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arXiv · 2104.11190

Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems

Moritz Hauck·Daniel Peterseim
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Abstract

We introduce a novel multi-resolution Localized Orthogonal Decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets (gamblets) and proves its applicability for a class of complex-valued, non-hermitian and indefinite problems. It computes hierarchical bases that block-diagonalize the Helmholtz operator and thereby decouples the discretization scales. Sparsity is preserved by a novel localization strategy that improves stability properties even in the elliptic case. We present a rigorous stability and a-priori error analysis of the proposed method for homogeneous media. In addition, we investigate the fast solvability of the blocks by a standard iterative method. A sequence of numerical experiments illustrates the sharpness of the theoretical findings and demonstrates the applicability to scattering problems in heterogeneous media.

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Moritz Hauck, Daniel Peterseim. 2021-04-22. Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems. https://arxiv.org/abs/2104.11190

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