A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912343992041472 |
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| author | Luong, Toai Mengesha, Tadele Wise, Steven M. Wong, Ming Hei |
| author_facet | Luong, Toai Mengesha, Tadele Wise, Steven M. Wong, Ming Hei |
| contents | 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 $\varepsilon$, 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 $\varepsilon \to 0$. 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 $\varepsilon \to 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence Luong, Toai Mengesha, Tadele Wise, Steven M. Wong, Ming Hei Analysis of PDEs 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 $\varepsilon$, 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 $\varepsilon \to 0$. 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 $\varepsilon \to 0$. |
| title | A Diffuse Domain Approximation with Transmission-Type Boundary Conditions II: Gamma--Convergence |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.17148 |