A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma-Convergence

Abstract

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness $ε,$ which scales with the minimum grid size. This reformulation extends the problem to a regular domain, incorporating boundary conditions via singular source terms. In this work, we analyze the convergence of a DDM approximation problem with transmission-type Neumann boundary conditions. We prove that the energy functional of the diffuse domain problem $Γ$-converges to the energy functional of the original problem as $ε → 0.$ Additionally, we show that the solution of the diffuse domain problem strongly converges in $H^1 (Ω),$ up to a subsequence, to the solution of the original problem, as $ε → 0.$