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, 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 --converges to the energy functional of the original problem as 0. Additionally, we show that the solution of the diffuse domain problem strongly converges in H¹ (), up to a subsequence, to the solution of the original problem, as 0.
Luong et al. (Wed,) studied this question.
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