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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 , is governed by an elliptic equation and the other, say , by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the - and -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.
Mots-clés : finite element method, energy norm, a posteriori error analysis, hydro-mechanical coupling, poroelasticity
@article{M2AN_2009__43_2_353_0,
author = {Ern, Alexandre and Meunier, S\'ebastien},
title = {A posteriori error analysis of {Euler-Galerkin} approximations to coupled elliptic-parabolic problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {353--375},
publisher = {EDP-Sciences},
volume = {43},
number = {2},
year = {2009},
doi = {10.1051/m2an:2008048},
zbl = {1166.76036},
mrnumber = {2512500},
language = {en},
url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2008048/}
}
TY - JOUR AU - Ern, Alexandre AU - Meunier, Sébastien TI - A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2009 SP - 353 EP - 375 VL - 43 IS - 2 PB - EDP-Sciences UR - https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2008048/ DO - 10.1051/m2an:2008048 LA - en ID - M2AN_2009__43_2_353_0 ER -
%0 Journal Article %A Ern, Alexandre %A Meunier, Sébastien %T A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2009 %P 353-375 %V 43 %N 2 %I EDP-Sciences %U https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2008048/ %R 10.1051/m2an:2008048 %G en %F M2AN_2009__43_2_353_0
Ern, Alexandre; Meunier, Sébastien. A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 2, pp. 353-375. doi : 10.1051/m2an:2008048. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2008048/
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