Analysis of a preconditioner for a membrane diffusion problem with the Kedem–Katchalsky transmission condition

Abstract

A composite h - p discontinuous Galerkin finite element discretization of a diffusion problem, where subdomains are separated by thin membranes, modeled by the Kedem–Katchalsky transmission condition, is considered. A preconditioner based on the additive Schwarz method is proved to have the condition number bounded independently of the mesh size, the membrane permeability and the diffusion coefficient, provided the subspace solvers have their condition numbers bounded as well. Numerical experiments confirm these findings and additionally indicate the convergence rate is only weakly dependent on the degree of the approximating polynomials.