TY - JOUR
T1 - A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma-Convergence 
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 - 728
EP - 744
PY - 2025
DA - 2025/05
SN - 22
DO - http://doi.org/10.4208/ijnam2025-1031
UR - https://global-sci.org/intro/article_detail/ijnam/24083.html
KW - Partial differential equations, phase-field approximation, diffuse domain method, diffuse interface approximation, transmission boundary conditions, gamma-convergence, reaction-diffusion equation.
AB - <p style="text-align: justify;">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 $ε → 0.$</p>