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Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Published online by Cambridge University Press:  15 February 2007

Stefano Berrone
Stefano Berrone*
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. sberrone@calvino.polito.it
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Abstract

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 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (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.

Keywords

A posteriori error estimatesparabolic problemsdiscontinuous coefficients.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006 , pp. 991 - 1021
Copyright
© EDP Sciences, SMAI, 2007

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References

Babuška, I. and Rheinboldt, W.C., Error estimates for adaptive finite element method. SIAM J. Numer. Anal. 15 (1978) 736–754. CrossRef
R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Num. (2001) 1–102.
Bergam, A., Bernardi, C. and Mghazli, Z., A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2004) 1117–1138. CrossRef
Bernardi, C. and Verfürth, R., Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579–608. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978).
Clément, P., Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. 9 (1975) 77–84.
Dörfler, W., A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. CrossRef
Dryja, M., Sarkis, M.V. and Widlund, O.B., Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313–348. CrossRef
Eriksson, K. and Johnson, C., Adaptive finite element methods for parabolic problems. V. Long-time integration. SIAM J. Numer. Anal. 32 (1995) 1750–1763. CrossRef
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Num. (1995) 105–158.
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.
Morin, P., Nocetto, R.H. and Siebert, K.G., Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658. CrossRef
Petzoldt, M., A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16 (2002) 47–75. CrossRef
Picasso, M., Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237. CrossRef
R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Ruhr-Universität Bochum, Report 180/1995.
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996).
Verfürth, R., A posteriori error estimates for finite element discretization of the heat equations. Calcolo 40 (2003) 195–212. CrossRef
Zienkiewicz, O.C. and Zhu, J.Z., A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24 (1987) 337–357. CrossRef