Konstantin Brenner

Ophélie Angelini, Konstantin Brenner, Danielle Hilhorst

We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection-reaction-diffusion equation. Equations of this type arise in many contexts, such as the modeling of contaminant transport in porous media. We discretize the diffusion term, which can be anisotropic and heterogeneous, via a hybrid finite volume scheme. We construct a partially upwind scheme for the convection term. We consider a wide range of unstructured possibly non-matching polygonal meshes in arbitrary space dimension. The only assumption on the mesh is that the volume elements must be star-shaped. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet--Kolmogorov compactness theorem.

Konstantin Brenner, Clément Cancès

The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to Richards equation a strategy that consists in defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of Richards equation is then discretized thanks to a monotone Finite Volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate non-degeneracy conditions on the parametrization that Newtonâs method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach.