Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies
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arXiv
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| Format: | Preprint |
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2023
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| author | Averseng, Martin Spence, Euan A. Galkowski, Jeffrey |
| author_facet | Averseng, Martin Spence, Euan A. Galkowski, Jeffrey |
| contents | For $h$-FEM discretisations of the Helmholtz equation with wavenumber $k$, we obtain $k$-explicit analogues of the classic local FEM error bounds of [Nitsche, Schatz 1974], [Wahlbin 1991], [Demlow, Guzmán, Schatz 2011], showing that these bounds hold with constants independent of $k$, provided one works in Sobolev norms weighted with $k$ in the natural way.
We prove two main results: (i) a bound on the local $H^1$ error by the best approximation error plus the $L^2$ error, both on a slightly larger set, and (ii) the bound in (i) but now with the $L^2$ error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the $k$-explicit analogue of the main result of [Demlow, Guzmán, Schatz, 2011]. The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of $k^{-1}$) and is the $k$-explicit analogue of the results of [Nitsche, Schatz 1974], [Wahlbin 1991].
Since our Sobolev spaces are weighted with $k$ in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies $\lesssim k$). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14737 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies Averseng, Martin Spence, Euan A. Galkowski, Jeffrey Numerical Analysis For $h$-FEM discretisations of the Helmholtz equation with wavenumber $k$, we obtain $k$-explicit analogues of the classic local FEM error bounds of [Nitsche, Schatz 1974], [Wahlbin 1991], [Demlow, Guzmán, Schatz 2011], showing that these bounds hold with constants independent of $k$, provided one works in Sobolev norms weighted with $k$ in the natural way. We prove two main results: (i) a bound on the local $H^1$ error by the best approximation error plus the $L^2$ error, both on a slightly larger set, and (ii) the bound in (i) but now with the $L^2$ error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the $k$-explicit analogue of the main result of [Demlow, Guzmán, Schatz, 2011]. The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of $k^{-1}$) and is the $k$-explicit analogue of the results of [Nitsche, Schatz 1974], [Wahlbin 1991]. Since our Sobolev spaces are weighted with $k$ in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies $\lesssim k$). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error. |
| title | Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2304.14737 |