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In this paper, a generalized finite element method (GFEM) with optimal localapproximation spaces for solving high-frequency heterogeneous Helmholtzproblems is systematically studied. The local spaces are built from selectedeigenvectors of carefully designed local eigenvalue problems defined ongeneralized harmonic spaces. At both continuous and discrete levels, $(i)$wavenumber explicit and nearly exponential decay rates for local and globalapproximation errors are obtained without any assumption on the size ofsubdomains; $(ii)$ a quasi-optimal convergence of the method is established byassuming that the size of subdomains is $O(1/k)$ ($k$ is the wavenumber). Anovel resonance effect between the wavenumber and the dimension of local spaceson the decay of error with respect to the oversampling size is implied by theanalysis. Furthermore, for fixed dimensions of local spaces, the discrete localerrors are proved to converge as $h\rightarrow 0$ ($h$ denoting the mesh size)towards the continuous local errors. The method at the continuous level extendsthe plane wave partition of unity method [I. Babuska and J. M. Melenk,Int.\;J.\;Numer.\;Methods Eng., 40 (1997), pp.~727--758] to theheterogeneous-coefficients case, and at the discrete level, it delivers anefficient non-iterative domain decomposition method for solving discreteHelmholtz problems resulting from standard FE discretizations. Numericalresults are provided to confirm the theoretical analysis and to validate theproposed method.

Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems