No abstract

In the present work, we deal with the convergence of cell-centered nonlinear finite volume schemes for anisotropic and heterogeneous diffusion operators. A general framework for the convergence study of finite volume methods is provided and used to establish the convergence of the new methods. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with linear finite volume schemes is provided.

We propose an efficient nonlinear solver for the resolution of the Richards equation. It is based on variable switching and is easily implemented thanks to a fictitious variable allowing to describe both the saturation and the pressure. Numerical experiments show that our method enables to use Newton's method with large time steps, reasonable number of iterations and in regions where the pressure-saturation relationship is given by a graph.

In this paper, we consider a dead-oil model where the capillary pressure is neglected. In a multidimensional space and for a phase-by-phase upstream weigthing cell-centered finite volume scheme, we prove the pressure estimates, the existence of solutions to the discrete equations and the stability of the saturation calculation. This is done in the explicit case as well as in the implicit case. Some numerical tests show the convergence of the scheme.

This work proposes an a priori error estimate of a multiscale finite element method to solve convection-diffusion problems where both velocity and diffusion coefficient exhibit strong variations at a scale which is much smaller than the domain of resolution. In that case, classical discretization methods, used at the scale of the heterogeneities, turn out to be too costly. Our method, introduced in [3], aims at solving this kind of problems on coarser grids with respect to the size of the heterogeneities by means of particular basis functions. These basis functions are defined using cell problems and are designed to reproduce the variations of the solution on an underlying fine grid. Since all cell problems are independent from each other, these problems can be solved in parallel, which makes the method very efficient when used on parallel architectures. This article focuses on the proof of an a priori error estimate of this method.