The finite element method and its reliability / Ivo Babuška and Theofanis Strouboulis.

Format:

Book

Author/Creator:

Babuška, Ivo.

Contributor:

Strouboulis, Theofanis.

Series:

Numerical mathematics and scientific computation

Language:

English

Subjects (All):

Finite element method.

Reliability (Engineering).

Physical Description:

xi, 802 pages : illustrations ; 25 cm.

Place of Publication:

Oxford, [England] : Clarendon Press ; New York : Oxford University Press, 2001.

Summary:

The finite element method is a numerical method widely used in engineering. Experience shows that unreliable computation can lead to very serious consequences. Hence reliability questions stand are at the forefront of engineering and theoretical interests. This book presents the mathematical theory of the finite element method and is the first to focus on the questions of how reliable computed results really are. It addresses among other topics the local behaviour, errors caused by pollution, superconvergence, and optimal meshes. Many computational examples illustrate the importance of the theoretical conclusions for practical computations. Graduate students, lecturers, and researchers in mathematics, engineering, and scientific computation will benefit from the clear structure of the book, and will find this a very useful reference.

Contents:

1.1 A brief history of the finite element method 2

1.2 On computational engineering 3

2 Mathematical formulation of the model problem 8

2.1 The one-dimensional model problem 8

2.1 The two-dimensional model problem 13

2.2 The variational formulation of the model problem 19

2.2 The variational formulation of the model problem in two dimensions 37

2.3 Smoothness of the solution of the model problem 41

2.3 Smoothness of the solution of the two-dimensional model problem 46

2.4 Historical and bibliographical remarks 49

3 The finite element method 52

3.1 The Galerkin method and the properties of the approximate solution 54

3.1 The Galerkin method in two dimensions 64

3.2 The finite element method 66

3.2 The finite element method in two dimensions 80

3.3 The finite element interpolation and approximation error 93

3.3 The finite element interpolation and approximation error in two dimensions 130

3.4 The finite element interpolation and approximation: a special case 140

3.4 The interpolation error for the function r[superscript [alpha open phi]([theta]) in two dimensions 156

3.5 The description of the mesh and analyses of its optimality 158

3.5 The description of the mesh and analysis of its optimality in two dimensions 180

3.6 Historical and bibliographical comments 180

4 Local behavior in the finite element method 186

4.1 A-priori estimates for the nodal errors and the maximum error 189

4.1 A-priori estimates for the nodal errors and the maximum error in two dimensions 200

4.2 A-priori estimates for negative norms of the error 201

4.2 A-priori estimates for negative norms of the error in two dimensions 223

4.3 Local and pollution error in the finite element solution 227

4.3 The splitting of the error in the two-dimensional case 249

4.4 A-priori estimates for the pollution error 257

4.4 A-priori estimates for the pollution error in two dimensions 279

4.5 Further analysis of the pollution error: Interior estimates 289

4.5 The interior estimates in two dimensions 301

4.6 A-priori estimates for the local error: superconvergence 302

4.6 Superconvergence in two dimensions 318

4.7 A general approach for the analysis of local behavior of finite element methods for locally periodic meshes and its applications 320

4.7 A general approach for local asymptotic analysis and analysis of the supercovergence in two dimensions 338

4.8 Superconvergence via local averaging 381

4.8 Superconvergence based on local recovery: the two-dimensional case 398

4.9 Historical and bibliographical comments 411

5 A-posteriori estimation of the error 415

5.1 The residuum and the construction of lower and upper estimators for the energy norm of the error 418

5.1 The residuum and the construction of lower and upper estimators for the energy norm of the error: the two dimensional case 444

5.2 Analysis of the Dirichlet element residual estimator 451

5.2 Analysis of the Dirichlet element residual estimator in two dimensions 475

5.3 Analysis of the Neumann element residual estimator 481

5.3 Analysis of the Neumann element residual estimator in two dimensions 500

5.4 Subdomain residual estimates 522

5.4 Subdomain residual estimates in two dimensions 535

5.5 Analysis of the explicit element residual estimator 543

5.5 Analysis of the explicit element residual estimator in two dimensions 551

5.6 A-posteriori error estimates based on local averaging 563

5.6 A-posteriori estimates based on local averaging: the two-dimensional case 568

5.7 The effectivity of the error indicators: the principles for their comparison and the detailed analysis of their quality 569

5.7 The asymptotic analysis of the effectiveness of the element error indicators in two dimensions 597

5.8 Historical and bibliographical comments 643

6 Guaranteed a-posteriori error estimation and a-posteriori estimation of the pollution error 652

6.1 Upper and lower bounds for the energy norm of the error 655

6.1 Upper and lower bounds for the energy norm of the error in two dimensions 690

6.2 Upper and lower bounds for the error in the output, and a-posteriori estimation of the pollution 717

6.2 Upper and lower bounds for the error in the outputs, and a-posteriori estimation of the pollution error in two dimensions 745

6.3 Historical and bibliographical comments 777.

Notes:

Includes bibliographical references and indexes.

ISBN:

0198502761

OCLC:

45683357