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Duality Based A Posteriori Error Estimator for the Dirichlet Problem
Author(s) -
Repin S.,
Sauter S.,
Smolianski A.
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310238
Subject(s) - estimator , mathematics , sobolev space , duality (order theory) , orthogonality , stein's unbiased risk estimate , a priori and a posteriori , dirichlet distribution , discretization , norm (philosophy) , mathematical optimization , bias of an estimator , boundary value problem , mathematical analysis , minimum variance unbiased estimator , statistics , geometry , discrete mathematics , philosophy , epistemology , political science , law
The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. All the derivations are done on continuous level, and the estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H 1 , independently of the discretization method chosen. In particular, we make no use of the Galerkin orthogonality, which enables us to implement the estimator for measuring the error of the fictitious domain/penalty finite element method. The estimator is easily computable, and the only constant present in the estimator is the one from Friedrichs' inequality; the constant depends solely on the domain geometry, and the estimator is quite non‐sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.