Super-Localized Orthogonal Decomposition for High-Frequency Helmholtz Problems
Philip Freese, Moritz Hauck, Daniel Peterseim · SIAM Journal on Scientific Computing · 2024
Abstract. We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber [Formula: see text]. On a coarse mesh of width [Formula: see text], the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width [Formula: see text] and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multiscale methods, two- and three-dimensional numerical computations show that the localization error decays super-exponentially as the oversampling parameter [Formula: see text] is increased. This suggests that optimal convergence is observed under the substantially relaxed oversampling condition [Formula: see text] with [Formula: see text] denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.