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A Posteriori Control of Dimension Reduction Errors on Long Domains
Author(s) -
Repin Sergey,
Sauter Stefan,
Smolianski Anton
Publication year - 2004
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200410338
Subject(s) - estimator , a priori and a posteriori , dimension (graph theory) , reduction (mathematics) , mathematics , domain (mathematical analysis) , upper and lower bounds , work (physics) , matrix (chemical analysis) , distribution (mathematics) , mathematical optimization , mathematical analysis , statistics , combinatorics , geometry , physics , materials science , epistemology , composite material , thermodynamics , philosophy
In this work, we consider linear elliptic problems posed in long domains, i.e. the domains whose size in one coordinate direction is much greater than the size in the other directions. If the variation of the coefficients and right‐hand side along the emphasized direction is small, the original problem can be reduced to a lower‐dimensional one that is supposed to be much easier to solve. The a‐posteriori estimation of the error stemming from the model reduction constitutes the goal of the present work. For general coefficient matrix and right‐hand side of the equation, the reliable and efficient error estimator is derived that provides a guaranteed upper bound for the modelling error, exhibits the optimal asymptotics as the size of the domain tends to infinity and correctly indicates the local error distribution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)