This paper is devoted to the study of an error estimate of the nite volume approximation to the solution u 2 L 1 (IR N IR) of the equation ut + div(vf(u)) = 0, where v is a vector function depending on time and space. A \h 1=4 " error estimate for an initial value in BV (IR N) is shown for a large variety of nite volume monotoneous ux schemes, with an explicit or implicit time discretization. For this purpose, the error estimate is given for the general setting of approximate entropy solutions, where the error is expressed in terms of measures in IR N and IR N IR. The study of the implicit schemes involves the study of the existence and uniqueness of the approximate solution. The cases where an \h 1=2 " error estimate can be achieved are also discussed.

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We are concerned with the approximation of solutions to a compressible two-phase flow model in porous media thanks to an enhanced control volume finite element discretization. The originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh. Contrary to the ideas of [23], the point of the current scheme relies on a phaseby-phase "sub"-unpwinding approach so that we can recover the coercivity-like property. It allows on a second place for the preservation of the physical bounds on the discrete saturation. The convergence of the numerical scheme is therefore performed using classical compactness arguments. Numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method.

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth.

A generalized thermistor model is discretized thanks to a fully implicit vertex-centered finite volume scheme on simplicial meshes. An assumption on the stiffness coefficients is mandatory to prove a discrete maximum principle on the electric potential. This property is fundamental to handle the stability issues related to the Joule heating term. Then the convergence to a weak solution is established. Finally, numerical results are presented to show the efficiency of the methodology and to illustrate the behavior of the temperature together with the electric potential within the medium.

Models of two-phase flows in porous media, used in petroleum engineering, lead to a coupled system of two equations, one elliptic, the other degenerate parabolic, with two unknowns: the saturation and the pressure. In view of applications in hydrogeology, we are interested at the singular limit of this model, as the ratio µ of air/liquid mobility goes to infinity, and in a comparison with the one-phase Richards model. We construct a robust finite volume scheme that can apply for large values of the parameter µ. This scheme is shown to satisfy a priori estimates (the saturation is shown to remain in a fixed interval, and a discrete L 2 (0, T ; H 1 (Ω)) estimate is proved for both the pressure and a function of the saturation) which are sufficient to derive the convergence of a subsequence to a weak solution of the continuous equations, as the size of the discretization tends to zero. At the limit as the mobility of the air phase tends to infinity, we obtain the two-phase flow model introduced in the work Henry, Hilhorst and Eymard [14] (see also [13]) which we call the quasi-Richards equation.

We study the convergence of general finite volume schemes for the diphasic flow problem in porous media ut − div(u∇p) = 0 and ∆p = 0 in a bounded domain. A general formula for the numerical flux on a triangular mesh is given. The stability and an estimate of the total variation of the approximate solutions are obtained by means of a variational method; the convergence follows easily for general data.Résumé. Onétudie la convergence de schémas aux volumes finis pour le problème d'écoulement diphasique en milieu poreux ut − div(u∇p) = 0 et ∆p = 0 dans un domaine borné. Une formule générale pour le flux numérique est présentée sur un maillage triangulaire. Le fluxà deux points tel que celuià décentrage amont ou celui de Lax-Friedrichs en sont des cas particuliers. La stabilité et une estimation de la variation totale des solutions approchées sont obtenues par une méthode variationnelle; la convergence en découle facilement pour des données générales.