A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma-Convergence
Year: 2025
Type: article
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 $ε → 0.$
Authors: Toai Luong, Tadele Mengesha, Steven M. Wise, Ming Hei Wong
Institution: University of Tennessee at Knoxville
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4410724282