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

Ern, Alexandre; Meunier, Sébastien

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 353-375.

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say $u$, is governed by an elliptic equation and the other, say $p$, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the $u$- and $p$-components to obtain optimally convergent a priori bounds for all the terms in the error energy norm. Then, a residual-type a posteriori error analysis is performed. Upon extending the ideas of Verfürth for the heat equation [Calcolo 40 (2003) 195-212], an optimally convergent bound is derived for the dominant term in the error energy norm and an equivalence result between residual and error is proven. The error bound can be classically split into time error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction technique introduced by Makridakis and Nochetto [SIAM J. Numer. Anal. 41 (2003) 1585-1594], an optimally convergent bound is derived for the remaining terms in the error energy norm. Numerical results are presented to illustrate the theoretical analysis.

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