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

DOI : 10.1051/m2an:2008048
Classification : 65M60, 65M15, 74F10
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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {353--375},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {2},
     year = {2009},
     doi = {10.1051/m2an:2008048},
     mrnumber = {2512500},
     zbl = {1166.76036},
     language = {en},
     url = {http://www.numdam.org/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: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 353
EP  - 375
VL  - 43
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/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: Modélisation mathématique et analyse numérique
%D 2009
%P 353-375
%V 43
%N 2
%I EDP-Sciences
%U http://www.numdam.org/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: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 2, pp. 353-375. doi : 10.1051/m2an:2008048. http://www.numdam.org/articles/10.1051/m2an:2008048/

[1] I. Babuška, M. Feistauer and P. Šolín, On one approach to a posteriori error estimates for evolution problems solved by the method-of-lines. Numer. Math. 89 (2001) 225-256. | MR | Zbl

[2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | MR | Zbl

[3] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2005) 1117-1138. | MR | Zbl

[4] M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155-169. | JFM

[5] C. Chavant and A. Millard, Simulation d'excavation en comportement hydro-mécanique fragile. Technical report, EDF R&D/AMA and CEA/DEN/SEMT (2007) http://www.gdrmomas.org/ex_qualifications.html.

[6] Z. Chen and J. Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comp. 73 (2004) 1167-1193. | MR | Zbl

[7] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | MR | Zbl

[8] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43-77. | MR | Zbl

[9] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems 32 (1995) 706-740. | MR | Zbl

[10] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004). | MR | Zbl

[11] O. Lakkis and Ch. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp. 75 (2006) 1627-1658. | MR | Zbl

[12] Ch. Makridakis and R.H. Nochetto, Ellitpic reconstruction and a posteriori error estimates for elliptic problems. SIAM J. Numer. Anal. 41 (2003) 1585-1594. | MR | Zbl

[13] S. Meunier, Analyse d'erreur a posteriori pour les couplages hydro-mécaniques et mise en œuvre dans Code_Aster. Ph.D. Thesis, École nationale des ponts et chaussées, France (2007). | Zbl

[14] M.A. Murad and A.F.D. Loula, Improved accuracy in finite element analysis of Biot's consolidation problem. Comput. Meth. Appl. Mech. Engrg. 95 (1992) 359-382. | MR | Zbl

[15] M.A. Murad and A.F.D. Loula, On stability and convergence of finite element approximations of Biot's consolidation problem. Internat. J. Numer. Methods Engrg. 37 (1994) 645-667. | MR | Zbl

[16] M.A. Murad, V. Thomée and A.F.D. Loula, Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem. SIAM J. Numer. Anal. 33 (1996) 1065-1083. | MR | Zbl

[17] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237. | MR | Zbl

[18] R.L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | MR | Zbl

[19] R.E. Showalter, Diffusion in deformable media. IMA Volumes in Mathematics and its Applications 131 (2000) 115-130. | MR | Zbl

[20] R.E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310-340. | MR | Zbl

[21] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997). | MR | Zbl

[22] R. Verfürth, A posteriori error estimations and adaptative mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67-83. | Zbl

[23] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Chichester, UK (1996). | Zbl

[24] R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195-212. | MR | Zbl

[25] K. Von Terzaghi, Theoretical Soil Mechanics. Wiley, New York (1936).

[26] M. Wheeler, A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723-759. | MR | Zbl

Cité par Sources :