In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate of order h, assuming only the W 2,p (for p > 2) regularity of the continuous solution, on a mesh of quadrangles. The proof is based on an extension of the ideas developed in [12]. Some new difficulties arise here, due to the weak regularity of the solution, and the necessity to approximate the entire gradient, and not only its normal component, as in [12]. Résumé. Dans cet article, onétudie une classe de schémas volumes finis sur des maillages stucturés généraux, pour un problème linéaire de convection diffusion. La convection est approchée par un schéma décentré amont, et la diffusion par un schéma dit "des cellules diamants" [4]. On démontre une estimation d'erreur d'ordre h pour une solution continue dans W 2,p (p > 2), sur des maillages de quadrangles. La démonstration est une généralisation des idées de [12]. Les nouvelles difficultés sont la régularité plus faible de la solution exacte et la nécessité de construire une approximation du gradient et pas seulement de sa composante normale aux interfaces.

We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch models.