A hybrid finite volume -- finite element method for bulk--surface coupled problems

TL;DR

This paper introduces a flexible hybrid finite volume--finite element method for coupled bulk-surface advection-diffusion problems, capable of handling complex geometries without fitted meshes, demonstrated through contaminant transport modeling.

Contribution

It presents a novel hybrid approach combining finite volume and trace finite element methods for bulk-surface problems on unfitted meshes, enhancing flexibility and applicability.

Findings

Effective handling of complex embedded geometries.

Accurate modeling of contaminant transport in fractured media.

Robust performance in steady and unsteady simulations.

Abstract

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.