Bulk-Surface Coupled PDE with an Open Boundary

TL;DR

This paper investigates a coupled bulk-surface PDE with an open boundary, reformulating it as an integro-differential equation, analyzing singularities, and developing a finite element method with proven convergence.

Contribution

It introduces a novel finite element approach that accounts for edge singularities in bulk-surface PDEs with open boundaries, supported by rigorous analysis.

Findings

Existence and uniqueness of solutions established.

Asymptotic expressions for edge singularities derived.

Numerical experiments confirm theoretical convergence rates.

Abstract

We study a bulk-surface coupled Laplace system involving an embedded open boundary. The problem is reformulated as an integro-differential equation using boundary integral representations, for which we establish existence and uniqueness of the solution. A Wiener-Hopf technique is employed to study the solution regularity and derive asymptotic expressions for the edge singularity. Building on these results, we develop a finite element method that incorporates the singularity structure and provide a rigorous error analysis. Numerical experiments confirm the theoretical convergence rates.