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Volume 22, Issue 5
A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma-Convergence

Toai Luong, Tadele Mengesha, Steven M. Wise & Ming Hei Wong

DOI: 10.4208/ijnam2025-1031

Int. J. Numer. Anal. Mod., 22 (2025), pp. 728-744.

Published online: 2025-05

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  • 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 $ε → 0.$

  • Keywords

Partial differential equations, phase-field approximation, diffuse domain method, diffuse interface approximation, transmission boundary conditions, gamma-convergence, reaction-diffusion equation.

  • AMS Subject Headings

35A15, 35J20, 35J25, 49J45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-728, author = {Luong , ToaiMengesha , TadeleWise , Steven M. and Wong , Ming Hei}, title = {A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma-Convergence }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {5}, pages = {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 $ε → 0.$

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1031}, url = {http://global-sci.org/intro/article_detail/ijnam/24083.html} }
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 -

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 $ε → 0.$

Luong , ToaiMengesha , TadeleWise , Steven M. and Wong , Ming Hei. (2025). A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma-Convergence . International Journal of Numerical Analysis and Modeling. 22 (5). 728-744. doi:10.4208/ijnam2025-1031
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