Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies

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, 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.