The paper studies the local behavior of the error in Helmholtz finite element method (FEM) solutions. The main results are:
Theorem 1.1 provides a bound on the local H1 error in terms of the best approximation error and the L2 error on a slightly larger set, with constants independent of the wavenumber k. This result holds for shape-regular meshes.
Theorem 1.2 provides a similar bound, but with the L2 error replaced by the error in a negative Sobolev norm. This result holds when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of k^-1).
The key insight is that these bounds in k-weighted norms imply that Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies (i.e., frequencies ≲k). This is illustrated through numerical experiments that show the error is dominated by low frequencies away from the source, while high frequencies dominate near the source.
The paper also discusses the relationship of these results to previous work on local error bounds for second-order elliptic PDEs, and how the Helmholtz case requires the use of k-weighted norms.
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