Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies

IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED Advances in Computational Mathematics Pub Date : 2024-11-18 DOI:10.1007/s10444-024-10193-w
M. Averseng, J. Galkowski, E. A. Spence
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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.

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亥姆霍兹有限元求解在低频局部准最优
对于波长为 k 的 Helmholtz 方程的 h-FEM 离散化,我们获得了 Nitsche 和 Schatz 的经典局部 FEM 误差边界的 k-explicit analoges(Math.Comput.28(128), 937-958 1974)、Wahlbin(1991,§9)、Demlow 等人(Math.Comput.80(273),1-9 2011),证明只要以自然的方式用 k 加权的索波列夫规范计算,这些界值以与 k 无关的常数成立。我们证明了两个主要结果:(i) 通过最佳逼近误差加上\(L^2\)误差对局部\(H^1\)误差的约束,两者都在一个稍大的集合上;(ii) (i)中的约束,但现在\(L^2\)误差被负Sobolev规范中的误差所取代。结果(i)适用于形状规则的三角剖分,是 Demlow 等人的主要结果(Math.Comput.80(273), 1-9 2011).当网格在波长尺度上局部准均匀(即在 \(k^{-1}\) 的尺度上)时,结果(ii)是有效的,并且是 Nitsche 和 Schatz(Math.Comput.28(128), 937-958 1974)、Wahlbin (1991, §9)的结果。由于我们的索波列夫空间是以自然方式用k加权的,结果(ii)表明亥姆霍兹有限元求解在低频(即频率(\lesssim k\ ))时是局部准最优的。数值实验证实了这一特性,同时也突出了亥姆霍兹有限元误差中有趣的传播现象。
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来源期刊
CiteScore
3.00
自引率
5.90%
发文量
68
审稿时长
3 months
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.
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