The Finite Element Method and Its Reliability

by Babuska, Ivo; Strouboulis, Theofanis

• ISBN13: 9780198502760
• ISBN10: 0198502761
• Format: Hardcover
• Copyright: 2001-11-08
• Publisher: Oxford University Press

Summary

The finite element method is a numerical method widely used inengineering. Experience shows that unreliable computation can lead to veryserious consequences. Hence reliability questions stand more and more at theforefront of engineering and theoretical interests. The present book presentsthe mathematical theory of the finite element method and focuses on the questionof how reliable computed results really are. It addresses among other topics thelocal behaviour, errors caused by pollution, superconvergence, and optimalmeshes. Many computational examples illustrate the importance of the theoreticalconclusions for practical computations. Graduate students, lecturers, andresearchers in mathematics, engineering, and scientific computation will benefitfrom the clear structure of the book, and will find this a very usefulreference.

Table of Contents

Introduction	1	(7)
Introduction	1	(1)
A brief history of the finite element method	2	(1)
On computational engineering	3	(5)
Matchematical formulation of the model problem	8	(44)
Introduction	8	(1)
The one-dimensional model problem	8	(5)
The two-dimensional model problem	13	(6)
The variational formulation of the model problem	19	(18)
The variational formulation of the model problem in two dimensions	37	(4)
Smoothness of the solution of the model problem	41	(5)
Smoothness of the solution of the two-dimendional model problem	46	(3)
Historical and bibliographical remarks	49	(3)
The finite element method	52	(134)
Introduction	52	(2)
The Galerkin method and the properties of the approximate solution	54	(10)
The Galerkin method in two dimensions	64	(2)
The finite element method	66	(14)
The finite element method in two dimensions	80	(13)
The finite element interpolation and approximation error	93	(37)
The finite element interpolation and approximation error in two dimensions	130	(10)
The finite element interpolation and approximation: a special case	140	(16)
The interpolation error for the function rαϕ(&thetas;) in two dimensions	156	(2)
The description of the mesh and analysis of its optimality	158	(22)
The description of the mesh and analysis of its optimality in two dimensions	180	(1)
Historical and bibliographical comments	180	(6)
Local behavior in the finite element method	186	(229)
Introduction	186	(3)
A-priori estimates for the nodal errors and the maximum error	189	(11)
A-priori estimates for the nodal errors and the maximum error in two dimensions	200	(1)
A-priori estimates for negative norms of the error	201	(22)
A-priori estimates for negative norms of the error in two dimensions	223	(4)
Local and pollution error in the finite element solution	227	(22)
The splitting of the error in the two-dimensional case	249	(8)
A-priori estimates for the pollution error	257	(22)
A-priori estimates for the pollution error in two dimensions	279	(10)
Further analysis of the pollution error: Interior estimates	289	(12)
The interior estimates in two dimensions	301	(1)
A-priori estimates for the local error: superconvergence	302	(16)
Superconvergence in two dimensions	318	(2)
A general approach for the analysis of local behavior of finite element methods for locally periodic meshes and its applications	320	(18)
A general approach for local asymptotic analysis and analysis of the superconvergence in two dimensions	338	(43)
Superconvergence via local averaging	381	(17)
Superconvergence based on local recovery: the two-dimensional case	398	(13)
Historical and bibliographical comments	411	(4)
A-posteriori estimation of the error	415	(237)
Introduction	415	(3)
The Residuum and the construction of lower and upper estimators for the energy norm of the error	418	(26)
The residuum and the construction of lower and upper estimators for the energy norm of the error: the two dimensional case	444	(7)
Analysis of the Dirichlet element residual estimator	451	(24)
Analysis of the Dirichlet element residual estimator in two dimensions	475	(6)
Analysis of the Neumann element residual estimator	481	(19)
Analysis of the Neumann element residual estimator in two dimensions	500	(22)
Subdomain residual estimates	522	(13)
Subdomain residual estimators in two dimensions	535	(8)
Analysis of the explicit element residual estimator	543	(8)
Analysis of the explicit element residual estimator in two dimensions	551	(12)
A-posteriori error estimates based on local averaging	563	(5)
A-posteriori error estimates based on local averaging: the two-dimensional case	568	(1)
The effectivity of the error indicators: the principles for their comparison and the detailed analysis of their quality	569	(28)
The asymptotic analysis of the effectiveness of the element error indicators in two dimensions	597	(46)
Historical and bibliographical comments	643	(9)
Guaranteed a-posteriori error estimation and a-posteriori estimation of the pollution error	652	(128)
Introduction	652	(3)
Upper and lower bounds for the energy norm of the error	655	(35)
Upper and lower bounds for the energy norm of the error in two dimensions	690	(27)
Upper and lower bounds for the error in the output, and a-posteriori estimation of the pollution	717	(28)
Upper and lower bounds for the error in the output, and a-posteriori estimation of the pollution error in two dimensions	745	(32)
Historical and bibliographical comments	777	(3)
Epilogue	780	(1)
Appendix to chapter 2	781	(6)
Appendix to chapter 3	787	(2)
Glossary of symbols	789	(3)
Author Index	792	(3)
Subject Index	795