no. 3
Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 3, pp. 455-484.

In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time dependent problems.

DOI : 10.1051/m2an/2010009
Classification : 65N30, 65N15, 65N50, 65J15
Mots clés : a posteriori error estimates, transition condition, parabolic problems 
@article{M2AN_2010__44_3_455_0,
     author = {Berrone, Stefano},
     title = {Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {455--484},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {3},
     year = {2010},
     doi = {10.1051/m2an/2010009},
     mrnumber = {2666651},
     zbl = {1195.65117},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010009/}
}
TY  - JOUR
AU  - Berrone, Stefano
TI  - Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2010
SP  - 455
EP  - 484
VL  - 44
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010009/
DO  - 10.1051/m2an/2010009
LA  - en
ID  - M2AN_2010__44_3_455_0
ER  - 
%0 Journal Article
%A Berrone, Stefano
%T Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2010
%P 455-484
%V 44
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010009/
%R 10.1051/m2an/2010009
%G en
%F M2AN_2010__44_3_455_0
Berrone, Stefano. Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 3, pp. 455-484. doi : 10.1051/m2an/2010009. http://www.numdam.org/articles/10.1051/m2an/2010009/

[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 Numer. 10 (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] S. Berrone, Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients. ESAIM: M2AM 40 (2006) 991-1021. | Numdam | Zbl

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

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

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

[9] 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

[10] 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

[11] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numer. 4 (1995) 105-158. | Zbl

[12] 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.

[13] B.P. Lamichhane and B.I. Wohlmuth, Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. Calcolo 39 (2002) 219-237. | Zbl

[14] B.P. Lamichhane, R.P. Stevenson and B.I. Wohlmuth, Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math. 102 (2005) 93-121. | Zbl

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

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

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

[18] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | Zbl

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

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

[21] B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38 (2000) 989-1012. | Zbl

[22] 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 :