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A Posteriori Error Analysis via Duality TheoryA Posteriori Error Analysis for Idealizations in Linear Problems

A Posteriori Error Analysis via Duality Theory: A Posteriori Error Analysis for Idealizations in... Chapter 3 A POSTERIORI ERROR ANALYSIS FOR IDEALIZATIONS IN LINEAR PROBLEMS This chapter is devoted to a posteriori error estimates for idealizations of linear elliptic problems on nonsmooth domains. The goal is to provide easily computable, efficient estimates once solutions of the idealized problems are found. The duality theory of convex analysis, reviewed in Chapter 2, is used to de- rive a posteriori error estimates for coefficient, boundary condition and domain idealizations. The estimates obtained through the duality technique involve auxiliary functions (i.e. dual variables) subjected to certain constraints. Selec- tion of auxiliary functions influences the accuracy of an estimate dramatically. We discuss in length, especially for coefficient idealization, various selections of auxiliary functions. Numerical examples show that our selections lead to efficient error bounds. For quantitative error estimates of right-hand side idealization, only elemen- tary calculus techniques are involved. Nevertheless, for the sake of complete- ness, a detailed derivation of error bounds for the right-hand side idealization is also given. In the first four sections, we study the effects on solutions of idealizations in coefficients, the right-hand side, the boundary condition and the domain for model second-order linear elliptic boundary value problems. The model problems discussed are taken http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Posteriori Error Analysis via Duality TheoryA Posteriori Error Analysis for Idealizations in Linear Problems

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Publisher
Springer US
Copyright
© Springer Science+Business Media, Inc. 2005
ISBN
978-0-387-23536-3
Pages
67 –125
DOI
10.1007/0-387-23537-X_3
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 3 A POSTERIORI ERROR ANALYSIS FOR IDEALIZATIONS IN LINEAR PROBLEMS This chapter is devoted to a posteriori error estimates for idealizations of linear elliptic problems on nonsmooth domains. The goal is to provide easily computable, efficient estimates once solutions of the idealized problems are found. The duality theory of convex analysis, reviewed in Chapter 2, is used to de- rive a posteriori error estimates for coefficient, boundary condition and domain idealizations. The estimates obtained through the duality technique involve auxiliary functions (i.e. dual variables) subjected to certain constraints. Selec- tion of auxiliary functions influences the accuracy of an estimate dramatically. We discuss in length, especially for coefficient idealization, various selections of auxiliary functions. Numerical examples show that our selections lead to efficient error bounds. For quantitative error estimates of right-hand side idealization, only elemen- tary calculus techniques are involved. Nevertheless, for the sake of complete- ness, a detailed derivation of error bounds for the right-hand side idealization is also given. In the first four sections, we study the effects on solutions of idealizations in coefficients, the right-hand side, the boundary condition and the domain for model second-order linear elliptic boundary value problems. The model problems discussed are taken

Published: Jan 1, 2005

Keywords: Linear Problem; Duality Theory; Auxiliary Function; Posteriori Error; Lipschitz Domain

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