A posteriori analysis of the finite element discretization of some parabolic equations

Bergam, A.; Bernardi, C.; Mghazli, Z.

doi:10.1090/S0025-5718-04-01697-7

A posteriori analysis of the finite element discretization of some parabolic equations
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by A. Bergam, C. Bernardi and Z. Mghazli PDF

Math. Comp. 74 (2005), 1117-1138 Request permission

Abstract:

We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes. Résumé. Nous considérons la discrétisation d’équations paraboliques, soit linéaires soit semi-linéaires, par un schéma d’Euler implicite en temps et par éléments finis en espace. L’idée de cet article est de construire des indicateurs d’erreur liés à l’approximation en temps et en espace et de prouver leur équivalence avec l’erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d’éléments finis adaptés à la solution.

References

Hans Wilhelm Alt and Stephan Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), no. 3, 311–341. MR 706391, DOI 10.1007/BF01176474
A. Bergam, C. Bernardi, F. Hecht, Z. Mghazli, Error indicators for the mortar finite element discretization of a parabolic problem, Numerical Algorithms 34 (2003), 187–201.
C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), no. 5, 1893–1916. MR 1639966, DOI 10.1137/S0036142995293766
C. Bernardi, B. Métivet, Indicateurs d’erreur pour l’équation de la chaleur, Revue européenne des éléments finis 9 (2000), 425–438.
C. Bernardi, B. Métivet, R. Verfürth, Analyse numérique d’indicateurs d’erreur, in Maillage et adaptation, P.-L. George ed., Hermès (2001), 251–278.
C. Bernardi and G. Raugel, Approximation numérique de certaines équations paraboliques non linéaires, RAIRO Anal. Numér. 18 (1984), no. 3, 237–285 (French, with English summary). MR 751759, DOI 10.1051/m2an/1984180302371
M. Bieterman and I. Babuška, The finite element method for parabolic equations. I. A posteriori error estimation, Numer. Math. 40 (1982), no. 3, 339–371. MR 695602, DOI 10.1007/BF01396451
M. Bieterman and I. Babuška, The finite element method for parabolic equations. I. A posteriori error estimation, Numer. Math. 40 (1982), no. 3, 339–371. MR 695602, DOI 10.1007/BF01396451
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43–77. MR 1083324, DOI 10.1137/0728003
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), no. 6, 1729–1749. MR 1360457, DOI 10.1137/0732078
Mostafa Gabbouhy and Zoubida Mghazli, Un résultat d’existence de solution faible d’un système parabolique-elliptique non linéaire doublement dégénéré, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 3, 533–546 (French, with English and French summaries). MR 1923689
V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
Claes Johnson, Yi Yong Nie, and Vidar Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), no. 2, 277–291. MR 1043607, DOI 10.1137/0727019
J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Band 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (French). MR 0153974
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
Ricardo H. Nochetto, Giuseppe Savaré, and Claudio Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525–589. MR 1737503, DOI 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M
Marco Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), no. 3-4, 223–237. MR 1673951, DOI 10.1016/S0045-7825(98)00121-2
J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. Math. 69 (1994), no. 2, 213–231. MR 1310318, DOI 10.1007/s002110050088
Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170, DOI 10.1007/978-3-662-03359-3
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley & Teubner (1996).
R. Verfürth, A posteriori error estimates for nonlinear problems: $L^r(0,T;W^{1,\rho }(\Omega ))$-error estimates for finite element discretizations of parabolic equations, Numer. Methods Partial Differential Equations 14 (1998), no. 4, 487–518. MR 1627578, DOI 10.1002/(SICI)1098-2426(199807)14:4<487::AID-NUM4>3.0.CO;2-G
R. Verfürth, A posteriori error estimates for nonlinear problems. $L^r(0,T;L^\rho (\Omega ))$-error estimates for finite element discretizations of parabolic equations, Math. Comp. 67 (1998), no. 224, 1335–1360. MR 1604371, DOI 10.1090/S0025-5718-98-01011-4
Rüdiger Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 695–713 (English, with English and French summaries). MR 1726480, DOI 10.1051/m2an:1999158
R. Verfürth, A posteriori error estimation techniques for non-linear elliptic and parabolic pdes, Revue européenne des éléments finis 9 (2000), 377–402.