Finite Elements An Introduction to the Method and Error Estimation

by Babuska, Ivo; Whiteman, John; Strouboulis, Theofanis

ISBN13: 9780198506706

ISBN10: 0198506708

Format: Paperback

Pub. Date: 2010-12-30

Publisher(s): Oxford University Press

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Summary

Most of the many books on finite elements are devoted either to mathematical theory or to engineering applications, but not to both. This book seeks to bridge the gap by presenting the main theoretical ideas of the finite element method and the analysis of its errors in an accessible way. At the same time it presents computed numbers which not only illustrate the theory but can only be analysed using the theory. This approach, both dual and interacting between theory and computation makes this book unique. Much research is currently being done into reliability in computational modelling, involving both validation of the mathematical models and verification of the numerical schemes. By treating finite element error analysis in this way this book is a significant contribution to the verification process of finite element modelling in the context of reliability.

Author Biography

Ivo Babuka is the Robert B. Trull Chair of Engineering, Department of Aerospace Engineering, University of Texas at Austin, USA John R. Whiteman is Distinguished Professor and Director of BICOM, Institute of Computational Mathematics, Brunel University, UK Theofanis Strouboulis is a Professor in the Department of Aerospace Engineering, Texas AM University, USA

Introduction	p. 1
The finite element method	p. 1
Mathematical model	p. 1
Validation and verification	p. 2
The finite element method, error analysis and estimation and its role in the processes of verification and validation	p. 3
The purpose of this book and its layout	p. 4
Literature	p. 4
Formulations of the problems	p. 5
One-dimensional deformation of an elastic bar and one-dimensional heat conduction	p. 6
Classical differential equation formulation of the bar problem	p. 6
The principle of virtual work and weak formulation	p. 15
The principle of minimization of energy	p. 25
One-dimensional heat transfer	p. 28
Engineering application, one-dimensional heat-transfer problem	p. 29
Two-dimensional heat-conduction problem	p. 34
Classical partial differential equation formulation	p. 34
Weak formulation	p. 40
Engineering application; two-dimensional heat-transfer problem	p. 50
Finite element methods	p. 53
Introduction	p. 53
The Galerkin method	p. 54
One-dimensional finite element method	p. 60
The finite element method with piecewise linear functions	p. 61
Implementation: one-dimensional problem with piecewise linear basis functions	p. 64
Complete process for one-dimensional problem	p. 72
The finite element method with piecewise quadratic functions	p. 74
Engineering application: one-dimensional heat-transfer problem	p. 78
Two-dimensional finite element method	p. 90
The finite element method with piecewise linear functions	p. 90
The finite element method with piecewise quadratic functions	p. 96
Two benchmark problems	p. 99
Engineering application: two-dimensional heat transfer-problems	p. 104
Best approximation property of the finite element solutions	p. 118
Interpolation and its error	p. 121
Estimate of interpolation error on a single element in one dimension	p. 121
Estimate of interpolation error on a single element in two dimensions	p. 132
a priori estimates of the error of the finite element solution in the energy norm	p. 145
Introduction to a priori error analysis	p. 145
Error of the finite element solution in one dimension	p. 146
Error analysis for the one-dimensional engineering problem of Section 3.3.5	p. 164
Two-dimensional problems	p. 166
Error of the finite element solution in two dimensions	p. 166
Error analysis for Benchmark Problems 1 and 2	p. 172
Error analysis for the two-dimensional heat-transfer problem; 2D Eng Problem	p. 173
Functionals and superconvergence	p. 175
One-dimensional problems	p. 175
Error in the functionals in one dimension	p. 175
Local character of the error and pollution	p. 184
Superconvergence in one dimension	p. 189
Engineering application; one-dimensional heat-transfer problem	p. 199
Two-dimensional problems	p. 201
The error in the functional	p. 201
Local character of the error	p. 204
Superconvergence in two dimensions	p. 207
Engineering application: two-dimensional heat-transfer problem	p. 214
a posteriori error estimates	p. 216
Error indicators and estimators in one dimension	p. 217
The Dirichlet element-based error estimator	p. 217
The Neumann element-based error estimator	p. 225
The performance of the Neumann element-based error estimator	p. 228
The Dirichlet subdomain (patch) estimator	p. 229
The Neumann subdomain (patch) estimator	p. 243
The performance of the Neumann subdomain estimators for the one-dimensional engineering problems	p. 245
Averaging-based error indicators and estimators	p. 247
The performance of the ZZ-estimator for the one-dimensional engineering problems	p. 259
The Richardson error estimator	p. 259
The performance of the Richardson estimator	p. 267
Error indicators and estimators in two dimensions	p. 272
The Dirichlet element-based error estimator	p. 272
The Neumann element-based error estimator	p. 274
The performance of the Neumann element-based estimator	p. 278
The Dirichlet subdomain (patch)"estimator	p. 278
The Neumann subdomain (patch) estimator	p. 281
The performance of the Neumann subdomain (patch) estimator	p. 283
Averaging-based indicators and estimators (ZZ)	p. 285
The performance of the ZZ-estimator	p. 286
The Richardson error estimator and its performance	p. 286
Comparison of the various error estimates	p. 290
The Neumann element error estimator	p. 290
The Neumann subdomain error estimator	p. 290
Averaging-based error estimators	p. 290
The Richardson error estimator	p. 291
a posteriori error estimations for the 2D engineering problem	p. 291
The Neumann element-based estimator	p. 291
Performance of the Neumann estimator	p. 295
Performance of the ZZ-estimator	p. 297
Performance of the Dirichlet subdomain estimator	p. 299
Performance of the Richardson estimator	p. 300
The performance of the a posteriori error estimators	p. 300
Recommendations for approaching error estimation	p. 304
a posteriori estimation of errors in the functional	p. 305
Adaptive finite element methods	p. 306
A note on verification	p. 308
Epilogue	p. 309
Appendix: A	p. 311
Linear spaces, normed linear spaces, linear functionals, bilinear forms	p. 311
Linear space	p. 311
Normed linear space	p. 311
Inner product spaces	p. 311
Schwaxz inequality	p. 312
Convergence, completeness and Hilbert spaces	p. 312
Convergence	p. 312
Cauchy sequence	p. 312
Hilbert space	p. 312
Linear functionals and bilinear forms	p. 313
Linear functionals	p. 313
Bilinear forms	p. 313
The Lax-Milgram lemma	p. 313
Bibliography	p. 314
Index	p. 317
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