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Volume 22, Issue 5
A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics

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

Int. J. Numer. Anal. Mod., 22 (2025), pp. 694-727.

Published online: 2025-05

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  • 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.

  • AMS Subject Headings

35B40, 35C20, 65N55, 35J25, 65H05, 65N12

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-694, author = {Luong , ToaiMengesha , TadeleWise , Steven M. and Wong , Ming Hei}, title = {A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {5}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1030}, url = {http://global-sci.org/intro/article_detail/ijnam/24082.html} }
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 -

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.

Luong , ToaiMengesha , TadeleWise , Steven M. and Wong , Ming Hei. (2025). A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics. International Journal of Numerical Analysis and Modeling. 22 (5). 694-727. doi:10.4208/ijnam2025-1030
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