Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, p. 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 : https://doi.org/10.1051/m2an:2006034
Classification:  65M60,  65M15,  65M50
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {6},
     year = {2006},
     pages = {991-1021},
     doi = {10.1051/m2an:2006034},
     zbl = {1121.65098},
     mrnumber = {2297102},
     language = {en},
     url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, pp. 991-1021. doi : 10.1051/m2an:2006034. http://www.numdam.org/item/M2AN_2006__40_6_991_0/

[1] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element method. SIAM J. Numer. Anal. 15 (1978) 736-754. | Zbl 0398.65069

[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 1105.65349

[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 1072.65124

[4] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579-608. | Zbl 0962.65096

[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[6] P. Clément, Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl 0368.65008

[7] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | Zbl 0854.65090

[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 0857.65131

[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 0835.65117

[10] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Num. (1995) 105-158. | Zbl 0829.65122

[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 1016.65074

[13] M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16 (2002) 47-75. | Zbl 0997.65123

[14] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237. | Zbl 0935.65105

[15] R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Ruhr-Universität Bochum, Report 180/1995. | Zbl 0974.65087

[16] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996). | Zbl 0853.65108

[17] R. Verfürth, A posteriori error estimates for finite element discretization of the heat equations. Calcolo 40 (2003) 195-212. | Zbl pre02216993

[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 0602.73063