MATHEMATICS OF COMPUTATION Volume 74, Number 251, Pages 1117–1138 S 0025-5718(04)01697-7 Article electronically published on August 10, 2004 A POSTERIORI ANALYSIS OF THE FINITE ELEMENT DISCRETIZATION OF SOME PARABOLIC EQUATIONS A. BERGAM, C. BERNARDI, AND Z. MGHAZLI Abstract. We are interested in the discretization of parabolic equations, ei- ther 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. R´ esum´ e. Nous consid´ erons la discr´ etisation d’´ equations paraboliques, soit lin´ eaires soit semi-lin´ eaires, par un sch´ ema d’Euler implicite en temps et par ´ el´ ements finis en espace. L’id´ ee de cet article est de construire des indica- teurs d’erreur li´ es `a l’approximation en temps et en espace et de prouver leur ´ equivalence avec l’erreur, dans le but de travailler avec des pas de temps adap- tatifs et des maillages d’´ el´ ements finis adapt´ es `a la solution. 1. Introduction For more than twenty years, an impressive amount of work has been done con- cerning a posteriori analysis and mesh adaptivity for the finite element discretiza- tion of elliptic problems; see [Ve1] and references therein. The main results consist of exhibiting local error indicators which can be computed explicitly as a function of the discrete solution and the data and such that their Hilbertian sum is equivalent to the error. Moreover, since they are local, they provide a good representation of the error distribution; hence they are very efficient tools for mesh adaptivity. However, it seems that the analogous results concerning parabolic problems are presently not complete. They most often deal either only with time scheme adaptivity (see [JNT] or [NSV]) or space finite element adaptivity (see for instance [BB1], [BB2], [BBHM] or [BM]) or with space-time finite element adaptivity (see [EJ1], [EJ2], [Ve2], [Ve3] and [Ve5]): the finite element discretization in these references relies on the space-time variational formulation of the equation and this leads to a family of error indicators which represent the combined space and time errors. In this paper, we are interested in the discretization of some parabolic equations, which relies on a standard finite element method with respect to the space variables and Euler’s implicit scheme with respect to time. Even if the same discrete problem is already considered in [Pi] and [Ve5], our adaptivity strategy is rather different. Received by the editor January 19, 2002 and, in revised form, January 27, 2004. 2000 Mathematics Subject Classification. Primary 65N30, 65N50. Key words and phrases. Parabolic equations, finite elements, a posteriori analysis. Recherche men´ ee dans le cadre du projet AUPELF-UREF n 0 2000/PAS/38 et de l’A.I. France- Maroc n 0 221/STU/00. c 2004 American Mathematical Society 1117 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use