A cut discontinuous Galerkin method for coupled bulk-surface problems

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 demonstrated 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.