more self-contained This necessity led to some minor additions in Chapters I-IV of the original version, and to the introduction of a chapter (namely, Chapter Y of this book) on relaxation methods, since these methods play an important role in various parts of this book For the same reasons we decided to add an appendix (Appendix I) introducing linear variational problems and their approximation, since many of the methods discussed in this book try to reduce the solution of a nonlinear problem to a succession of linear ones (this is true for Newton's method, but also for the augmented Lagrangian, preconditioned conjugate gradient, alternating-direction methods, etc., discussed in several parts of this book) Significant progress has been achieved these last years in computational fluid dynamics, using finite element methods It was clear to us that this second version had to include some of the methods and results whose efficiency has been proved in the above important applied field This led to Chapter VII, which completes and updates Chapter VI of the original version, and in which approximation and solution methods for some important problems in fluid dynamics are discussed, such as transonic flows for compressible inviscid fluids and the Navier-Stokes equations viii Preface for incompressible viscous fluids Like the original version, the main goal of this book is to serve as an introduction to the study of nonlinear variational problems, and also to provide tools which may be used for their numerical solution We sincerely believe that many of the methods discussed in this book will be helpful to those physicists, engineers, and applied mathematicians who are concerned with the solution of nonlinear problems involving differential operators Actually this belief is supported by the fact that some of the methods discussed in this book are currently used for the solution of nonlinear problems of industrial interest in France and elsewhere (the last illustrations of the book represent a typical example of such situations) The numerical integration of nonlinear hyperbolic problems has not been considered in this book; a good justification for this omission is that this subject is in the midst of an important evolution at the moment, with many talented people concentrating on it, and we think that several more years will be needed in order to obtain a clear view of the situation and to see which methods take a definitive lead, particularly for the solution of multidimensional problems Let us now briefly describe the content of the book Chapters I and II are concerned with elliptic variational inequalities (EVI), more precisely with their approximation (mostly by finite element methods) and their iterative solution Several examples, originating from continuum mechanics, illustrate the methods which are described in these two chapters Chapter III is an introduction to the approximation of parabolic variational inequalities (PVI); in addition, we discuss in some detail a particular PVI related to the unsteady flow of some viscous plastic media (Bingham fluids) in a cylindrical pipe In Chapter IV we show how variational inequality concepts and methods may be useful in studying some nonlinear boundary-value problems which can be reduced to nonlinear variational equations In Chapters V and VI we discuss the iterative solution of some variational problems whose very specific structure allows their solution by relaxation methods (Chapter V) and by decomposition-coordination methods via augmented Lagrangians (Chapter VI); several iterative methods are described and illustrated with examples taken mostly from mechanics Chapter VII is mainly concerned with the numerical solution of the full potential equation governing transonic potential flows of compressible inviscid fluids, and of the Navier-Stokes equations for incompressible viscous fluids We discuss the approximation of the above nonlinear fluid flow problems by finite element methods, and also iterative methods of solution of the approximate problems by nonlinear least-squares and preconditioned conjugate gradient algorithms In Chapter VII we also emphasize the solution of the Stokes problem by either direct or iterative methods The results of Preface ix numerical experiments illustrate the possibilities of the solution methods discussed in Chapter VII, which also contains an introduction to arc-lengthcontinuation methods (H B Keller) for solving nonlinear boundary-value problems with multiple solutions As already mentioned, Appendix I is an introduction to the theory and numerical analysis of linear variational problems, and one may find in it details (some being practical) about the finite element solution of such important boundary-value problems, like those of Dirichlet, Neumann, Fourier, and others In Appendix II we describe a finite element method with upwinding which may be helpful for solving elliptic boundary-value problems with large first-order terms Finally, Appendix III, which contains various information and results useful for the practical solution of the Navier-Stokes equations, is a complement to Chapter VII, Sec (Actually the reader interested in computational fluid mechanics will find much useful theoretical and practical information about the numerical solution of fluid flow problems—Navier-Stokes equations, in particular—in the following books: Implementation of Finite Element Methods for Navier-Stokes Equations by F Thomasset, and Computational Methods for Fluid Flow by R Peyret and T D Taylor, both published in the Springer Series in Computational Physics.) Exercises (without answers) have been scattered throughout the text; they are of varying degrees of difficulty, and while some of them are direct applications of the material in this book, many of them give the interested reader or student the opportunity to prove by him- or herself either some technical results used elsewhere in the text, or results which complete those explicitly proved in the book Concerning references, we have tried to include all those available to us and which we consider relevant to the topics treated in this book It is clear, however, that many significant references have been omitted (due to lack of knowledge and/or organization of the author) Also we apologize in advance to those authors whose contributions have not been mentioned or have not received the attention they deserve Large portions of this book were written while the author was visiting the following institutions: the Tata Institute of Fundamental Research (Bombay and Bangalore), Stanford University, the University of Texas at Austin, the Mathematical Research Center of the University of Wisconsin at Madison, and the California Institute of Technology We would like to express special thanks to K G Ramanathan, G H Golub, J Oliger, J T Oden, J H Nohel, and H B Keller, for their kind hospitality and the facilities provided for us during our visits We would also like to thank C Baiocchi, P Belayche, J P Benque, M Bercovier, H Beresticky, J M Boisserie, H Brezis, F Brezzi, J Cea, T F Chan, P G Ciarlet, G Duvaut, M Fortin, D Gabay, A Jameson, G x Preface Labadie, C Lemarechal, P Le Tallec, P L Lions, B Mercier, F Mignot, C S Moravetz, F Murat, J C Nedelec, J T Oden, S Osher, R Peyret, J P Puel, P A Raviart, G Strang, L Tartar, R Temam, R Tremolieres, V Girault, and O Widlund, whose collaboration and/or comments and suggestions were essential for many of the results presented here We also thank F Angrand, D Begis, M Bernadou, J F Bourgat, M O Bristeau, A Dervieux, M Goursat, F Hetch, A Marrocco, O Pironneau, L Reinhart, and F Thomasset, whose permanent and friendly collaboration with the author at INRIA produced a large number of the methods and results discussed in this book Thanks are due to P Bohn, B Dimoyat, Q V Dinh, B Mantel, J Periaux, P Perrier, and G Poirier from Avions Marcel Dassault/Breguet Aviation, whose faith, enthusiasm, and friendship made (and still make) our collaboration so exciting, who showed us the essence of a real-life problem, and who inspired us (and still do) to improve the existing solution methods or to discover new ones We are grateful to the Direction des Recherches et Etudes Techniques (D.R.E.T.), whose support was essential to our researches on computational fluid dynamics We thank Mrs Francoise Weber, from INRIA, for her beautiful typing of the manuscript, and for the preparation of some of the figures in this book, and Mrs Frederika Parlett for proofreading portions of the manuscript Finally, we would like to express our gratitude to Professors W Beiglbock and H B Keller, who accepted this book for publication in the Springer Series in Computational Physics, and to Professor J L Lions who introduced us to variational methods in applied mathematics and who constantly supported our research in this field Chevreuse September 1982 ROLAND GLOWINSKI Contents Some Preliminary Comments xiv CHAPTER I Generalities on Elliptic Variational Inequalities and on Their Approximation Introduction Functional Context Existence and Uniqueness Results for EVI of the First Kind Existence and Uniqueness Results for EVI of the Second Kind Internal Approximation of EVI of the First Kind Internal Approximation of EVI of the Second Kind Penalty Solution of Elliptic Variational Inequalities of the First Kind References 1 12 15 26 CHAPTER II Application of the Finite Element Method to the Approximation of Some Second-Order EVI Introduction An Example of EVI of the First Kind: The Obstacle Problem A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem A Third Example of EVI of the First Kind: A Simplified Signorini Problem An Example of EVI of the Second Kind: A Simplified Friction Problem A Second Example of EVI of the Second Kind: The Flow of a Viscous Plastic Fluid in a Pipe On Some Useful Formulae 27 27 27 41 56 68 78 96 CHAPTER III On the Approximation of Parabolic Variational Inequalities Introduction: References Formulation and Statement of the Main Results Numerical Schemes for Parabolic Linear Equations Approximation of PVI of the First Kind 98 98 98 99 101 479 Subject Index incompressible finite elasticity 194 incompressible fluids See fluid(s) indefinite matrix 196, 288, 289 indicator functional 2, 114, 167, 185, 192, 294 inequality(ies) elliptic variational See elliptic variational inequality(ies) and E.V.I Holder 52 hyperbolic variational 104 Korn 393 parabolic variational See parabolic variational inequality(ies) and P.V.I quasivariational Schwarz 32, 69, 84, 86, 107, 181, 340, 342, 363 triangle 158 variational See variational inequality(ies) inextensible finite elasticity 194 infinity condition at 246 flow uniform at 220 Mach number at 226,231,433 initial condition 200, 246 initial value problem 200, 318 injection 168, 171, 173, 176 canonical 114 compact 124, 325, 344, 345, 368 continuous 344,368 operator 325 injective operator 182 inner product 6, 8, 128, 167, 169 intake (air) 310, 311, 312, 313, 314, 315, 316 integral boundary 266, 267, 276 boundary integral equation 378, 387 integration numerical See numerical integration Simpson integration formula 34, 62, 72, 75, 378 theory 335 time 308 interior nodes of a triangulation 128,209 of a triangle 365 penalty functional 224 penalty method(s) 221, 222, 223, 226, 231, 232, 233, 234, 235, 236, 240, 241, 242 nonlinearly weighted 225 internal approximation 8, 13, 321, 326, 328 elastic forces 394 work of the 394 interpolate (linear) 49, 51, 52, 86, 370 interpolation operator 369, 373, 375, 390 linear 36, 125, 368, 373 quadratic 36 intersection of convex sets 191 inviscid fluids vii, viii, 110, 195, 199, 211, 212 irrotational flow(s) 28,134 isentropic (quasi-isentropic flows) 212 isomach lines 231 isoparametric finite elements 85 isopressure lines 311, 315 isotropic material 42 iteration(s) 161, 162, 170, 220, 262, 265, 274, 275, 291, 298, 308, 314 conjugate gradient 209, 310 Newton 164 iterative method viii, 12, 16, 27, 53, 63, 76, 102, 128, 139, 159, 176, 187, 198, 199, 207, 266, 271, 275, 285, 288, 289, 302,315, 330 dual 53, 57 process 140, 320 Stokes solvers 276 solution viii, 40, 94, 186, 188, 245, 289, 294, 301 of variational problems viii solvers 273 Jacobian matrix 197 jet of electrons 385 jet of protons 385 John-Kuhn-Tucker multipliers 24, 25, 54, 68 Joukowsky (Kutta-Joukowsky condition) See Kutta-Joukowsky condition jump condition 226 of density 226 of entropy 222 of normal derivatives 223 of velocity 222 Karman Haar-Karman principle 42 Von Karman equations 211 Ker (abbreviation of kernel) 19, 20, 22, 271,288,290,295 kernel 348 Korn airfoil 214,230,231,240 480 Korn inequality 393 Kuhn (John-Kuhn-Tucker multipliers) 24, 25, 54, 68 Kutta-Joukowsky condition 135, 213, 214, 215,220 three-dimensional 214 Kutta (Runge-Kutta method) 208 Lagrange multiplier(s) 20, 23, 270 Lagrangian augmented See augmented Lagrangian functional 54, 55, 63, 77, 167, 269, 293 laminar flow(s) 79 Lamme coefficients 393 Lanczos method 289, 404 Laplace operator 337, 349 law (Bernoulli) 214 Lax-Milgram lemma (or theorem) 3, 4, 322, 324, 325, 400 layer boundary 405,413 free 405 leading edge 312 least-squares 211 formulation 202, 203, 215, 217, 218, 230, 259,262,263,264,281,282,285, 317 nonlinear 317 functional 322 linear least-squares problem(s) 196 method(s) 195, 206, 211, 244, 245, 257, 273, 318, 319 nonlinear viii, 208, 215, 217 approach 206 formulation 317 problems 196 problem(s) 196, 197, 204, 217, 219, 260, 264, 282 solution 195, 198, 215, 217, 230, 259, 263,281,310 of nonlinear problems 195 Lebesgue dominated convergence theorem 115, 116, 117, 118, 119, 126 lemma Fatou 113,115,116,118,125 Lax-Milgram 3, 4, 322, 324, 325, 400 length arc See arc-length characteristic 310 lifting airfoil 213 limit of regularity 44 limitation of regularity 47, 84 Subject Index line(s) isopressure 311,315 search 205,219 shock 214 stream 222,226,308,314 linear approximation of linear variational problems vii, 332 approximation (piecewise linear) 39, 68, 85,94, 159,223,366 boundary value problem(s) 318, 319 elliptic 385 convergence 303, 306, 428 constraint(s) 174, 189, 193 equality 18,20 Dirichlet problem 54, 205 elasticity 321, 393 equations 392 operator 300, 321 problem 393 system 307 three-dimensional 392 elliptic boundary value problems 385 partial differential operator 321, 349, 354,355 equality constraints 18,20 equations 41 variational 429 finite dimensional linear variational problems 374 first-order system 259 functional (or form) See functional (linear) hyperbolic problem 321 interpolate 49,51,52,86,370 interpolation operator 36, 125, 368, 373 least squares problem(s) 196 numerical analysis of linear variational problems ix operator 323, 336, 339 parabolic problem 321 partial differential equations 321, 343 piecewise linear approximation 39, 68, 85,94, 159,223,366 problem(s) vii, 182, 275, 299, 300, 323, 364, 384 programming 318 system(s) 101, 129, 132, 133, 140, 190, 271, 285, 287, 288, 289, 292, 293, 294, 295, 319, 330, 334, 375, 376, 381, 383, 384, 385, 392, 398, 404, 416, 429 Subject Index sparse 330, 383 triangular 272, 292 variational equation 64, 230, 260, 267, 282, 429 variational formulation 321 variational problem(s) vii, ix, 206, 270, 321, 322, 325, 340, 341, 343, 352, 357, 374, 397, 398, 416 approximation of vii, 332 finite-dimensional 374 numerical analysis of 374 linearly constrained minimization problem 269, 270 linearly independent constraints 269 Lions-Stampacchia theorem 324 Lipschitz continuous 31, 69, 152, 199, 216, 380,381,390 boundary 44,61,73,87,301,339 locally 128 operator 199 uniformly 29, 137 liquids (nitration of) 28 local approximation results 373 error estimates 368, 369 vector basis 319 lower semicontinuity (weak) 17 lower semicontinuous See l.s.c functional See functional (l.s.c.) lower triangular matrix 384 l.s.c (abbreviation of lower semicontinuous) 1,2,6, 7, 12, 15,23, 69,78,98, 112, 141, 148, 151, 166, 168, 171, 173, 176,294 lubrication 28 L+ (approximation of) 55 Mach distribution 220, 221, 431, 433 Mach number 226,227,431 at infinity 226,231,433 magnetic (ferro-magnetic media) 165 magnetic state 131,165 magnetosphere 386 magneto-static problem 188 material elastic 42 plastic 42 matrix(ces) 392,416 band structure of 384 bandwidth of 404 column of 272 481 condition number of 272 conditioning of 272 diagonal 101, 102, 103, 128, 129, 188, 272 2p + 384 full 271, 301 Hessian 91, 371 ill-conditioned 23 indefinite 196, 288, 289 Jacobian 197 lower triangular 272, 292, 383 nonsingular 272, 288 nonsymmetric 18, 130, 188, 189, 196, 385, 404 positive-definite 18, 21, 40, 128, 129, 130, 132, 147, 148, 182, 188, 189,272, 288, 289, 292, 317, 319, 330, 376, 381, 383, 384, 385,398,404,416 positive-semidefinite 188, 196, 271, 273, 274, 290 preconditioning 196, 197, 385 rank of a 288,291 scaling 196, 197, 385 sparse 288,319,377,383,398,416 spectral radius of a 23,91,371 sub- 272 symmetric 18, 21, 40, 128, 129, 132, 147, 148, 188, 196,271,272,273,274, 288, 289, 290, 292, 317, 319, 330, 376, 381, 383, 384, 385, 398,404, 416 triangular 384 transpose 18, 196 maximum principle 31 measurable set 332 measure (Dirac) 344 mechanics continuum viii fluid 194, 345, 349, 414 of continuous media 392 medium (a) elastic continuous 392 ferro-magnetic 165 porous 28 rigid 79 viscous-plastic viii membrane (elastic) 28 mesh (finite element) 228,431,432 method of characteristics 259 mildly nonlinear elliptic equations 110 mildly nonlinear elliptic problems 191 mildly nonline^ systems 188 minimal residu A method 305 minimal surfac, problem(s) 165 Subject Index 482 minimization constrained 152 method(s) 196 of convex functional 140 of functionals 140 of nondifferentiable functionals 230 of quadratic convex functionals 196 problem(s) 4, 6, 17, 19, 23, 43, 56, 65, 69, 79,85, 112, 137, 141, 142, 150, 152, 159, 177, 188, 195,269,294, 326, 340, 353, 357, 358, 388 nondifferentiable 76 one-dimensional 204 single variable 205, 219, 275 min-max theorem 82 Mises (Von) criterion 42 mixed boundary value problem(s) 354, 355, 356, 361 variational formulation of 361 finite element approximations 245, 246, 248,251,267,276,277 of the Navier-Stokes equations 277 of the Stokes problem 247 finite element methods 263, 276 MODULEF 362 mollifiers 37 mollifying sequence 48,61, 120 moment of torsion 42 monotone operator 122, 141, 142, 181, 209, 224 monotonicity 144 multidimensional problem(s) viii multi-index 367 multiple solutions ix multiplier(s) 70, 82 John-Kuhn-Tucker 24, 25, 54, 68 Lagrange 20, 23, 270 multiply connected domain 135 multiprocessor computer 193 multi-step methods 208 NACA 0012 airfoil 225, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239 bi 241, 242 Navier-Stokes equations vii, viii, 211, 246, 257, 262, 276, 285, 301, 308, 319, 370, 395, 399, 414, 417, 424, 425, 430 approximation of the 251, 252, 277, 423 finite element 318,422,423 mixed 277 semidiscrete 252, 278 compressible 300, 319 discrete 254, 259, 263, 281 fully 308 semi- 252, 278 steady 253,268 unsteady 268 finite element approximation of the 423 mixed 277 formulation of the 430 variational 276 mixed finite element approximation of the 277 nonlinear term of the 415, 416, 417, 420 nonstationary 245 semidiscrete approximation of the unsteady 252, 278 solution of the ix, 195, 244, 262, 318 splitting methods for the incompressible 318 stationary 245,251,259,263 steady 202, 245, 251, 259, 266, 285, 317, 415 discrete 253,268 stream-function-vorticity formulation of the 246 time-dependent 252, 253, 258, 418 unsteady 245, 252, 254, 256, 258, 266, 278,280,285,317 discrete 268 semidiscrete 278 variational formulation of the 276, 422 Navier-Stokes problem decomposition properties of the approximate 424 discrete 254,263,281 Neumann boundary conditions 213, 354 Neumann-Dirichlet boundary conditions 428 Neumann-Dirichlet problems 354, 361, 379 discrete 362 Neumann problem(s) ix, 64, 65, 66, 69, 77, 335,336,337,338, 339, 341, 342,345, 346, 347, 356, 358, 364, 386, 389 approximate 219 approximation of the 335 finite element 364 discrete 362 one-dimensional 344 variational formulation of the 337 Neumann variational problem 77 Newtonian fluid 195, 245 Newton iteration 164 Subject Index method vii, 16, 129, 130, 164, 200, 207, 320 one-dimensional 131 quasi- 130,207 Nicholson (Crank-Nicholson scheme) 100, 102, 103,253,279,312 nodes (boundary) See boundary nodes nodes of a finite element triangulation See triangulation noncompact set 152 nonconvex variational problems 194 nondifferentiable functional(s) 25, 96, 149, 230 minimization of 230 nondifferentiable minimization 76 nonempty set See set (nonempty) nonfactorable set 149,150 nonhomogeneous Dirichlet problem 358 variational formulation of the 358 nonlinear approximation of nonlinear flow problems viii boundary value problem(s) viii, ix, 204, 206, 318, 319 compressible flow problem(s) 318 Dirichlet problem 185, 186, 195, 198, 202, 207 discrete fourth-order nonlinear operator 224 elliptic equations 110, 131, 165,211,212 equations 140, 163, 195, 212 finite-dimensional problems 253, 257, 278,317 finite-dimensional systems 218 fourth-order problems 194 hyperbolic equations 212 hyperbolic problems viii least squares viii, 208, 215, 217 approach 206 formulation 317 problems 196 solution of nonlinear problems 195 mildly nonlinear elliptic equations 110 mildly nonlinear elliptic problems 191 mildly nonlinear systems 188 operator 198 discrete fourth-order 224 partial differential 136 problem(s) vii, viii, 1,11, 128, 195, 196, 198,211,257,259,318,319,384 programming problems 47, 72 single variable nonlinear equation 187, 190 483 steady nonlinear problem 254 step 310,417 system(s) 24, 134, 140, 163, 187, 188, W0, 207, 209, 215 term of the Navier-Stokes equations 415, 416, 417, 420 time-dependent problem 313 variational equations viii, 16, 83, 217, 259 variational problems viii, 194, 321 weight 225, 226 nonlinearity 257 nonlinearly weighted interior penalty method 225 nonphysical shocks 221, 224, 227 nonsingular matrix 272, 288 nonsymmetric bilinear form 1, 28, 110, 194, 322 matrix See matrix (nonsymmetric) non-well-posed problem 337 norm 1,18,98,167,330 Euclidean 20, 196, 197, 274 H\a) 348, 350, 354, 355, 361 (H\C1))3 393 Sobolev 367, 368, 369 normal derivative(s) 337 jump of 223 equation 196,404 stress 424 nozzle 135, 231, 312 at high incidence 308, 310 cross section of a 312 number condition 22, 272 Mach 226,227,431 at infinity 226,231,433 Reynold 211,245,310,313,314, 315,316 high 317 numerical analysis of linear variational problems ix diffusion 413 integration 218, 334, 375, 377, 378, 381, 392 formula 72, 377 of nonlinear hyperbolic problems viii procedure 9, 85, 122, 300, 326, 382 quadrature schemes 34 schemes for parabolic linear equations 99 step-by-step 99 solution of nonlinear variational problems viii 484 numerical solution (Continued) of the compressible Navier-Stokes equations 319 stability 317 obstacle problem 27, 28, 41, 45, 159 approximate 63 discrete 24,40 one-dimensional Newton method 131 one-dimensional problem 262, 265, 285 one-sided finite difference scheme 215 open set 121 operator(s) approximate divergence 300 automorphic 301 compact 325 continuous 174, 323, 339 contraction 3, 5, 6, 67, 95 differentiable 200 differential viii partial 330, 334, 349, 397 pseudo- 267 Dirichlet 273 discrete fourth-order nonlinear 224 discrete Poisson 317 discrete Stokes 317 divergence (approximate) 300 elasticity (linear) 300, 321 elliptic partial differential operajor 321, 349, 355, 397 second-order 335, 349, 354, 355, 374 strongly 199, 301 injective 182 interpolation 369, 373, 375, 390 linear 36, 125, 368, 373 quadratic 36 Laplace 337, 349 linear 323, 336, 339 elasticity 300, 321 interpolation 36, 125, 368, 373 Lipschitz continuous 199 monotone 122, 141, 142, 181, 209, 224 strictly 141, 142, 174, 175, 176 nonlinear 198 discrete fourth-order 224 partial differential 136 partial differential 330, 334, 349, 397 elliptic 321,349,355,374,397 second-order 335, 349, 354, 355, 374 nonlinear 136 Subject Index second-order 27, 349, 354 Poisson (discrete) 317 projection 3, 66, 77, 94, 153, 154, 329 pseudo-differential 267 quadratic interpolation 36 range of 18 scaling 317 second-order partial differential 27, 349, 354 elliptic 335, 349, 354, 355, 374 self-adjoint 153,182,301 Stokes (discrete) 317 symmetric 323 trace 339, 350, 357 truncation 215 optimal control problem 202, 204 optimization method(s) problem(s) 7, 153, 168, 194 theory 148 in Hilbert spaces 137 ordinary differential equation(s) 200 first-order 208 system of 252, 278 orthogonal complement 70 oscillations 413 spurious 400 Ostrogradsky (Green-Ostrogradsky formula) See Green-Ostrogradsky formula out of core procedure 275 over-relaxation block over-relation methods with projection 193 method(s) 63, 130, 131, 152, 383, 384 method(s) with projection 40, 63, 158 parameter 76, 148, 177 successive See S.O.R and S.S.O.R pairing (duality) See duality panel methods 270 parabolic equations 101 linear 99 variational, 99 linear parabolic problem 321 variational inequalities (see also P.V.I.) approximation of viii, 98, 101, 103 parallel computing 193 parameter discretization 301 Subject Index over-relaxation 76, 148, 177 under-relaxation 148 viscosity 79,245 partial differential equation(s) 318, 392 approximation of 330 elliptic 343, 362 first-order system of 397 linear 321, 343 second-order 350 solution methods for 398 system(s) of 307 time-dependent 258 partial differential operator(s) See operator(s) Peaceman-Rachford algorithm 201 Peaceman-Rachford alternating direction method 254, 280 penalized problem 15, 16, 20, 21, 24 penalty approximation 391 approximation by penalty methods 15, 362, 391 interior penalty functional 224 interior penalty method(s) 221, 222, 223, 226, 231, 232, 233, 234, 235, 236, 240,241,242 nonlinearly weighted 225 methods 15, 20, 23, 24, 25, 26, 221, 359, 362, 391 solution 15,25 of elliptic variational inequalities 1, 110,191 of variational problems 25 perfect fluid 28, 134 performances (aerodynamical) 431 periodic functions 389 periodicity condition 391 phase distorsion 413 physical flow(s) 135,216 plane 135 shock(s) 221,222,226,231 non- 221,224,227 space 244 piecewise linear or quadratic approximation See approximation finite elements See finite element(s) plane hodograph 134 physical 135 plastic elasto-plastic torsion problem 41, 54, 136, 158, 167, 184 485 material 42 region(s) 161, 162 plasticity problem 42 plasticity yield 42, 79 plate 211 clamped plate problem 394 variational formulation of the 394 problem 392 thin 394 thickness of a 394 Poincare-Friedrichs lemma 29 Poiseuille flow 308 Poisson coefficient 394 discrete operator 317 problem 317 solver 220 solver 206 discrete 220 Polak-Ribiere algorithm 198 method 133,204 strategy 261,283 polar coordinate system 243 polygonal boundary 380 domain See domain (polygonal) polyhedral domain 368 polynomials second-order 403 space of 217, 249, 365 porous medium (a) 28 positive cone 64 positive definite bilinear form 269 matrix See matrix(ces) positive density 216 positive semidefmite bilinear form 273, 345, 346 matrix See matrix(ces) potential electrical 349, 386 flow(s) viii, 134, 195, 199, 211, 212 full potential equation viii, 212 stress 42 velocity 134, 212, 248, 349 precision (double) 133 preconditioned conjugate gradient algorithm(s) 219, 306 method(s) vii preconditioning (conjugate gradient method with) 317, 319 486 preconditioning matrix 196, 197, 385 pressure 214, 245, 302, 317, 318, 395, 396, 418,419.429 discontinuities 212 discrete 249,251,277,284,424,429 distribution 231,310,312 iso-pressure lines 311, 315 trace of 429 -velocity formulation of the Navier-Stokes equations 245, 317 primal problem 293, 294 primitive variables 318 principle conservation 337 Haar-Karman 42 maximum 31 processor (multi-processor computer) 193 product inner 6,8,128,167,169 scalar See scalar product programming linear 318 nonlinear 47, 72 quadratic 34, 60 projection gradient-projection method operator 3, 66, 77, 94, 153, 154, 329 over-relaxation methods with 40, 63, 158, 193 S.O.R methods with 162 projector 154 proper functional(s) See functional(s) protons (jet of) 385 pseudo-arc-length continuation method 195, 198 pseudo-differential operator 267 P.V.I, (abbreviation of Parabolic Variational Inequality(ies)) viii, 1, 98, 104, 105, 109 of the first kind 101 of the second kind 103 quadratic approximation (piecewise quadratic) 39, 68, 308 constraint(s) 47 convergence 317 finite element(s) 273 piecewise 33, 39, 59, 68, 72, 85, 308, 416 functional 152, 160, 162 convex 196 Subject Index interpolation operator 36 minimization of quadratic convex functionals 196 programming 34,60 quadrature (numerical) 34 quadrilateral finite elements 215, 227, 366 qualification condition 24 quasi-isentropic flow 212 quasi-Newton method(s) 130,207 quasi-variational inequality(ies) Q.V.I, (abbreviation of Quasi-Variational Inequality(ies)) range of an operator 18 rank of a matrix 288,291 Rankine-Hugoniot conditions 214 rarefaction shock(s) 212, 221, 226 ratio of convergence 297, 307 ratio of specific heats 134,212 -R(B) (abbreviation of range of operator B) 18, 168, 171, 173, 174 recirculation region 308 Reeves (Fletcher-) See Fletcher-Reeves reflecting boundary conditions 417, 424 reflexive Banach space 140, 152 regularity limit 44 limitation 47, 84 of the solution 30, 57, 69, 364 properties 43, 79, 88, 364 result(s) 31,43,122 theorem 30 regularization effect 226 elliptic 224 method 25, 26 process 225 technique 82 regularized problem 83 relaxation (see also S.O.R and S.S.O.R.) algorithm(s) 131, 143, 151, 152, 158 block relaxation algorithms 152, 155 block relaxation methods 151, 177, 184, 187, 190, 191 methods vii, viii, 76, 130, 140, 142, 149, 163, 177,215,384 over- See over-relaxation under- See under-relaxation Rellich theorem 345 residual minimal residual method 305 weighted residual method 318 Subject Index restarting procedure 133 Reynold number 211, 245, 310, 313, 314, 315,316 high 317 Ribiere (Polak-) See Polak-Ribiere Riesz representation theorem 3, 110, 153, 154,322,323, 331 Riesz-Schauder theory 325 rigid medium 79 roundoff errors 133, 176, 196, 274, 291, 297, 385 Runge-Kutta method 208 saddle-point(s) 54, 55, 64, 65, 68, 77, 94, 168, 169, 170, 171, 172, 174, 176, 178, 179, 182, 193, 194, 270, 293, 294 scalar product 1, 18, 56, 98, 101, 290, 293, 299, 330, 348, 385, 387, 396, 400 Euclidean 274 L2(Q)- 299 scaling (conjugate gradient method with) 317, 319, 320 scaling matrix 196, 197, 385 scaling operator 317 Schauder (Riesz-Schauder theory) 325 scheme 99 alternating-direction 308, 310, 313 backward Euler 100 backward finite difference 215 backward implicit 200, 308 centered finite difference 215 Crank-Nicholson 100, 102, 103, 253, 279,312 discretization 208 for parabolic linear equations 99 Euler 99 backward 100 explicit 99, 101, 102, 103, 317 finite difference 215, 228, 245 backward 215 centered 215 five-point 382 one-sided 215 upwinded 215 finite element 399 with upwinding 414 five-point finite difference 382 fully implicit 253,317 implicit 100, 102, 103, 107, 253, 278 fully 253, 317 487 semi- 253,317 two-step 100, 102, 104, 254, 279 numerical finite difference scheme for parabolic linear equations 99 numerical finite difference scheme (step by step) 99 numerical quadrature 34 one-sided finite difference 215 semidiscrete 258 semi-implicit 253, 317 time discretization 252, 278 two-step implicit 100, 102, 104, 254, 279 Schwarz inequality 32, 69, 84, 86, 107, 181, 340, 342, 363 sciences (decision) search (line) 205,219 second kind elliptic variational inequalities of the See E.V.I, parabolic variational inequalities of the See P.V.I second-order approximation 401 accurate 401 elliptic equations 404 elliptic partial differential operator 335, 349, 354, 355, 374 elliptic problems 355, 364 E.V.I 27 finite elements 85 finite element approximation of second-order E.V.I 27 partial differential equations 350 partial differential operator 27, 349, 354 polynomial 403 section (cross) See cross-section Seidel (Gauss-Seidel method(s)) 140, 148 self-adjoint operator 153, 182, 301 semiconductors 414 semicontinuity (weak lower) 17 semicontinuous (lower) See l.s.c functional(s) See functional(s) semidefinite (positive semidefinite bilinear form) See bilinear form(s) semidefinite (positive semidefinite matrix) See matrix(ces) semidiscrete approximation 252,278 of the Navier-Stokes equations 252, 278 scheme 258 semi-implicit scheme 253, 317 separable Hilbert space 328 488 separated flow 317 sequence (mollifying) 48, 61, 120 sequential computing 193 sesquilinear bilinear form 325 set(s) or subset(s) bounded 121, 141, 144, 152, 345 closed 1, 2, 9, 29, 33, 41, 46, 56, 60, 84, 98, 114, 137, 140, 142, 150, 151, 152, 153, 154, 167, 191, 345 weakly 43,49 compact 27, 38, 41, 43, 49, 93, 175, 325 non- 152 weakly 43, 84, 345 connected 332 convex 1, 2, 9, 23, 29, 33, 38, 41, 46, 56, 60, 84, 98, 114, 117, 134, 137, 140, 142, 148, 149, 150, 151, 153, 154, 167, 191, 345 approximation of 9, 33, 46, 60 intersection of 191 diameter of 332 measurable 332 noncompact 152 nonempty 1, 2, 9, 23, 33, 41, 46, 56, 60, 98, 114, 137, 140, 142, 148, 191 nonfactorable 149, 150 open 121 unisolvent 366 shock(s) 212, 214, 231 condition 214 expansion 221 line 214 nonphysical 221, 224, 227 physical 221, 222, 226, 231 rarefaction 212,221,226 surface 214 weak 222 Signorini problem 56 simplicial finite element(s) 227 simply connected cross-section 42 simply connected domain 135 Simpson integration formula 34, 62, 72, 75, 378 simulation (flow) 231 simulation (transonic flow) 231, 433 single valued potential 213 single variable minimization problem 205, 219, 275 single variable nonlinear equation 187, 190 skew-symmetric bilinear form 351 slit of a domain 213 smooth boundary See boundary Subject Index Sobolev imbedding theorem 52 norm(s) 367, 368, 369 spaces 27, 216, 247, 266, 338, 363 solution(s) approximate See approximate (solution) conjugate gradient See conjugate gradient direct 285, 288, 319 discrete 218, 224 finite element ix, 318, 374, 379, 394, 395 iterative viii, 40, 94, 186, 188, 245, 289, 294, 301 of elliptic variational inequalities viii of variational problems viii least squares See least squares methods for partial differential equations 398 multiple ix numerical See numerical (solution) of the Navier-Stokes equations ix, 195, 244, 262,318 penalty 15,25 penalty solution of elliptic variational inequalities 15 penalty solution of variational problems 25 regularity of the 30, 57, 69, 364 solver(s) direct Stokes 276 discrete Poisson 220 iterative 273 Stokes 276 Poisson 206 discrete 220 Stokes 262, 266, 267, 273, 276, 285 direct 276 iterative 273 sonic transition 224 S.O.R (abbreviation of Successive Over-Relaxation) 140, 161 with projection 162 space(s) or subspace(s) 33, 88, 355, 379 Banach 152, 251, 367 reflexive 140, 152 boundary 268 closed 9, 12, 99, 111, 326, 348, 350, 355, 396 complementary 268 discrete 248, 365, 374, 391, 422 discretization 308,312 dual 1,16,140,203,247,318 topological 198, 203, 321 Subject Index finite dimensional 9, 12, 33, 99, 174, 176, 193, 326, 329, 365 finite element 248,379,415,423 Hilbert See Hilbert space(s) inclusion of 368 of periodic functions 389 of polynomials 217,249,365 of traces 416 physical 244 Sobolev See Sobolev (space(s)) topological dual 198, 203, 321 topological vector 166 vector 58 approximation of 9, 33, 46, 60 space variable approximation 200 sparse linear systems 330, 383 sparse matrix(ces) 288, 319, 377, 383, 398, 416 specific heats (ratio of) 134,212 spectral radius of a matrix 23, 91, 371 speed of convergence 76, 128, 148, 177, 296, 305, 384 splitting methods for the incompressible Navier-Stokes equations 318 spurious oscillations 400 S.S.O.R (abbreviation of Symmetrized Successive Over-Relaxation) 140 stability 317 condition 101, 253 conditional 99, 102, 103, 104 numerical 317 properties 337 unconditional 100, 102, 103, 104, 253, 254, 255, 257, 278, 279 Stampacchia (Lions-Stampacchia theorem) 324 starting procedure 100 state equation 202, 217, 260, 262, 263, 282, 285 discrete variational 218,224 state (magnetic) 131, 165 stationary flow(s) 79 stationary Navier-Stokes equations 245, 251,259,263 non- 245 steady discrete Navier-Stokes equations 253, 268 steady flow 418 steady Navier-Stokes equations See Navier-Stokes equations steady nonlinear problem 254 steady Stokes problem 246, 248, 395 489 steepest descent method 129, 302, 305 step arc-length 207 by step numerical scheme 99 cavity with a 404 discretization 161 flow in a channel with a 308 fractional step method 258, 318 multi-step method 208 nonlinear 310,417 time 99, 200, 201, 313, 314, 315, 316, 417 discretization 253 stiff phenomenon 101 stiff problems 101, 182, 196, 200 Stokes operator (discrete) 317 Stokes problem viii, 248, 249, 259, 260, 262, 266, 267, 276, 277, 289, 298, 300, 303, 305, 306, 310, 312, 370, 392, 396, 420, 424, 429 approximate 276,428 approximation of the 248,397 finite element 397 decomposition of the 266, 273, 276, 415, 425, 428, 429 discrete 245, 257, 265, 267, 277, 282, 284, 285, 287, 288, 289, 297, 298, 299, 300,317,319,425,426,429 finite element approximation of the 397 mixed formulation of the 247 steady 246, 248, 395 three-dimensional 275 two-dimensional 275 variational formulation of the 247, 261, 395, 425 mixed 247 Stokes solver(s) 262, 266, 267, 273, 276, 285 direct 276 iterative 276 stopping criterion 161,176 stopping test 177, 198 strain-stress relation 393 strain tensor 393 stream direction 227 function 139,246,312 lines 222,226,308,314 -vorticity formulation of the Navier-Stokes equations 246 stress field 42 normal 424 potential 42 490 stress (Continued) -strain relation 393 tensor 393,430 zero stress initial state 42 stretching (exponential stretching method) 319 strictly convex functional See functional(s) strictly monotone operator See operator (monotone) strictly subsonic flow 214 strong convergence See convergence strongly elliptic operator 199,301 subset See set(s) or subset(s) subsonic flow See flow (subsonic) approximation of subsonic flow 138 strictly 214 subsonic problem (variational formulation of the) 135 subspace See space or subspace(s) subtriangle 249,423 suction phenomenon 312 summation convention of repeated indices 393 superlinear convergence 133, 291, 318, 320 supersonic flow(s) 134, 212, 215, 224, 431 supp (abbreviation of support) 27, 37, 38, 48 support 376 compact 36, 37, 48, 120, 247, 338, 339 surface forces 392 minimal surface problem 165 shock 214 symmetric airfoil 134,215 bilinear form See bilinear form (symmetric) non- 1,28,110,194,322 skew- 351 flow 215 non- 135 matrix See matrix(ces) (symmetric) non- See matrix(ces) (nonsymmetric) operator 323 symmetrization process 257, 318 system(s) discrete elliptic 428 first-order system of partial differential equations 397 linear See linear (systems) elasticity 307 first-order 259 sparse 330, 383 Subject Index triangular 272, 292 nonlinear See nonlinear (system(s)) finite-dimensional 218 mildly 188 of coupled variational inequalities 170, 193 of ordinary differential equations 252, 278 of partial differential equations 307 first-order 397 Taylor expansion 371 Tchebycheff acceleration 305 tensor (strain) 393 tensor (stress) 393, 430 termination (finite) 291 test function 338, 419 test (stopping) 177, 198 tetrahedral finite elements 215,227 thickness of a plate 394 thin plate problem 394 three-dimensional finite element mesh 431 flow(s) 214 Kutta-Joukowsky condition 214 linear elasticity 392 Stokes problem 275 time accuracy 99, 100 C.P.U 161 dependent bilinear form 101 equations 98 flow of a Bingham fluid 104 Navier-Stokes equations 252, 253, 258,418 nonlinear problems 313 partial differential equations 258 problems 108, 200, 246, 252, 254, 258 discretization 99, 245, 252, 253, 278, 312, 317,418 schemes 252, 278 step 253 integration 308 step 99, 200, 201, 253, 313, 314, 315, 316, 417 truncation error 253, 254, 255, 257, 278, 279 topological dual space 198, 203, 321 topological vector space 166 topology (weak) 11 Subject Index topology (weak *) 119, 120, 126, 335 torsion 41 angle 42 elasto-plastic torsion problem 41, 54, 136, 158, 167, 184 moment 42 problem 43,150 trace(s) 28, 56, 73, 267, 270, 419 of the pressure 429 operator 339, 350, 357 space of 416 trailing edge 213 transfer (heat-transfer phenomenon) 355 transform (hodograph) 134 transition (sonic) 224 transonic flow(s) See flow(s) (transonic) transpose matrix 18, 196 transverse force 394 trapezoidal rule 34, 72, 123, 300, 377, 378, 382 triangle(s) 223, 227, 228, 230, 249, 300, 309, 312, 370, 377, 378, 403, 405, 406, 407 adjacent 225,226,401,402 angle(s) of 34, 49, 372 centroid of 33, 35, 377 closed 365 diameter of 52 edge(s) of 33, 92, 269, 286, 368, 401, 402 inflow 229 inequality 158 interior of 365 sub- 249,423 vertex(ices) of 33, 52, 228, 229, 286, 300, 365, 366, 370, 378, 389, 400, 402, 403 triangular finite element(s) 215, 227 mesh 228 triangular linear systems 272, 292 triangular matrix 384 lower 272, 292, 383 triangulation(s) 88, 365, 373 angle(s) of 52, 61, 73, 85, 90, 124, 159, 250, 369, 400 edge(s) 225, 377, 416 family of 248, 365, 369, 374, 379, 400, 422 regular 369, 373, 375, 380, 390 finite element See finite element (triangulation) node(s) of 123, 159, 308, 309, 312, 405, 406, 407, 416 interior 128, 209 491 refined 314 refinement 308 regular 413 vertex(ices) of 90, 218, 228, 229, 299, 366, 379, 381, 390, 391,422 tri-jet engine 431 truncation 222 operator 215 technique 113 time truncation error 253, 254, 255, 257, 278, 279 Tucker (John-Kuhn-Tucker multipliers) 24, 25, 54, 68 turning point 209,211,318 two-dimensional flow(s) 213,245 two-dimensional Stokes problem 275 two-piece airfoil 230,231 two-step implicit scheme 100, 102, 104, 254, 279 unbounded domain 138, 330, 335, 338 unbounded regions (flows in unbounded regions) 417 unconditional stability See stability under-relaxation algorithm(s) 131, 158 methods 152, 384 parameter 148 uniform continuity 35, 75, 173, 174, 334 uniform (flow uniform at infinity) 220 uniformly continuous operator 147 convex functional(s) 144,152,171 Lipschitz continuous function(s) 29, 137 proper functional(s) 12 unisolvent set 366 unsteady discrete Navier-Stokes equations 268 flow(s) 310,313,314,315,316 Navier-Stokes equations See Navier-Stokes equations upwinded approximation 228 discrete upwinded continuity equation 230 finite difference scheme 215 first-order approximation 404 upwinding ix, 317, 404 finite element approximation with 400 finite element method with 399 finite element scheme with 414 method 231,404,405 492 upwinding (Continued) of the density 221, 226, 227, 237, 238, 239 technique 227 Uzawa algorithm 54, 55, 66, 77, 94, 170, 289, 292, 293, 294 variables (primitive) 318 variation (of entropy) 214 variational Dirichlet problem (discrete) 268 variational equation(s) 2, 25, 69, 217, 218 linear 64, 230, 260, 267, 282, 429 nonlinear viii, 16, 83, 217, 259 parabolic 99 variational formulation(s) 43, 55, 69, 79, 205,215, 218,261,283, 317, 321, 337, 379, 386, 389, 397, 399, 418, 420, 427 equivalent 45, 216 linear 321 mixed variational formulation of the Stokes problem 247 of flow problems 136 of mixed boundary value problems 361 of problems of continuum mechanics 392 of the clamped plate problem 394 of the continuity equation 215 of the Dirichlet problem 350, 351, 358 of the Fourier problem 356,374 of the linear elasticity problem 393 of the Navier-Stokes equations 276, 422 of the Neumann problem 337 of the nonhomogeneous Dirichlet problem 358 of the Stokes problem 247, 261, 395, 425 of the subsonic flow problem 135 variational inequality(ies) 1, 26, 28, 56, 69, 78, 111, 114, 141 approximation of 26 elliptic See elliptic (variational inequality(ies)) and E.V.I, hyperbolic 104 parabolic See parabolic (variational inequality(ies)) and P.V.I, quasi- system of coupled 170,193 variational methods 335, 354, 355, 391, 397 variational problem(s) 23, 42, 166, 194, 247, 248, 265, 342, 343, 346, 348, 353, 355, 357, 388, 393, 394, 395, 396 Subject Index discrete Dirichlet 268 elliptic 334 iterative solution of viii linear See linear (variational problem(s)) Neumann 77 nonconvex 194 nonlinear viii, 194, 321 numerical solution of viii penalty solution of 25 variational state equation (discrete) 218, 224 variations (calculus of) 112, 217, 341 vector basis 218, 271, 286, 296, 299, 329, 381, 391, 392 local 319 space(s) 58 approximation of 9, 33, 46, 60 topological 166 stress 42 velocity vector of a flow 134 velocity 317, 318, 349 critical 134,212 discontinuities of 212 distribution 312, 313 jump 222 potential 134,212,248,349 pressure-velocity formulation of the Navier-Stokes equations 245,317 variables 223 vector of a flow 134 vertex of a cone 58 vertex(ices) of a triangle See triangle(s) vertex(ices) of a triangulation See triangulation(s) virtual work theorem 394 viscosity (artificial) 221 viscosity parameter 79, 245 viscous fluid(s) See fluid(s) (viscous) viscous-plastic fluid 78 viscous-plastic medium viii Von Karman equations 211 Von Mises criterion 42 vorticity 246 distribution 312 iso-vorticity lines 316 stream-function-vorticity formulation of the Navier-Stokes equations 246 wake 212 problems 28 493 Subject Index weak cluster point(s) 9, 172, 173, 179, 182 convergence 10, 14, 17, 35, 124, 174, 345 * convergence 84, 121, 125, 335, 360 formulation 214 lower semicontinuity 17 shock 222 topology 11 * topology 119,120,126,335 weakly closed set 43, 49 weakly compact set 43, 84, 345 weight (nonlinear) 225, 226 weighted (nonlinearly weighted interior penalty method) 225 weighted residual method 318 well-posed problem 148, 149, 337, 417 non- 337 wing 431 work of the external forces 394 of the internal elastic forces 394 virtual work theorem 394 yield (plasticity) 42, 79 Young method 41 zero stress initial state 42 .. .Scientific Computation Editorial Board J.-J Chattot, Davis, CA, USA P Colella, Berkeley, CA, USA W E, Princeton,... Navier-Stokes Equations by F Thomasset, and Computational Methods for Fluid Flow by R Peyret and T D Taylor, both published in the Springer Series in Computational Physics.) Exercises (without... of Fundamental Research in 1977, and then have it published in the collection Springer Series in Computational Physics, we thought, at first, that it would be an easy task Actually, we realized