Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 6, pp. 991-1021.

In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2θ1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237; Verfürth, Calcolo 40 (2003) 195-212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313-348; Petzoldt, Adv. Comput. Math. 16 (2002) 47-75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

DOI : 10.1051/m2an:2006034
Classification : 65M60, 65M15, 65M50
Mots clés : a posteriori error estimates, parabolic problems, discontinuous coefficients
@article{M2AN_2006__40_6_991_0,
     author = {Berrone, Stefano},
     title = {Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {991--1021},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {6},
     year = {2006},
     doi = {10.1051/m2an:2006034},
     mrnumber = {2297102},
     zbl = {1121.65098},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2006034/}
}
TY  - JOUR
AU  - Berrone, Stefano
TI  - Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2006
SP  - 991
EP  - 1021
VL  - 40
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2006034/
DO  - 10.1051/m2an:2006034
LA  - en
ID  - M2AN_2006__40_6_991_0
ER  - 
%0 Journal Article
%A Berrone, Stefano
%T Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2006
%P 991-1021
%V 40
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2006034/
%R 10.1051/m2an:2006034
%G en
%F M2AN_2006__40_6_991_0
Berrone, Stefano. Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 6, pp. 991-1021. doi : 10.1051/m2an:2006034. http://archive.numdam.org/articles/10.1051/m2an:2006034/

[1] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element method. SIAM J. Numer. Anal. 15 (1978) 736-754. | Zbl

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

[3] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2004) 1117-1138. | Zbl

[4] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579-608. | Zbl

[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978). | MR | Zbl

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

[7] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | Zbl

[8] M. Dryja, M.V. Sarkis and O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313-348. | Zbl

[9] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. V. Long-time integration. SIAM J. Numer. Anal. 32 (1995) 1750-1763. | Zbl

[10] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Num. (1995) 105-158. | Zbl

[11] B.S. Kirk, J.W. Peterson, R. Stogner and S. Petersen, LibMesh. The University of Texas, Austin, CFDLab and Technische Universität Hamburg, Hamburg. http://libmesh.sourceforge.net.

[12] P. Morin, R.H. Nocetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631-658. | Zbl

[13] M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16 (2002) 47-75. | Zbl

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

[15] R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Ruhr-Universität Bochum, Report 180/1995. | Zbl

[16] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996). | Zbl

[17] R. Verfürth, A posteriori error estimates for finite element discretization of the heat equations. Calcolo 40 (2003) 195-212.

[18] O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24 (1987) 337-357. | Zbl

Cité par Sources :