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arXiv · 2504.17148

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

Toai Luong·Tadele Mengesha·Steven M. Wise·Ming Hei Wong
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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 $\varepsilon$, 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 $\varepsilon \to 0$. 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 $\varepsilon \to 0$.

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Toai Luong, Tadele Mengesha, Steven M. Wise, Ming Hei Wong. 2025-04-23. A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence. https://arxiv.org/abs/2504.17148

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