Convergence of a Control Volume Finite Element scheme for a cross-diffusion system modeling ion transport

Abstract

An approximation of a system coupling the cross-diffusion of chemical species within a solvent, subjected to an electric field, is obtained through a control volume finite element (CVFE) scheme on general simplicial meshes in two or three space dimensions. The discrete unknowns of the numerical scheme are derived from the chemical potential of the species. The scheme is designed in order to fulfill entropy inequalities, yielding compactness properties for the discrete solutions and convergence to a weak solution of the continuous problem. Numerical illustrations of the convergence properties are provided in situations where diffusion of ionic species degenerates.

Figures (6)

• Figure 1: Triangle $S\in \mathcal{T}$ (solid line) and dual cell $K\in \mathcal{M}$ (dashed line).
• Figure 2: Relative error under space grid refinement using the arithmetic mean
• Figure 3: Concentrations of ion species, solvent and electric potential at time $T=1$ and $y=0$
• Figure 4: Initial meshes used in numerical Experiments
• Figure 5: Relative error under space grid refinement using the arithmetic mean

Theorems & Definitions (43)

• Definition 1.1: Weak Solution
• Theorem 2.1
• Definition 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4