On the linear convergence of additive Schwarz methods for the $p$-Laplacian
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909893403869184 |
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| author | Lee, Young-Ju Park, Jongho |
| author_facet | Lee, Young-Ju Park, Jongho |
| contents | We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper, we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. Firstly, we present a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behavior akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements, we establish the asymptotic linear convergence of additive Schwarz methods for the $p$-Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09183 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the linear convergence of additive Schwarz methods for the $p$-Laplacian Lee, Young-Ju Park, Jongho Numerical Analysis 65N55, 65J15, 65K15 We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper, we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. Firstly, we present a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behavior akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements, we establish the asymptotic linear convergence of additive Schwarz methods for the $p$-Laplacian. |
| title | On the linear convergence of additive Schwarz methods for the $p$-Laplacian |
| topic | Numerical Analysis 65N55, 65J15, 65K15 |
| url | https://arxiv.org/abs/2210.09183 |