Convergence rates of the fractional to the local Dirichlet problem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918532832296960 |
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| author | Bungert, Leon del Teso, Félix |
| author_facet | Bungert, Leon del Teso, Félix |
| contents | We prove non-asymptotic rates of convergence in the $W^{s,2}(\mathbb R^d)$-norm for the solution of the fractional Dirichlet problem to the solution of the local Dirichlet problem as $s\uparrow 1$. For regular enough boundary values we get a rate of order $\sqrt{1-s}$, while for less regular data the rate is of order $\sqrt{(1-s)|\log(1-s)|}$. We also obtain results when the right hand side depends on $s$, and our error estimates are true for all $s\in(0,1)$. The proofs use variational arguments to deduce rates in the fractional Sobolev norm from energy estimates between the fractional and the standard Dirichlet energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03299 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rates of the fractional to the local Dirichlet problem Bungert, Leon del Teso, Félix Analysis of PDEs 35A15, 35B30, 35B40, 35R11 We prove non-asymptotic rates of convergence in the $W^{s,2}(\mathbb R^d)$-norm for the solution of the fractional Dirichlet problem to the solution of the local Dirichlet problem as $s\uparrow 1$. For regular enough boundary values we get a rate of order $\sqrt{1-s}$, while for less regular data the rate is of order $\sqrt{(1-s)|\log(1-s)|}$. We also obtain results when the right hand side depends on $s$, and our error estimates are true for all $s\in(0,1)$. The proofs use variational arguments to deduce rates in the fractional Sobolev norm from energy estimates between the fractional and the standard Dirichlet energy. |
| title | Convergence rates of the fractional to the local Dirichlet problem |
| topic | Analysis of PDEs 35A15, 35B30, 35B40, 35R11 |
| url | https://arxiv.org/abs/2408.03299 |