Discontinuous Galerkin Methods for Mass Transfer through Semipermeable Membranes

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multicompartment partial differential equation models, interconnected with interface conditions, is presented and analyzed. The study of interface problems is motivated by models of mass transfer of solutes through semipermeable 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 modeling 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 analyzed 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.