Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912606743166976 |
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| author | Graham, Ivan G. Spence, Euan A. |
| author_facet | Graham, Ivan G. Spence, Euan A. |
| contents | We analyse two-level Schwarz domain-decomposition GMRES preconditioners -- both the classic additive Schwarz preconditioner and a hybrid variant -- for finite-element discretisations of the Helmholtz equation with wavenumber $k$, where the coarse space consists of piecewise polynomials.
We prove results for fixed polynomial degree (in both the fine and coarse spaces), as well as for polynomial degree increasing like $\log k$. In the latter case, we exhibit choices of fine and coarse spaces such that -- up to factors of $\log k$ -- the fine and coarse spaces are both pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES iterations is bounded independently of $k$.
These are the first convergence results about a two-level Schwarz preconditioner for high-frequency Helmholtz with a coarse space that is pollution free (up to factors of $\log k$) and does not consist of problem-adapted basis functions. Additionally, these are the first $k$-explicit convergence results about any two-level completely-additive Schwarz preconditioner for high-frequency Helmholtz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation Graham, Ivan G. Spence, Euan A. Numerical Analysis We analyse two-level Schwarz domain-decomposition GMRES preconditioners -- both the classic additive Schwarz preconditioner and a hybrid variant -- for finite-element discretisations of the Helmholtz equation with wavenumber $k$, where the coarse space consists of piecewise polynomials. We prove results for fixed polynomial degree (in both the fine and coarse spaces), as well as for polynomial degree increasing like $\log k$. In the latter case, we exhibit choices of fine and coarse spaces such that -- up to factors of $\log k$ -- the fine and coarse spaces are both pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES iterations is bounded independently of $k$. These are the first convergence results about a two-level Schwarz preconditioner for high-frequency Helmholtz with a coarse space that is pollution free (up to factors of $\log k$) and does not consist of problem-adapted basis functions. Additionally, these are the first $k$-explicit convergence results about any two-level completely-additive Schwarz preconditioner for high-frequency Helmholtz. |
| title | Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.15976 |