Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies
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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, Guzmán, 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, Guzmán, 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 ≲ k). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error.
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