Analysis of the diffuse domain approach for a bulk-surface coupled PDE system

TL;DR

This paper analyzes the diffuse domain approach for coupled bulk-surface PDEs, proving convergence and well-posedness, and covers various boundary conditions and surface equations.

Contribution

It provides a rigorous mathematical analysis demonstrating convergence and well-posedness of the diffuse domain approximation for coupled bulk-surface PDE systems.

Findings

Weak convergence of the solution to the coupled system

Strong convergence of the bulk quantity

Norm and strong convergence in weighted Sobolev spaces

Abstract

We analyse a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic partial differential system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approximation parameter tends to zero. Moreover, we can show strong convergence for the bulk quantity, while for the surface quantity, we can show norm convergence and strong convergence in a weighted Sobolev space. Our analysis also covers a second order surface elliptic partial differential equation and a bulk elliptic partial differential equation with Dirichlet, Neumann and Robin boundary condition.