Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This approach reformulates the original equations on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM for a two-sided problem with transmission Robin boundary conditions. Our results show that, in one dimension, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in $\varepsilon$. We provide numerical simulations that validate and illustrate the analytical result. Furthermore, for the Neumann boundary condition case, we show that the associated energy functional of the diffuse domain approximation $\Gamma-$convergences to the energy functional of the original problem, and the solution of the diffuse domain approximation strongly converges, up to a subsequence, to the solution of the original problem in $H^1(\Omega)$, as $\varepsilon \to 0$.

翻译：扩散域方法在复杂几何上逼近偏微分方程解方面受到广泛关注。该方法通过将尖锐边界界面替换为厚度为$\varepsilon$的扩散层来隐式表示几何结构，该厚度与最小网格尺寸成比例。此方法将原始方程重构于扩展的正则域上，并通过奇异源项纳入边界条件。本文对具有传输Robin边界条件的双边问题进行了扩散域方法的匹配渐近分析。结果表明，在一维情形下，扩散域近似解渐近收敛于原问题解，且关于$\varepsilon$具有精确的一阶精度。我们提供了数值模拟以验证并阐明该解析结果。此外，对于Neumann边界条件情形，我们证明了扩散域近似对应的能量泛函$\Gamma-$收敛于原问题的能量泛函，且当$\varepsilon \to 0$时，扩散域近似解在$H^1(\Omega)$空间中强收敛（至子序列）于原问题解。