Nowadays, mathematical and numerical modeling has become an essential compo- nent of the general scientific process. Ever since the 1960s, numerical analysis and scientific computation have made up the most rapidly growing part of mathemat- ics. One of the challenging problems in this area is the creation of fully reliable computer simulation methods, which could become an adequate complement to ex- perimental sciences. This book aims to give an overview of mathematical methods and computer technologies focused on reliable verification of computed solutions and present recently developed methods. We hope that it will be useful for an audi- ence much larger than just advanced specialists in numerical analysis and computer simulation methods. In actuality, the book can be used in three different ways. For engineers and specialists in natural sciences interested in quantitative analy- sis of mathematical models, it is best to concentrate on algorithms and prescriptions, which explain how to measure the accuracy of a numerical solution. In Chap. 2,we discuss various error indicators, which are used in mesh adaptive numerical algo- rithms in order to achieve proper restructuring (refinement) of the computational mesh (or changing the set of trial functions). We suggest a unified approach to this question and discuss different error indicators. Chapter 3 is concerned with the ques- tion: “how can guaranteed and computable bounds of errors associated with approx- imations of differential equations be derived?”. We tried to explain this in simple terms without a deep excursion into the mathematical background. In other words, the reader whose main purpose is to use the results (estimates) will find the corre- sponding detailed recommendations. Certainly, they are given for a limited amount of typical problems. Other cases can be found in the literature cited or require addi- tional analysis (in the latter case, a good understanding of the mathematical theory is necessary). For advanced specialists interested in the development of new error estimation methods, Chaps. 3–5 are the most interesting. Here, we discuss mathematical tech- nologies that provide guaranteed error control and applications to analysis of prob- lems with uncertain data. These chapters essentially use materials exposed in the books P. Neittaanmäki and S. Repin [NR04] and S. Repin [Rep08](in[NR04]the reader can find a complete set of a posteriori error estimation theory generated by
Accuracy Verification Methods CuuDuongThanCong.com Computational Methods in Applied Sciences Volume 32 Series Editor E Oñate International Center for Numerical Methods in Engineering (CIMNE) Technical University of Catalonia (UPC) Edificio C-1, Campus Norte UPC Gran Capitán, s/n 08034 Barcelona, Spain onate@cimne.upc.edu url: http://www.cimne.com For further volumes: www.springer.com/series/6899 CuuDuongThanCong.com Olli Mali r Pekka Neittaanmäki r Sergey Repin Accuracy Verification Methods Theory and Algorithms CuuDuongThanCong.com Olli Mali Department of Mathematical Information Technology University of Jyväskylä Jyväskylä, Finland Pekka Neittaanmäki Department of Mathematical Information Technology University of Jyväskylä Jyväskylä, Finland Sergey Repin Steklov Institute of Mathematics Russian Academy of Sciences St Petersburg, Russia and University of Jyväskylä Jyväskylä, Finland ISSN 1871-3033 Computational Methods in Applied Sciences ISBN 978-94-007-7580-0 ISBN 978-94-007-7581-7 (eBook) DOI 10.1007/978-94-007-7581-7 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2013951642 © Springer Science+Business Media Dordrecht 2014 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright 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(www.springer.com) CuuDuongThanCong.com Preface Nowadays, mathematical and numerical modeling has become an essential component of the general scientific process Ever since the 1960s, numerical analysis and scientific computation have made up the most rapidly growing part of mathematics One of the challenging problems in this area is the creation of fully reliable computer simulation methods, which could become an adequate complement to experimental sciences This book aims to give an overview of mathematical methods and computer technologies focused on reliable verification of computed solutions and present recently developed methods We hope that it will be useful for an audience much larger than just advanced specialists in numerical analysis and computer simulation methods In actuality, the book can be used in three different ways For engineers and specialists in natural sciences interested in quantitative analysis of mathematical models, it is best to concentrate on algorithms and prescriptions, which explain how to measure the accuracy of a numerical solution In Chap 2, we discuss various error indicators, which are used in mesh adaptive numerical algorithms in order to achieve proper restructuring (refinement) of the computational mesh (or changing the set of trial functions) We suggest a unified approach to this question and discuss different error indicators Chapter is concerned with the question: “how can guaranteed and computable bounds of errors associated with approximations of differential equations be derived?” We tried to explain this in simple terms without a deep excursion into the mathematical background In other words, the reader whose main purpose is to use the results (estimates) will find the corresponding detailed recommendations Certainly, they are given for a limited amount of typical problems Other cases can be found in the literature cited or require additional analysis (in the latter case, a good understanding of the mathematical theory is necessary) For advanced specialists interested in the development of new error estimation methods, Chaps 3–5 are the most interesting Here, we discuss mathematical technologies that provide guaranteed error control and applications to analysis of problems with uncertain data These chapters essentially use materials exposed in the books P Neittaanmäki and S Repin [NR04] and S Repin [Rep08] (in [NR04] the reader can find a complete set of a posteriori error estimation theory generated by v CuuDuongThanCong.com vi Preface the variational duality approach and [Rep08] is mainly devoted to the method using transformations of integral identities, which define generalized solutions of boundary value problems) We recommend them for further study of the mathematical theory of a posteriori error estimation However, in this book (unlike the abovementioned publications) we pay more attention to computational aspects and try to supply the reader with practical prescriptions Chapter is devoted to a special but important topic: analysis of effects caused by indeterminacy (incomplete knowledge) of problem data It contains many new results We show that studying problems with incompletely known data leads to conceptions and methods, which differ from those used in “classical” error analysis In particular, they lead to the notion of an a priori limited accuracy, which leads to a new perspective on quantitative analysis of mathematical models Chapter and Sects 4.1.2 and 4.1.3 (related to beams) use materials of the Ph.D thesis of O Mali [Mal11] The material exposed in Chaps and may be especially interesting for specialists in computational mechanics interested in finding bounds of the accuracy generated by approximation errors and data indeterminacy The entire book (maybe with the exception of Chaps 4–6) can also be considered as a textbook for undergraduate and postgraduate students studying applied mathematics and mathematics of computations For these reasons, we append three chapters (Appendices A, B, and C), in which basic mathematical knowledge is summarized These chapters present a concise lecture course “Numerical analysis of differential equations” (which has been developed by the authors for graduate and undergraduate students of the University of Jyväskylä) It discusses the main methods used for quantitative analysis of partial differential equations Chapters and are also written in the textbook style Here, we have used materials from lecture courses on a posteriori error estimation methods that have been delivered to undergraduate and postgraduate students by S Repin in Jyväskylä, Radon Institute of Computational and Applied Mathematics in Linz, Helsinki University of Technology, and University of Saarbrucken We would like to express our gratitude to the University of Jyväskylä and to the Academy of Finland for their support We are especially grateful to I Anjam and S Matculevich for contributions to the material exposed in the book, discussions, and proofreading and to M.-L Rantalainen for her help in preparing the electronic version of our book Many materials related to theoretical justification and practical implementation of new a posteriori error estimation methods are results of joint research exposed in joint publications with our colleagues, which are referred to in the respective parts of the book We express sincere gratitude to all of them for the cooperation and interesting discussions Finally, we would like to thank Springer-Verlag publishing group for the friendly cooperation Jyväskylä, Finland 2013 CuuDuongThanCong.com Olli Mali Pekka Neittaanmäki Sergey Repin Contents Errors Arising in Computer Simulation Methods 1.1 General Scheme 1.2 Errors of Mathematical Models 1.3 Approximation Errors 1.4 Numerical Errors Indicators of Errors for Approximate Solutions of Differential Equations 2.1 Error Indicators and Adaptive Numerical Methods 2.1.1 Error Indicators for FEM Solutions 2.1.2 Accuracy of Error Indicators 2.2 Error Indicators for the Energy Norm 2.2.1 Error Indicators Based on Interpolation Estimates 2.2.2 Error Indicators Based on Approximation of the Error Functional 2.2.3 Error Indicators of the Runge Type 2.3 Error Indicators for Goal-Oriented Quantities 2.3.1 Error Indicators Relying on the Superconvergence of Averaged Fluxes in the Primal and Adjoint Problems 2.3.2 Error Indicators Using the Superconvergence of Approximations in the Primal Problem 2.3.3 Error Indicators Based on Partial Equilibration of Fluxes in the Original Problem Guaranteed Error Bounds I 3.1 Ordinary Differential Equations 3.1.1 Derivation of Guaranteed Error Bounds 3.1.2 Computation of Error Bounds 3.2 Partial Differential Equations 3.2.1 Maximal Deviation from the Exact Solution 3.2.2 Minimal Deviation from the Exact Solution 3.2.3 Particular Cases 1 3 7 16 17 23 30 35 36 38 40 45 45 46 49 53 54 57 58 vii CuuDuongThanCong.com viii Contents 3.3 3.4 3.5 3.6 3.2.4 Problems with Mixed Boundary Conditions 3.2.5 Estimates of Global Constants Entering the Majorant 3.2.6 Error Majorants Based on Poincaré Inequalities 3.2.7 Estimates with Partially Equilibrated Fluxes Error Control Algorithms 3.3.1 Global Minimization of the Majorant 3.3.2 Getting an Error Bound by Local Procedures Indicators Based on Error Majorants Applications to Adaptive Methods 3.5.1 Runge’s Type Estimate 3.5.2 Getting Approximations with Guaranteed Accuracy by an Adaptive Numerical Algorithm Combined (Primal-Dual) Error Norms and the Majorant 60 62 64 67 69 70 74 77 85 85 85 87 Guaranteed Error Bounds II 4.1 Linear Elasticity 4.1.1 Introduction 4.1.2 Euler–Bernoulli Beam 4.1.3 The Kirchhoff–Love Arch Model 4.1.4 The Kirchhoff–Love Plate 4.1.5 The Reissner–Mindlin Plate 4.1.6 3D Linear Elasticity 4.1.7 The Plane Stress Model 4.1.8 The Plane Strain Model 4.2 The Stokes Problem 4.2.1 Divergence-Free Approximations 4.2.2 Approximations with Nonzero Divergence 4.2.3 Stokes Problem in Rotating System 4.3 A Simple Maxwell Type Problem 4.3.1 Estimates of Deviations from Exact Solutions 4.3.2 Numerical Examples 4.4 Generalizations 4.4.1 Error Majorant 4.4.2 Error Minorant 93 93 93 95 98 107 110 116 120 121 123 125 127 129 132 134 140 145 147 150 Errors Generated by Uncertain Data 5.1 Mathematical Models with Incompletely Known Data 5.2 The Accuracy Limit 5.3 Estimates of the Worst and Best Case Scenario Errors 5.4 Two-Sided Bounds of the Radius of the Solution Set 5.5 Computable Estimates of the Radius of the Solution Set 5.5.1 Using the Majorant 5.5.2 Using a Reference Solution 5.5.3 An Advanced Lower Bound 5.6 Multiple Sources of Indeterminacy 5.6.1 Incompletely Known Right-Hand Side 153 153 154 157 163 169 169 170 171 175 175 CuuDuongThanCong.com Contents ix 5.6.2 The Reaction Diffusion Problem 5.7 Error Indication and Indeterminate Data 5.7.1 Numerical Experiments 5.7.2 Results and Conclusions 5.8 Linear Elasticity with Incompletely Known Poisson Ratio 5.8.1 Sensitivity of the Energy Functional 5.8.2 Example: Axisymmetric Model 177 184 185 186 188 192 197 Overview of Other Results and Open Problems 6.1 Error Estimates for Approximations Violating Conformity 6.2 Linear Elliptic Equations 6.3 Time-Dependent Problems 6.4 Optimal Control and Inverse Problems 6.5 Nonlinear Boundary Value Problems 6.5.1 Variational Inequalities 6.5.2 Elastoplasticity 6.5.3 Problems with Power Growth Energy Functionals 6.6 Modeling Errors 6.7 Error Bounds for Iteration Methods 6.7.1 General Iteration Algorithm 6.7.2 A Priori Estimates of Errors 6.7.3 A Posteriori Estimates of Errors 6.7.4 Advanced Forms of Error Bounds 6.7.5 Systems of Linear Simultaneous Equations 6.7.6 Ordinary Differential Equations 6.8 Roundoff Errors 6.9 Open Problems 205 205 206 207 208 209 210 214 215 216 217 217 218 219 220 224 230 243 244 247 247 248 248 251 252 253 253 254 257 258 259 Appendix B Boundary Value Problems B.1 Generalized Solutions of Boundary Value Problems B.2 Variational Statements of Elliptic Boundary Value Problems B.3 Saddle Point Statements of Elliptic Boundary Value Problems B.3.1 Introduction to the Theory of Saddle Points 263 263 266 272 272 Appendix A Mathematical Background A.1 Vectors and Tensors A.2 Spaces of Functions A.2.1 Lebesgue and Sobolev Spaces A.2.2 Boundary Traces A.2.3 Linear Functionals A.3 Inequalities A.3.1 The Hölder Inequality A.3.2 The Poincaré and Friedrichs Inequalities A.3.3 Korn’s Inequality A.3.4 LBB Inequality A.4 Convex Functionals CuuDuongThanCong.com x Contents B.3.2 Saddle Point Statements of Linear Elliptic Problems B.3.3 Saddle Point Statements of Nonlinear Variational Problems B.4 Numerical Methods for Boundary Value Problems B.4.1 Finite Difference Methods B.4.2 Variational Difference Methods B.4.3 Petrov–Galerkin Methods B.4.4 Mixed Finite Element Methods B.4.5 Trefftz Methods B.4.6 Finite Volume Methods B.4.7 Discontinuous Galerkin Methods B.4.8 Fictitious Domain Methods 276 285 289 290 294 296 298 300 301 305 311 313 313 314 317 318 321 328 329 332 References 335 Index 353 Appendix C A Priori Verification of Accuracy C.1 Projection Error Estimate C.2 Interpolation Theory in Sobolev Spaces C.2.1 Interpolation Operators C.2.2 Interpolation on Polygonal Sets C.3 A Priori Convergence Rate Estimates C.4 A Priori Error Estimates for Mixed FEM C.4.1 Compatibility and Stability Conditions C.4.2 Projection Estimates for Fluxes CuuDuongThanCong.com 340 References [Dob03] M Dobrowolski, On the LBB constant on stretched domains Math Nachr 254/255, 64–67 (2003) [Dör96] W Dörfler, A convergent adaptive algorithm for Poisson’s equation SIAM J Numer Anal 33(3), 1106–1124 (1996) [DR98] W Dörfler, M Rumpf, An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation Math Comput 67(224), 1361–1382 (1998) [EGH00] R Eymard, T Gallouët, R Herbin, Finite volume methods, in Handbook of Numerical Analysis, vol VII, ed by P.G Ciarlet, J.L Lions (North-Holland, Amsterdam, 2000), pp 713–1020 [EJ88] K Eriksson, C 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conforming, 289 nonconforming, 289 Aspect ratio, 319 Aubin–Nitsche estimate, 327 B Banach theorem, 218 Biharmonic problem, 109 Bilinear V-elliptic form, 264 C Compatibility conditions, 329 Contractive mapping, 218 D Dual mixed method, 299 Dual variational problem, 275, 278 E Efficiency index, 9, 40, 50, 140 Element Courant, 79, 140, 161 Hsieh–Clough–Tocher, 38, 126 Raviart–Thomas, 26 refinement, 10 Energy estimate, 265 Energy functional Euler–Bernoulli beam, 95 generalized model, 145 Kirchhoff–Love arch, 100 quadratic, 57 Reissner-Mindlin plate, 110 Error approximation, 1, caused by program defects, indeterminacy best case, 156 maximal, 156 minimal, 156 worst case, 156 numerical, 1, of a mathematical model, 1, 3, 216 defeaturing, 217 dimension reduction methods, 216 Error indicator, 8, accuracy, accuracy with respect to a marker, 14 O Mali et al., Accuracy Verification Methods, Computational Methods in Applied Sciences 32, DOI 10.1007/978-94-007-7581-7, © Springer Science+Business Media Dordrecht 2014 CuuDuongThanCong.com 353 354 Error indicator (cont.) based on averaging, 25 based on error majorant, 77 based on local subproblems, 30 based on post processing, 23 based on superconvergence, 26 classification, 17 explicit residual, 17 global averaging, 28 goal-oriented, 35, 40, 43 hierarchical, 32 least squares fitting, 28 local RT averaging, 26 local subproblems, 29 partial equilibration, 27 Runge type, 30 Error majorant, 47 abstract model, 147 algorithms, 69 based upon Poincaré estimates, 64 biharmonic problem, 110, 207 combined error norm, 87 equilibration of fluxes, 67 global minimization, 70 FEM, 70 multigrid methods, 72 NURBS, 72 indicators of local errors, 52, 77 inverse problems, 209 Kirchhoff–Love arch, 102 linear elasticity, 118 coupled problems, 119 isotropic media, 119 plane strain model, 121 plane stress model, 121 mixed Dirichlét–Robin boundary conditions, 61 obstacle problem, 212 optimal control problems, 208 Poisson equation, 59 problems in exterior domains, 207 reaction diffusion problem, 54, 182 regularization of fluxes, 74 Reissner–Mindlin plate, 116 Stokes problem approximations with nonzero divergence, 127 divergence-free approximations, 125 Sturm–Liouville problem, 47, 48 time-dependent problems, 207 Error minorant, 57 abstract model, 150 diffusion equation, 58 CuuDuongThanCong.com Index Poisson equation, 59 reaction diffusion problem, 181 F Functional bidual (bipolar), 260 coercive, 269 compound, 262 conjugate, 287 convex, 259 dual (polar), 260 forcing, 210 Gâteaux differentiable, 262 lower semicontinuous, 267 residual, 253 subgradient, 260 uniformly convex, 260, 295 G Galerkin approximation, 75, 264, 296, 299, 314 convergence, 297 Generalized derivative, 249 Generalized solution, 264 Gradient averaging, 24 H Hooke’s law, 94 Hypercircle estimate, 67 I Incompletely known data, 153 multiple sources, 175 probabilistic approach, 153 right-hand side, 175 Inequality Clarkson, 215 Friedrichs, 48, 257 Hölder, 254 Korn, 257 LBB, 258 Mikhlin, 212 Poincaré, 256 strengthened Cauchy, 34 variational, 287, 288 Young, 247 Infsup condition, 283 Interpolation Clement’s, 18, 321 in Sobolev spaces, 315 operator, 317 Iteration methods advanced estimates, 220 Chebyshev, 227 fixed point, 217 Index Iteration methods (cont.) for systems of linear simultaneous equations, 224 Gauss–Seidel, 225 Jacobi, 224 Ostrowski estimates, 220 Picard–Lindelöf, 232 SOR, 225 L Lagrangian, 272 Stokes problem, 124 Lemma Bramble–Hilbert, 315 Cea’s, 314 Deny–Lions, 315 Du–Bois–Reymond, 267 infsup stability, 258, 282 Lax–Milgram, 264, 267 properties of compound functionals, 261 Limit density property, 294 Linear elasticity 3D theory, 116 constitutive relations, 117 isotropic media, 118 plane strain model, 121 plane stress model, 120 M Machine accuracy, 244 Marker, 11 bulk criterium, 12 mean value criterium, 11 predefined percentage, 12 Method cell centered finite volume, 302 CuuDuongThanCong.com 355 discontinuous Galerkin, 305 lifting operator, 252, 308 dual mixed, 280 fictitious domain, 311 finite difference, 290 stability, 292 mixed finite element, 298 Petrov–Galerkin, 296 Trefftz, 300 variational difference, 294 P Post-processing, 23 R Roundoff errors, 243 Runge’s type estimate guaranteed error bound, 85 heuristic, 31 S Saddle point, 272 Saturation assumption, 33 Simplex, 318 Solution set, 155 radius, 155 Space Lebesgue, 249 of boundary traces, 251 of rigid deflections, 257 Sobolev, 250, 251 Stability conditions, 329 Stokes problem, 123 rotating frame, 129 Subdifferential, 260 Superconvergence, 37 ... Jyväskylä Jyväskylä, Finland ISSN 187 1-3 033 Computational Methods in Applied Sciences ISBN 97 8-9 4-0 0 7-7 58 0-0 ISBN 97 8-9 4-0 0 7-7 58 1-7 (eBook) DOI 10.1007/97 8-9 4-0 0 7-7 58 1-7 Springer Dordrecht Heidelberg... vector-valued functions with square-summable rotor subspace of L2 (Ω, Rd ) that contains vector-valued functions with square-summable divergence subspace of L2 (Ω, Rd×d ) that contains tensor-valued... of left-hand side limit and right-hand side limit set of boolean vectors space of all infinitely differentiable functions with compact supports in Ω spaces of k-times differentiable scalar-valued