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

TL;DR

This paper introduces a fully discrete Galerkin scheme for the Maxwell-Stefan system coupled with the Poisson equation, ensuring thermodynamic consistency, mass conservation, and nonnegativity, with proven convergence and numerical validation.

Contribution

It presents a novel Galerkin scheme that handles drift terms with electric fields and proves its convergence to the continuous solution.

Findings

The scheme preserves total mass and nonnegativity.

Solutions satisfy the second law of thermodynamics.

Numerical experiments reveal molar mass effects on densities and equilibration.

Abstract

A fully discrete Galerkin scheme for a thermodynamically consistent transient Max-well-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.