On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908354655289344 |
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| author | Imagawa, Masaki Kawagoe, Daisuke |
| author_facet | Imagawa, Masaki Kawagoe, Daisuke |
| contents | We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by $εΔ$, where $ε$ is a positive parameter and $Δ$ is the Laplacian. In this article, we show the $L^2$ strong convergence of solutions as the parameter $ε$ tends to $0$, and discuss its convergence rates assuming $H^1$ or $H^2$ regularity for original solutions. A key observation is that the convergence rate depends on the regularity of original solutions and a relation between the boundary and the advection vector field. Some numerical computations support optimality of our convergence estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_17904 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation Imagawa, Masaki Kawagoe, Daisuke Analysis of PDEs 35B25, 41A25, 35F15, 46E35 We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by $εΔ$, where $ε$ is a positive parameter and $Δ$ is the Laplacian. In this article, we show the $L^2$ strong convergence of solutions as the parameter $ε$ tends to $0$, and discuss its convergence rates assuming $H^1$ or $H^2$ regularity for original solutions. A key observation is that the convergence rate depends on the regularity of original solutions and a relation between the boundary and the advection vector field. Some numerical computations support optimality of our convergence estimates. |
| title | On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation |
| topic | Analysis of PDEs 35B25, 41A25, 35F15, 46E35 |
| url | https://arxiv.org/abs/2303.17904 |