Year: 2025
Author: Toai Luong, Tadele Mengesha, Steven M. Wise, Ming Hei Wong
International Journal of Numerical Analysis and Modeling, Vol. 22 (2025), Iss. 5 : pp. 728–744
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 $ε → 0.$
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2025-1031
International Journal of Numerical Analysis and Modeling, Vol. 22 (2025), Iss. 5 : pp. 728–744
Published online: 2025-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: Partial differential equations phase-field approximation diffuse domain method diffuse interface approximation transmission boundary conditions gamma-convergence reaction-diffusion equation.