On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914678185132032 |
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| author | Gwiazda, Piotr Skrzeczkowski, Jakub Trussardi, Lara |
| author_facet | Gwiazda, Piotr Skrzeczkowski, Jakub Trussardi, Lara |
| contents | It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $λ\to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrtλ$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $λ$ could be linked to the discretization parameters, yielding appropriate error estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12772 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation Gwiazda, Piotr Skrzeczkowski, Jakub Trussardi, Lara Numerical Analysis Analysis of PDEs It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $λ\to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrtλ$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $λ$ could be linked to the discretization parameters, yielding appropriate error estimates. |
| title | On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2306.12772 |