Convergence of the Waveholtz Iteration on $\mathbb{R}^d$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911216590389248 |
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| author | Runborg, Olof Backman, Elliot |
| author_facet | Runborg, Olof Backman, Elliot |
| contents | In this paper we analyse the Waveholtz method, a time-domain iterative method for solving the Helmholtz iteration, in the constant-coefficient case in all of $\mathbb{R}^d$. We show that the difference between a Waveholtz iterate and the outgoing Helmholtz solution satisfies a Helmholtz equation with a particular kind of forcing. For this forcing, we prove a frequency-explicit estimate in weighted Sobolev norms, that shows a decrease of the differences as $1/\sqrt{n}$ in terms of the iteration number $n$. This guarantees the convergence of the real parts of the Waveholtz iterates to the real part of the outgoing solution of the Helmholtz equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15606 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of the Waveholtz Iteration on $\mathbb{R}^d$ Runborg, Olof Backman, Elliot Numerical Analysis 65M12, 35J05 In this paper we analyse the Waveholtz method, a time-domain iterative method for solving the Helmholtz iteration, in the constant-coefficient case in all of $\mathbb{R}^d$. We show that the difference between a Waveholtz iterate and the outgoing Helmholtz solution satisfies a Helmholtz equation with a particular kind of forcing. For this forcing, we prove a frequency-explicit estimate in weighted Sobolev norms, that shows a decrease of the differences as $1/\sqrt{n}$ in terms of the iteration number $n$. This guarantees the convergence of the real parts of the Waveholtz iterates to the real part of the outgoing solution of the Helmholtz equation. |
| title | Convergence of the Waveholtz Iteration on $\mathbb{R}^d$ |
| topic | Numerical Analysis 65M12, 35J05 |
| url | https://arxiv.org/abs/2510.15606 |