Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes

TL;DR

This paper introduces and analyzes a discontinuous Galerkin method for simulating mass transfer through semi-permeable membranes, addressing complex interface conditions in multi-compartment PDE models with proven stability and optimal bounds.

Contribution

It presents a novel interior penalty discontinuous Galerkin method tailored for multi-compartment PDEs with nonlinear interface conditions, including analysis and numerical validation.

Findings

The method achieves optimal a priori bounds under smooth solutions.

Numerical experiments confirm stability in advection-dominated regimes.

The approach effectively models selective permeability and partial reflection.

Abstract

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime.