Helmholtz boundary integral methods and the pollution effect
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912976951312384 |
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| author | Galkowski, Jeffrey Rachh, Manas Spence, Euan A. |
| author_facet | Galkowski, Jeffrey Rachh, Manas Spence, Euan A. |
| contents | This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation methods -- with subspaces consisting of $\textit{either}$ piecewise polynomials (in 2-d for collocation, in any dimension for Galerkin) $\textit{or}$ trigonometric polynomials (in 2-d) -- as well as a fully discrete quadrature (Nyström) method based on trigonometric polynomials (in 2-d).
For each of these methods, we prove -- in many cases for the first time -- rigorous results about the fundamental question: how quickly must the number of degrees of freedom (the dimension of the approximation space) grow with $k$ to maintain accuracy of the computed solution? Importantly, we determine which of these methods suffer from $\textit{the pollution effect}$. That is, we address the question: must the number of points per wavelength $\to \infty$ to maintain accuracy as $k\to\infty$? |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Helmholtz boundary integral methods and the pollution effect Galkowski, Jeffrey Rachh, Manas Spence, Euan A. Numerical Analysis Analysis of PDEs This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation methods -- with subspaces consisting of $\textit{either}$ piecewise polynomials (in 2-d for collocation, in any dimension for Galerkin) $\textit{or}$ trigonometric polynomials (in 2-d) -- as well as a fully discrete quadrature (Nyström) method based on trigonometric polynomials (in 2-d). For each of these methods, we prove -- in many cases for the first time -- rigorous results about the fundamental question: how quickly must the number of degrees of freedom (the dimension of the approximation space) grow with $k$ to maintain accuracy of the computed solution? Importantly, we determine which of these methods suffer from $\textit{the pollution effect}$. That is, we address the question: must the number of points per wavelength $\to \infty$ to maintain accuracy as $k\to\infty$? |
| title | Helmholtz boundary integral methods and the pollution effect |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2507.22797 |