A hybrid finite volume -- finite element method for bulk--surface coupled problems
This work addresses the need for flexible numerical methods for bulk-surface coupled problems in applications like contaminant transport in fractured porous media, but the results are incremental as it combines existing methods without a new paradigm.
The paper develops a hybrid finite volume-finite element method for solving coupled advection-diffusion equations in a bulk domain and on an embedded surface, demonstrating flexibility in handling complex geometries like curvilinear or branching structures through numerical experiments.
The paper develops a hybrid method for solving a system of advection--diffusion equations in a bulk domain coupled to advection--diffusion equations on an embedded surface. A monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface. In our approach, the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way. Moreover, a triangulation of the surface into regular shaped elements is not required. The background mesh is an octree grid with cubic cells. As an example of an application, we consider the modeling of contaminant transport in fractured porous media. One standard model leads to a coupled system of advection--diffusion equations in a bulk (matrix) and along a surface (fracture). A series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach. The method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures.