Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces

Abstract

We study a sharp interface model in the context of seawater intrusion in an anisotropic unconfined aquifer. It is a degenerate parabolic system with cross‐diffusion modeling the flow of fresh and saltwater. We study a nonlinear control volume finite element scheme. This scheme ensures the nonnegativity of the discrete solution without any restriction on the transmissibility coefficients. Moreover, it also provides a control on the entropy. The existence of a discrete solution and the convergence of this scheme are obtained, based on nonlinear stability results.

A convergent finite volume scheme for dissipation driven models with volume filling constraint

“…Prior to discretizing the model under consideration, we discuss several aspects of the continuous model. First, we highlight in Section 2.1 that the continuous model can be interpreted as the generalized gradient flow of the energy E in a geometry related to optimal transportation constrained by the volume filling condition (2). A reformulation of the problem is then proposed in Section 2.2 in order to make explicit the corresponding Lagrange multiplier, while we introduce in Section 2.3 an entropy production estimate on which the definition for weak solutions and our numerical analysis will rely.…”

A convergent entropy diminishing finite volume scheme for a cross-diffusion system

“…The implementation of the discrete entropy method [9] for cross-diffusion systems is more recent. Let us cite [1,2] where upstream mobility finite volume and control volume finite elements schemes for a multiphase extension of the porous medium equation are studied. Upwinding is also used in [7] to approximate the solution of a system which is very close to the problem (1) under study, or in [8] for a problem in which nonlocal interactions are also considered.…”

“…Upwinding is also used in [7] to approximate the solution of a system which is very close to the problem (1) under study, or in [8] for a problem in which nonlocal interactions are also considered. As a consequence of the upwind choice for the mobility, the schemes presented in [1,2,7] and [8] are first order accurate in space. An natural solution to pass to order two is to rather consider mobilities given by arithmetic means [12].…”

A Convergent Entropy Diminishing Finite Volume Scheme for a Cross-Diffusion System

“…The implementation of the discrete entropy method [19] for cross-diffusion systems is more recent. Let us cite [1,2] where upstream mobility finite volume and control volume finite element schemes for a multiphase extension of the porous medium equation are studied. Upwinding is also used in [12] to approximate the solution of a system which is very close to the problem (1.1) under study, or in [17] for a problem in which nonlocal interactions are also considered.…”

“…Upwinding is also used in [12] to approximate the solution of a system which is very close to the problem (1.1) under study, or in [17] for a problem in which nonlocal interactions are also considered. As a consequence of the upwind choice for the mobility, the schemes presented in [1,2,12] and [17] are first-order accurate in space. A natural solution to pass to order two is to rather consider mobilities given by arithmetic means [23].…”