Partial Differential Equations and the Finite Element Method PURE AND APPLIED MATHEMATICS A Wiley-Interscience Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: MYRON B ALLEN 111, DAVID A COX, PETER HILTON, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume Partial Differential Equations and the Finite Element Method Pave1 Solin The University of Texas at El Paso Academy of Sciences ofthe Czech Republic @ZEicIENCE A JOHN WILEY & SONS, INC., PUBLICATION Copyright 02006 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 ofthe 1976 United States Copyright Act, without 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Differential equations, Partial-Numerical solutions Finite clement method Title QA377.S65 2005 18'.64-dc22 Printed in the United States of America I 200548622 To Dagmar CONTENTS List of Figures xv List of Tables xxi xxiii Preface Acknowledgments Partial Differential Equations 1.1 1.2 Selected general properties 1.1.1 Classification and examples 1.1.2 Hadamard’s well-posedness 1.1.3 General existence and uniqueness results 1.1.4 Exercises Second-order elliptic problems 1.2.1 Weak formulation of a model problem 1.2.2 Bilinear forms, energy norm, and energetic inner product 1.2.3 The Lax-Milgram lemma 1.2.4 Unique solvability of the model problem 1.2.5 Nonhomogeneous Dirichlet boundary conditions 1.2.6 Neumann boundary conditions 1.2.7 Newton (Robin) boundary conditions 1.2.8 Combining essential and natural boundary conditions xxv 2 11 13 13 16 18 18 19 21 22 23 vii Viii CONTENTS 1.3 1.4 1.5 1.2.9 Energy of elliptic problems 1.2.10 Maximum principles and well-posedness 1.2.11 Exercises Second-order parabolic problems 1.3.1 Initial and boundary conditions 1.3.2 Weak formulation Existence and uniqueness of solution I 3.3 1.3.4 Exercises Second-order hyperbolic problems 1.4.1 Initial and boundary conditions Weak formulation and unique solvability 1.4.2 1.4.3 The wave equation I 4.4 Exercises First-order hyperbolic problems 1.5.1 Conservation laws 1S Characteristics Exact solution to linear first-order systems 1S 1S Riemann problem Nonlinear flux and shock formation 1S 1S Exercises 45 Continuous Elementsfor 1D Problems 2.1 2.2 The general framework The Galerkin method I Orthogonality of error and CCa’s lemma 2.1.2 Convergence of the Cialerkin method 2.1.3 Ritz method for symmetric problems I Exercises I Lowest-order elements 2.2.1 Model problem Finite-dimensional subspace V,, C 2.2.2 Piecewise-affine basis functions 2.2.3 The system of linear algebraic equations 2.2.4 v 2.3 2.4 24 26 29 30 30 30 31 32 33 33 34 34 35 36 36 38 39 41 43 44 2.2.5 Element-by-element assembling procedure 2.2.6 Refinement and convergence 2.2.7 Exercises Higher-order numerical quadrature 2.3.1 Gaussian quadrature rules 2.3.2 Selected quadrature constants 2.3.3 Adaptive quadrature 2.3.4 Exercises Higher-order elements 45 46 49 50 51 51 51 52 52 53 54 55 56 51 59 59 61 63 65 66 CONTENTS 2.5 2.6 2.7 Motivation problem 2.4.1 Affine concept: reference domain and reference maps 2.4.2 Transformation of weak forms to the reference domain 2.4.3 Higher-order Lagrange nodal shape functions 2.4.4 Chebyshev and Gauss-Lobatto nodal points 2.4.5 Higher-order Lobatto hierarchic shape functions 2.4.6 Constructing basis of the space Vh,p 2.4.7 Data structures 2.4.8 2.4.9 Assembling algorithm 2.4.10 Exercises The sparse stiffness matrix Compressed sparse row (CSR) data format 2.5.1 2.5.2 Condition number 2.5.3 Conditioning of shape functions Stiffness matrix for the Lobatto shape functions 2.5.4 2.5.5 Exercises Implementing nonhomogeneous boundary conditions 2.6.1 Dirichlet boundary conditions Combination of essential and natural conditions 2.6.2 2.6.3 Exercises Interpolation on finite elements 2.7.1 The Hilbert space setting 2.7.2 Best interpolant 2.7.3 Projection-based interpolant 2.7.4 Nodal interpolant 2.7.5 Exercises General Concept of Nodal Elements 3.1 3.2 3.3 3.4 3.5 The nodal finite element 3.1.1 Unisolvency and nodal basis 3.1.2 Checking unisolvency Example: lowest-order Q' - and PI-elements 3.2.1 Q1-element 3.2.2 P1-element Invertibility of the quadrilateral reference map z~ 3.2.3 Interpolation on nodal elements 3.3.1 Local nodal interpolant 3.3.2 Global interpolant and conformity Conformity to the Sobolev space H' 3.3.3 Equivalence of nodal elements Exercises ix 66 67 69 70 71 74 76 77 79 82 84 84 84 86 88 89 89 89 91 92 93 93 94 96 99 102 103 103 104 106 107 108 110 113 114 115 116 119 120 122 X CONTENTS Continuous Elements for 2D Problems 4.1 4.2 4.3 Lowest-order elements 4.1.1 Model problem and its weak formulation Approximations and variational crimes 4.1.2 4.1.3 Basis of the space Vh,p 4.1.4 Transformation of weak forms to the reference domain 4.1.5 Simplified evaluation of stiffness integrals 4.1.6 Connectivity arrays 4.1.7 Assembling algorithm for Q'/P'-elements 4.1.8 Lagrange interpolation on Q'/P'-meshes 4.1.9 Exercises Higher-order numerical quadrature in 2D Gaussian quadrature on quads 4.2.1 4.2.2 Gaussian quadrature on triangles Higher-order nodal elements 4.3.1 Product Gauss-Lobatto points 4.3.2 Lagrange-Gauss-Lobatto Qp,'-elements Lagrange interpolation and the Lebesgue constant 4.3.3 4.3.4 The Fekete points 4.3.5 Lagrange-Fekete PP-elements Basis of the space 4.3.6 Data structures 4.3.7 4.3.8 Connectivity arrays 4.3.9 Assembling algorithm for QPIPp-elements 4.3.10 Lagrange interpolation on Qp/Pp-meshes 4.3.1 Exercises v7,Tl Transient Problems and ODE Solvers 5.1 5.2 Method of lines 5.1.1 Model problem 5.1.2 Weak formulation 5.1.3 The ODE system Construction of the initial vector 5.1.4 Autonomous systems and phase flow 5.1.5 Selected time integration schemes One-step methods, consistency and convergence 5.2.1 5.2.2 Explicit and implicit Euler methods 5.2.3 Stiffness 5.2.4 Explicit higher-order RK schemes Embedded RK methods and adaptivity 5.2.5 5.2.6 General (implicit) RK schemes 125 126 126 127 129 131 133 134 135 137 137 139 139 139 142 142 143 148 149 152 154 157 160 162 166 166 167 168 168 168 169 170 171 172 173 175 177 179 182 184 CONTENTS 5.3 5.4 5.5 Introduction to stability 5.3.1 Autonomization of RK methods Stability of linear autonomous systems 5.3.2 Stability functions and stability domains 5.3.3 Stability functions for general RK methods 5.3.4 Maximum consistency order of IRK methods 5.3.5 5.3.6 A-stability and L-stability Higher-order IRK methods 5.4.1 Collocation methods Gauss and Radau IRK methods 5.4.2 5.4.3 Solution of nonlinear systems Exercises Beam and Plate Bending Problems 6.1 Bending of elastic beams Euler-Bernoulli model 6.1.2 Boundary conditions 6.1.3 Weak formulation Existence and uniqueness of solution 6.1.4 Lowest-order Hermite elements in 1D 6.2.1 Model problem 6.2.2 Cubic Hermite elements Higher-order Hermite elements in 1D 6.3.1 Nodal higher-order elements 6.3.2 Hierarchic higher-order elements 6.3.3 Conditioning of shape functions 6.3.4 Basis of the space Vh,p 6.3.5 Transformation of weak forms to the reference domain 6.3.6 Connectivity arrays 6.3.7 Assembling algorithm 6.3.8 Interpolation on Hermite elements Hermite elements in 2D 6.4.1 Lowest-order elements 6.4.2 Higher-order Hermite-Fekete elements 6.4.3 Design of basis functions Global nodal interpolant and conformity 6.4.4 Bending of elastic plates 6.5.1 Reissner-Mindlin (thick) plate model 6.5.2 Kirchhoff (thin) plate model 6.5.3 Boundary conditions Weak formulation and unique solvability 6.5.4 BabuSka’s paradox of thin plates 6.5.5 6.1 I 6.2 6.3 6.4 6.5 xi 185 186 187 188 191 193 194 197 197 200 202 205 209 210 210 212 214 214 216 216 218 220 220 222 225 226 228 228 23 233 236 236 238 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Atlcrpriw Method ofLir1r.s Chapinan & HalVCRC Boca Raton FL 2001 12 I V Vogelsang: On the strong unique continuation principle for inequalities of Maxwell type Mtrth A J I K 289, 285-295 I99 122 I Webb: Hierarchical vector based functions of arbitrary order for triangular and tetrahedral finite elements, lEEE Trans Antennas and Propagation 47 1244-1 253, 1999 123 H Whitney: Grorirrtric /rirrgrcrriorz Theory Princeton University Press 1957 124 0.Zienkiewicz, R L Taylor: The Finite Elerwnt Mrthocl Vol - SolidMrc.htrriic \ 5th edition, Butterworth-Heinemann Woburn, MA, 2002 Partial Differential Equations and the Finite Element Method by Pave1 Solin Copyright © 2006 John Wiley & Sons, Inc INDEX Abelian group 172 algorithm adaptive quadrature 63 RKS(4) method, I83 assembling P'/Q'-elements i n 2D, 136 PPIQP-elements in 2D 163 element-by-element in ID 56 Hermite elements in ID, 23 I vertex-by-vertex i n ID, 56 enumeration of bubble DOF in 2D 162 DOF for Hermite elements in ID 210 DOF for Lagrange elements in D, 78 edge DOF in 2D I6 I edges in 2D, 159 vertex DOF in 134 area moment of inertla, I2 array Butcher's, 180 connectivity ID, 78 2D, 134 CSR, 84 autonomilation of RK methods, 186 autonomous system, I I I Babuika's paradox, 254 Banach Stefan, 367 basis, 329 468 bijection 332 Cauchy problem 37 Cauchy properry 366 characteristic variables 40 characteristics 38 compactness definition 407 of identity, 408 completion, 369 condition number 84 condition CFL, 176 cone 14 conditioning, 86 condition\ boundary clamped 249 combined, 23 denection, I Dirichlet general, 20 Dirichlet homogeneous 14 es\ential 22 hard-supported 2.50 impedance, 287 moment, 213 natural, 22 Neumann I Newton, 22 perfect conductor, 287 shear force I3 INDEX siinply \upported 249 \lope I3 \oft-supported 250 \ymmetry 287 traction, 249 truncation 286 collocation I97 entropy 38 initial 30 interface 277 286 conductivity electric, 274 thermal, conductor, 275 cone condition 414 conformity definition I 18 to the \pace H(cur1) 301 to the \pace H , I 19 con\crvation law 37 hyperbolic 37 consi\tency error I73 order 173 constitutive relation\, 274 continuation I72 coiit iniii t y Hdlder 423 Lipschiti ot huundary, 413 of operator\ 357 convergence Heine detinition 360 of a \equence, 3.52 of one-step method\ I74 Wong 408 weal 408 Courant Richard I Curie'\ polnt 277 data structure 1" in ~ I 34 f'1'IQl' i n 2D 157 i n ID 77 deflection I deiiwy argtiiiient 385 of electric Hux 270 of inagiietic flux 270 detrrmiiiaiit 339 dielectric\ 275 diinen\ioii 33 I Dirichlet Johann Petei- Gusts\, Dii-ichlet lilt 20 diwrete problem 46 di\trihution\ I4 doiii;iin boundary J I3 dehnition 41 dual \pace 343 ellective \hearing force 249 clgeilvalue 340 eigenvector 34 I electric charge 273 density 273 element Argyw higher-order 265 quintic 255 edge N6dklec 309 Whitney, 302 genei-a1 nodal I04 Hermite cubic in ID, 218 cubic in 2D 236 higher-order hierarchic i n ID, 222 higher-order nodal i n ID 220 238 higher-order nodal i n 2D, Lagrange biaftine (2' i n 2D.108 higher-order PI' in 2D IS2 higher-order Q".' in 2D 143 higher-order hierarchic in ID, 74 higher-order nodal in ID lineai- I" i n 2D 1 ellipticity uniform, 28 embedding compact 423 continuous 423 energy norm 16 entropy condition\ 38 equation Burger'\ continuity 274 elliptic heat transfer Hcliiiholt/ 28X hyperbolic parabolic Par\evaI 400 Pois\on potential wavc 2x3 equation\ Maxwell's d i Were n t i a I form 273 integral form 272 time-harinonic 2x7 equipotentials 280 eqiiiv;ilence of eleinent\ I2 I eri-oicoii\i\tency I73 di\creti/atioii, I73 Eulei- method explicit 175 implicit 176 expan\ion coefficient\ 33 I Fehlherg'\ trick 182 tield con\ei-vative 280 divergence-free 28 I 469 470 INDEX electric 270 irrotational, 279 magnetic, 270 tixed point iteration, 372 Rux function 36 force lines, 280 Fourier Jean Baptiste Joseph 399 Fourier series 399 Galerkin Bark Grigorievich, 46 c c