A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems
Provides a robust numerical method for solving coupled bulk-surface PDEs on complex geometries without conforming meshes, benefiting computational scientists in fluid-structure interaction and biology.
The paper develops a cut Discontinuous Galerkin method for coupled bulk-surface diffusion-reaction equations on embedded domains, achieving provable stability, optimal convergence, and well-conditioned system matrices regardless of mesh position.
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.