A trace finite element method for a class of coupled bulk-interface transport problems

TL;DR

This paper develops and analyzes a trace finite element method for coupled bulk-interface advection-diffusion problems, enabling accurate numerical solutions on unfitted meshes with optimal error estimates.

Contribution

It introduces a novel unfitted finite element approach for bulk-surface coupled PDEs with comprehensive error analysis and higher order discretization capabilities.

Findings

Proved well-posedness of the coupled bulk-surface system.

Established optimal error estimates for the finite element method.

Demonstrated the method's applicability with higher order elements and surface reconstructions.

Abstract

In this paper we study a system of advection-diffusion equations in a bulk domain coupled to an advection-diffusion equation on an embedded surface. Such systems of coupled partial differential equations arise in, for example, the modeling of transport and diffusion of surfactants in two-phase flows. The model considered here accounts for adsorption-desorption of the surfactants at a sharp interface between two fluids and their transport and diffusion in both fluid phases and along the interface. The paper gives a well-posedness analysis for the system of bulk-surface equations and introduces a finite element method for its numerical solution. The finite element method is unfitted, i.e., the mesh is not aligned to the interface. The method is based on taking traces of a standard finite element space both on the bulk domains and the embedded surface. The numerical approach allows an implicit definition of the surface as the zero level of a level-set function. Optimal order error estimates are proved for the finite element method both in the bulk-surface energy norm and the $L^2$-norm. The analysis is not restricted to linear finite elements and a piecewise planar reconstruction of the surface, but also covers the discretization with higher order elements and a higher order surface reconstruction.