$Γ$-convergence of Nonlocal Dirichlet Energies With Penalty Formulations of Dirichlet Boundary Data
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909327396175872 |
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| author | Gan, Weiye Du, Qiang Shi, Zuoqiang |
| author_facet | Gan, Weiye Du, Qiang Shi, Zuoqiang |
| contents | We study nonlocal Dirichlet energies associated with a class of nonlocal diffusion models on a bounded domain subject to the conventional local Dirichlet boundary condition. The goal of this paper is to give a general framework to correctly impose Dirichlet boundary condition in nonlocal diffusion model. To achieve this, we formulate the Dirichlet boundary condition as a penalty term and use theory of $\varGamma$-convergence to study the correct form of the penalty term. Based on the analysis of $\varGamma$-convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of $\varGamma$-convergence as long as the penalty term satisfies a few mild conditions. This work provides a theoretical foundation for approximate Dirichlet boundary condition in nonlocal diffusion model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10352 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $Γ$-convergence of Nonlocal Dirichlet Energies With Penalty Formulations of Dirichlet Boundary Data Gan, Weiye Du, Qiang Shi, Zuoqiang Analysis of PDEs Numerical Analysis We study nonlocal Dirichlet energies associated with a class of nonlocal diffusion models on a bounded domain subject to the conventional local Dirichlet boundary condition. The goal of this paper is to give a general framework to correctly impose Dirichlet boundary condition in nonlocal diffusion model. To achieve this, we formulate the Dirichlet boundary condition as a penalty term and use theory of $\varGamma$-convergence to study the correct form of the penalty term. Based on the analysis of $\varGamma$-convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of $\varGamma$-convergence as long as the penalty term satisfies a few mild conditions. This work provides a theoretical foundation for approximate Dirichlet boundary condition in nonlocal diffusion model. |
| title | $Γ$-convergence of Nonlocal Dirichlet Energies With Penalty Formulations of Dirichlet Boundary Data |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2309.10352 |