Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies
Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies
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DOI：
10.48550/arxiv.2304.14737
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发表时间：
2023-04
期刊：
影响因子：
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通讯作者：
Martin Averseng;E. Spence;J. Galkowski
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文献类型：
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作者：
Martin Averseng;E. Spence;J. Galkowski
对于波数为k的Helmholtz方程的h-有限元离散，我们得到了[Nitsche，Schatz 1974]，[Wahlbin 1991]，[Demlow，Guzm\'an，Schatz 2011]的经典局部有限元误差界的k-显式类似物，表明这些界与k无关的常数保持不变，只要人们以自然的方式在Sobolev范数中以k加权。我们证明了两个主要结果：（i）最佳逼近误差加上L^2误差对局部H^1误差的界，两者都是在稍大的集合上，（ii）（i）中的界，但现在L^2误差被负Sobolev范数的误差取代。结果（i）对于形状规则的三角剖分是有效的，并且是[Demlow，Guzm\'an，Schatz，2011]的主要结果的$k$-显式模拟。当网格在波长尺度上是局部准均匀的（即，规模为$k^{-1}$），并且是[Nitsche，Schatz 1974]、[Wahlbin 1991]结果的$k$-显式模拟。由于我们的Sobolev空间以自然的方式用$k$加权，结果（ii）表明亥姆霍兹FEM解是局部准最优模低频（即，频率$\lesssim k$）。数值实验证实了这一属性，也突出了有趣的传播现象的亥姆霍兹有限元误差。
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, Guzm\'an, 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, Guzm\'an, 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.