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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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References

Babuška, I., Feistauer, M. and Šolín, P., On one approach to a posteriori error estimates for evolution problems solved by the method-of-lines. Numer. Math. 89 (2001) 225–256. CrossRef
Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. CrossRef
Bergam, A., Bernardi, C. and Mghazli, Z., A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2005) 1117–1138. CrossRef
Biot, M.A., General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155–169. CrossRef
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.
Chen, Z. and Feng, J., An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comp. 73 (2004) 1167–1193. CrossRef
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84.
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.
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in Ɩ ∞ Ɩ 2 and Ɩ∞Ɩ∞ . SIAM J. Numer. Anal. 32 (1995) 706–740.
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004).
Lakkis, O. and Makridakis, Ch., Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp. 75 (2006) 1627–1658. CrossRef
Ch. Makridakis, R.H. Nochetto, Ellitpic reconstruction and a posteriori error estimates for elliptic problems. SIAM J. Numer. Anal. 41 (2003) 1585–1594. CrossRef
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).
Murad, M.A. and Loula, A.F.D., Improved accuracy in finite element analysis of Biot's consolidation problem. Comput. Meth. Appl. Mech. Engrg. 95 (1992) 359–382. CrossRef
Murad, M.A. and Loula, A.F.D., On stability and convergence of finite element approximations of Biot's consolidation problem. Internat. J. Numer. Methods Engrg. 37 (1994) 645–667. CrossRef
Murad, M.A., Thomée, V. and Loula, A.F.D., Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem. SIAM J. Numer. Anal. 33 (1996) 1065–1083. CrossRef
Picasso, M., Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237. CrossRef
Scott, R.L. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. CrossRef
Showalter, R.E., Diffusion in deformable media. IMA Volumes in Mathematics and its Applications 131 (2000) 115–130. CrossRef
Showalter, R.E., Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310–340. CrossRef
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997).
Verfürth, R., A posteriori error estimations and adaptative mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67–83. CrossRef
R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Chichester, UK (1996).
Verfürth, R., A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003) 195–212. CrossRef
K. von Terzaghi, Theoretical Soil Mechanics. Wiley, New York (1936).
Wheeler, M., A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. CrossRef