Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A posteriori analysis of the finite element discretization of some parabolic equations


Authors: A. Bergam, C. Bernardi and Z. Mghazli
Journal: Math. Comp. 74 (2005), 1117-1138
MSC (2000): Primary 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-04-01697-7
Published electronically: August 10, 2004
MathSciNet review: 2136996
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes.

RÉSUMÉ. Nous considérons la discrétisation d'équations paraboliques, soit linéaires soit semi-linéaires, par un schéma d'Euler implicite en temps et par éléments finis en espace. L'idée de cet article est de construire des indicateurs d'erreur liés à l'approximation en temps et en espace et de prouver leur équivalence avec l'erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d'éléments finis adaptés à la solution.


References [Enhancements On Off] (What's this?)

  • [AL] H.W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), 311-341. MR 85c:35059
  • [BBHM] A. Bergam, C. Bernardi, F. Hecht, Z. Mghazli, Error indicators for the mortar finite element discretization of a parabolic problem, Numerical Algorithms 34 (2003), 187-201.
  • [BG] C. Bernardi, V. Girault, A local regularization operator for triangular and quadrilateral finite elements, SIAM J. Numer. Anal. 35 (1998), 1893-1916.MR 99g:65107
  • [BM] C. Bernardi, B. Métivet, Indicateurs d'erreur pour l'équation de la chaleur, Revue européenne des éléments finis 9 (2000), 425-438.
  • [BMV] C. Bernardi, B. Métivet, R. Verfürth, Analyse numérique d'indicateurs d'erreur, in Maillage et adaptation, P.-L. George ed., Hermès (2001), 251-278.
  • [BR] C. Bernardi, G. Raugel, Approximation numérique de certaines équations paraboliques non linéaires, R.A.I.R.O. Anal. Numér. 18 (1984), 237-285. MR 86a:65097
  • [BB1] M. Bieterman, I. Babuska, The finite element method for parabolic equations. I. A posteriori error estimation, Numer. Math. 40 (1982), 339-371. MR 85d:65052a
  • [BB2] M. Bieterman, I. Babuska, The finite element method for parabolic equations. II. A posteriori error estimation and adaptive approach, Numer. Math. 40 (1982), 373-406. MR 85d:65052b
  • [Cl] P. Clément, Approximation by finite element functions using local regularization, R.A.I.R.O. Anal. Numér. 9 (1975), 77-84.MR 53:4569
  • [EJ1] K. Eriksson, C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), 43-77. MR 91m:65274
  • [EJ2] K. Eriksson, C. Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), 1729-1749.MR 96i:65081
  • [GM] M. Gabbouhy, Z. Mghazli, Un résultat d'existence de solution faible d'un système parabolique-elliptique non linéaire doublement dégénéré, Ann. Faculté des Sciences de Toulouse X (2001), 533-546.MR 2003g:35130
  • [GR] V. Girault, P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics 749, Springer-Verlag (1979). MR 83b:65122
  • [JNT] C. Johnson, Y.-Y. Nie, V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), 277-291.MR 91g:65199
  • [Li] J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Springer (1961). MR 27:3935
  • [LM] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1, Dunod (1968). MR 40:512
  • [NSV] R.H. Nochetto, G. Savaré, C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), 525-589. MR 2000k:65142
  • [Pi] M. Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), 223-237. MR 2000b:65188
  • [PR] J. Pousin, J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems, Numer. Math. 69 (1994), 213-231. MR 95k:65111
  • [Th] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics 25, Springer (1997).MR 98m:65007
  • [Ve1] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley & Teubner (1996).
  • [Ve2] R. Verfürth, A posteriori error estimates for nonlinear problems: $L^r(0,T;W^{1,\rho}(\Omega))$-error estimates for finite element discretizations of parabolic equations, Numer. Meth. for PDE 14 (1998), 487-518.MR 99g:65099
  • [Ve3] R. Verfürth, A posteriori error estimates for nonlinear problems: $ L^r(0,T;L^{\rho}(\Omega))$-error estimates for finite element discretizations of parabolic equations, Math. Comput. 67 (1998), 1335-1360.MR 99b:65120
  • [Ve4] R. Verfürth, Error estimates for some quasi-interpolation operators, Modél. Math. et Anal. Numér. 33 (1999), 695-713.MR 2001a:65149
  • [Ve5] R. Verfürth, A posteriori error estimation techniques for non-linear elliptic and parabolic pdes, Revue européenne des éléments finis 9 (2000), 377-402.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N50

Retrieve articles in all journals with MSC (2000): 65N30, 65N50


Additional Information

A. Bergam
Affiliation: Laboratoire SIANO, Département de Mathématiques et d’Informatique, Faculté des Sciences, Université Ibn Tofail, B.P. 133, Kénitra, Maroc

C. Bernardi
Affiliation: Analyse Numérique, C.N.R.S. & Université Pierre et Marie Curie,B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France

Z. Mghazli
Affiliation: Laboratoire SIANO, Département de Mathématiques et d’Informatique, Faculté des Sciences, Université Ibn Tofail, B.P. 133, Kénitra, Maroc

DOI: https://doi.org/10.1090/S0025-5718-04-01697-7
Keywords: Parabolic equations, finite elements, a posteriori analysis
Received by editor(s): January 19, 2002
Received by editor(s) in revised form: January 27, 2004
Published electronically: August 10, 2004
Additional Notes: Recherche menée dans le cadre du projet AUPELF-UREF n$^{0}$ 2000/PAS/38 et de l’A.I. France-Maroc n$^{0}$ 221/STU/00
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society