This paper presents several results concerning the vector potential which can be associated with a divergence‐free function in a bounded three‐dimensional domain. Different types of boundary conditions are given, for which the existence, uniqueness and regularity of the potential are studied. This is applied firstly to the finite element discretization of these potentials and secondly to a new formulation of incompressible viscous flow problems.

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We prove pointwise a posteriori error estimates for semi-and fullydiscrete finite element methods for approximating the solution u to a parabolic model problem. Our estimates may be used to bound the finite element error u − u h L ∞ (D) , where D is an arbitrary subset of the space-time domain of the definition of the given PDE. In contrast to standard global error estimates, these estimators de-emphasize spatial error contributions from space-time regions removed from D. Our results are valid on arbitrary shape-regular simplicial meshes which may change in time, and also provide insight into the contribution of mesh change to local errors. When implemented in an adaptive method, these estimates require only enough spatial mesh refinement away from D in order to ensure that local solution quality is not polluted by global effects.

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We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two-or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients.Résumé. Nous considérons uneéquation elliptique du second ordreà coefficients discontinus ou anisotropes dans un domaine borné en dimension 2 ou 3, et sa discrétisation paréléments finis. Le but de cet article est de démontrer des estimations d'erreur a priori et a posteriori dans une norme appropriée qui soient indépendantes de la variation des coefficients.