Bulk‐Surface Coupled PDE With an Open Boundary

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 we provide a rigorous error analysis. Numerical experiments confirm the theoretical convergence rates.