Convergence of the Semi-Discrete WaveHoltz Iteration
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908691764084736 |
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| author | Rotem, Amit Runborg, Olof Appelo, Daniel |
| author_facet | Rotem, Amit Runborg, Olof Appelo, Daniel |
| contents | In this paper we prove that for stable semi-discretizations of the wave equation for the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem, if it exists. We show that for certain classes of frequency domain problems, the WaveHoltz iteration without acceleration converges in $O(ω)$ iterations with the constant factor depending logarithmically on the desired tolerance. We conjecture that the Helmholtz problem in open domains with no trapping waves is one such class of problems and we provide numerical examples in one and two dimensions using finite differences and discontinuous Galerkin discretizations which demonstrate these converge results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of the Semi-Discrete WaveHoltz Iteration Rotem, Amit Runborg, Olof Appelo, Daniel Numerical Analysis 65N12, 65N22 In this paper we prove that for stable semi-discretizations of the wave equation for the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem, if it exists. We show that for certain classes of frequency domain problems, the WaveHoltz iteration without acceleration converges in $O(ω)$ iterations with the constant factor depending logarithmically on the desired tolerance. We conjecture that the Helmholtz problem in open domains with no trapping waves is one such class of problems and we provide numerical examples in one and two dimensions using finite differences and discontinuous Galerkin discretizations which demonstrate these converge results. |
| title | Convergence of the Semi-Discrete WaveHoltz Iteration |
| topic | Numerical Analysis 65N12, 65N22 |
| url | https://arxiv.org/abs/2407.06929 |