Convergence of an implicit Euler Galerkin scheme for Poisson–Maxwell–Stefan systems

Abstract

A fully discrete Galerkin scheme for a thermodynamically consistent transient Maxwell-Stefan system for the mass particle densities, coupled to the Poisson equation for the electric potential, is investigated. The system models the diffusive dynamics of an isothermal ionized fluid mixture with vanishing barycentric velocity. The equations are studied in a bounded domain, and different molar masses are allowed. The Galerkin scheme preserves the total mass, the nonnegativity of the particle densities, their boundedness, and satisfies the second law of thermodynamics in the sense that the discrete entropy production is nonnegative. The existence of solutions to the Galerkin scheme and the convergence of a subsequence to a solution to the continuous system is proved. Compared to previous works, the novelty consists in the treatment of the drift terms involving the electric field. Numerical experiments show the sensitive dependence of the particle densities and the equilibration rate on the molar masses.

Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport

“…It turned out that this approach is not working for the finite-element discretization. Furthermore, the computationally cheaper implementation used in [23] for a similar scheme in one space dimension, where a Newton and Picard iteration are combined, did not work well in the two-dimensional test cases presented in this paper.…”

“…A lumped finite-element method was analyzed in [13] for a reaction-cross-diffusion equation on a stationary surface with positive definite diffusion matrix. In [23], an implicit Euler Galerkin approximation in entropy variables for a Poisson-Maxwell-Stefan system was shown to converge. Recently, an abstract framework for the numerical approximation of evolution problems with entropy structure was presented in [10].…”

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

“…An natural solution to pass to order two is to rather consider mobilities given by arithmetic means [12]. The motivation of the finite element scheme proposed in [27] is also the same. However, the scheme proposed in [27] is expressed in entropy (or dual) variables (in our context log(u i )) leading to computational difficulties when the concentrations are close to 0.…”

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

“…The motivation of the finite element scheme proposed in [45] is also the same. However, the scheme proposed in [45] is expressed in entropy (or dual) variables (in our context log(u i )) leading to computational difficulties when the concentrations are close to 0. Other entropy stable numerical schemes have been proposed for cross-diffusion systems, as for instance discontinuous Galerkin schemes in [53], or finite volumes on staggered cartesian grids for Maxwell-Stefan cross-diffusion in [42].…”

“…A natural solution to pass to order two is to rather consider mobilities given by arithmetic means [23]. The motivation of the finite element scheme proposed in [45] is also the same. However, the scheme proposed in [45] is expressed in entropy (or dual) variables (in our context log(u i )) leading to computational difficulties when the concentrations are close to 0.…”