TY - JOUR
T1 - A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics
AU - Luong , Toai
AU - Mengesha , Tadele
AU - Wise , Steven M.
AU - Wong , Ming Hei
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 694
EP - 727
PY - 2025
DA - 2025/05
SN - 22
DO - http://doi.org/10.4208/ijnam2025-1030
UR - https://global-sci.org/intro/article_detail/ijnam/24082.html
KW - Partial differential equations, phase-field approximation, diffuse domain method, diffuse interface approximation, asymptotic analysis, numerical simulation, reaction-diffusion equation, transmission boundary conditions.
AB - <p style="text-align: justify;">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.</p>