A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems

TL;DR

This paper introduces a stable and optimally convergent cut Discontinuous Galerkin method for coupled bulk-surface diffusion-reaction problems, with proven stability and good conditioning regardless of domain positioning.

Contribution

The paper develops a novel cutDGM for coupled bulk-surface equations, ensuring stability, optimal convergence, and well-conditioned systems on unfitted meshes.

Findings

Method is stable and optimally convergent.

System matrix remains well-conditioned regardless of domain position.

Numerical experiments confirm theoretical properties.

Abstract

We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted background mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demon- strated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the background mesh.