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.

翻译：对于波数为$k$的亥姆霍兹方程$h$-FEM离散格式，本文建立了[Nitsche, Schatz 1974]、[Wahlbin 1991]及[Demlow, Guzmán, Schatz 2011]中经典局部有限元误差界的$k$显式类比，证明当按照自然方式在含$k$权重的Sobolev范数下工作时，这些误差界中的常数均与$k$无关。我们主要证明两个结论：(i) 局部$H^1$误差可由稍大区域上的最佳逼近误差与$L^2$误差共同界定；(ii) 在结论(i)的基础上，将$L^2$误差替换为负阶Sobolev范数下的误差。结论(i)适用于形状正则三角剖分，是[Demlow, Guzmán, Schatz, 2011]主要结果的$k$显式类比。结论(ii)在网格相对波长尺度（即$k^{-1}$尺度）满足局部拟均匀条件时成立，是[Nitsche, Schatz 1974]与[Wahlbin 1991]成果的$k$显式类比。由于本文的Sobolev空间按自然方式引入波数权重，结论(ii)表明亥姆霍兹有限元解关于低频模态（即频率$\lesssim k$）具有局部拟最优性。数值实验验证了这一性质，并揭示了亥姆霍兹有限元误差中值得关注的传播现象。