A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics
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. 694–727
Abstract
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 $ε,$ 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-type Robin boundary conditions. Our results show that, in the one dimensional space, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in $ε.$ Furthermore, we provide numerical simulations that validate and illustrate the analytical result.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2025-1030
International Journal of Numerical Analysis and Modeling, Vol. 22 (2025), Iss. 5 : pp. 694–727
Published online: 2025-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 34
Keywords: Partial differential equations phase-field approximation diffuse domain method diffuse interface approximation asymptotic analysis numerical simulation reaction-diffusion equation transmission boundary conditions.