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A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

Published online by Cambridge University Press:  05 December 2008

Alexandre Ern  and
Sébastien Meunier
Alexandre Ern
Affiliation:
Université Paris-Est, CERMICS, École des Ponts, 77455 Marne-la-Vallée Cedex 2, France. [email protected]
Sébastien Meunier
Affiliation:
EDF R&D, 1 avenue du Général de Gaulle, 92141 Clamart Cedex, France. [email protected]
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Abstract

We analyze Euler-Galerkin approximations (conforming finite elements inspace and implicit Euler in time) to coupled PDE systems in which one dependentvariable, say u, is governed by an elliptic equation and the other,say p, by a parabolic-like equation. The underlying application is theporoelasticity system within the quasi-static assumption. Differentpolynomial orders are used for the u- and p-components toobtain optimally convergent a priori bounds for allthe terms in the error energy norm.Then, a residual-type a posteriori error analysis is performed. Upon extending theideas of Verfürth for the heat equation [Calcolo40 (2003)195–212],an optimally convergent bound is derived for the dominant term in theerror energy norm and an equivalence result between residual anderror is proven. The error bound can be classically split intotime error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction techniqueintroduced by Makridakis and Nochetto [SIAM J. Numer. Anal.41 (2003) 1585–1594],an optimally convergent bound is derived for the remaining terms in theerror energy norm. Numerical results are presented toillustrate the theoretical analysis.

Keywords

Finite element methodenergy norma posteriori error analysishydro-mechanical couplingporoelasticity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 353 - 375
Copyright
© EDP Sciences, SMAI, 2009

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