{"title":"Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies","authors":"M. Averseng, J. Galkowski, E. A. Spence","doi":"10.1007/s10444-024-10193-w","DOIUrl":null,"url":null,"abstract":"<div><p>For <i>h</i>-FEM discretisations of the Helmholtz equation with wavenumber <i>k</i>, we obtain <i>k</i>-explicit analogues of the classic local FEM error bounds of Nitsche and Schatz (Math. Comput. <b>28</b>(128), 937–958 1974), Wahlbin (1991, §9), Demlow et al.(Math. Comput. <b>80</b>(273), 1–9 2011), showing that these bounds hold with constants independent of <i>k</i>, provided one works in Sobolev norms weighted with <i>k</i> in the natural way. We prove two main results: (i) a bound on the local <span>\\(H^1\\)</span> error by the best approximation error plus the <span>\\(L^2\\)</span> error, both on a slightly larger set, and (ii) the bound in (i) but now with the <span>\\(L^2\\)</span> error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the <i>k</i>-explicit analogue of the main result of Demlow et al. (Math. Comput. <b>80</b>(273), 1–9 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 <span>\\(k^{-1}\\)</span>) and is the <i>k</i>-explicit analogue of the results of Nitsche and Schatz (Math. Comput. <b>28</b>(128), 937–958 1974), Wahlbin (1991, §9). Since our Sobolev spaces are weighted with <i>k</i> in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies <span>\\(\\lesssim k\\)</span>). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 6","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10193-w.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10444-024-10193-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
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 and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9), Demlow et al.(Math. Comput. 80(273), 1–9 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 et al. (Math. Comput. 80(273), 1–9 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 and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9). 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.
期刊介绍:
Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis.
This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.