Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T03:03:14.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Skipping transition conditions in a posteriori error estimates for finite element discretizationsof parabolic equations

Published online by Cambridge University Press:  04 February 2010

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

Abstract

In this paper we derive a posteriori error estimates for theheat equation. The time discretizationstrategy is based on a θ-method and the mesh used for eachtime-slab is independent of the mesh used for the previoustime-slab. The novelty of this paper is an upper bound for theerror caused by the coarsening of the mesh used for computing thesolution in the previous time-slab. The technique applied forderiving this upper bound is independent of the problem and can begeneralized to other time dependent problems.

Keywords

A posteriori error estimatestransition conditionparabolic problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 455 - 484
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Babuška, I. and Rheinboldt, W.C., Error estimates for adaptive finite element method. SIAM J. Numer. Anal. 15 (1978) 736–754.
R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (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
Berrone, S., Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients. ESAIM: M2AM 40 (2006) 991–1021. 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 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
Eriksson, K., Estep, D., Hansbo, P. and Johnson, C., Introduction to adaptive methods for differential equations. Acta Numer. 4 (1995) 105–158. CrossRef
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.
Lamichhane, B.P. and Wohlmuth, B.I., Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. Calcolo 39 (2002) 219–237. CrossRef
Lamichhane, B.P., Stevenson, R.P. and Wohlmuth, B.I., Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math. 102 (2005) 93–121. CrossRef
Morin, P., Nochetto, 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
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. CrossRef
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
Wohlmuth, B.I., A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989–1012. 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