Localized Orthogonal Decomposition for two-scale Helmholtz-type problems

TL;DR

This paper introduces a localized orthogonal decomposition method for two-scale Helmholtz problems, effectively overcoming the pollution effect with a quasi-optimal approach that uses enriched test functions on local patches.

Contribution

It develops a Petrov-Galerkin LOD method for two-scale Helmholtz problems, addressing the pollution effect and coupling challenges in a multiscale setting.

Findings

Achieves quasi-optimal error bounds with logarithmic oversampling.

Overcomes pollution effect with few degrees of freedom per wavelength.

Handles coupling and periodic boundary conditions effectively.

Abstract

In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in (Ohlberger, Verf\"urth. A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016). There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution effect" in the finite element literature. Following ideas of (Gallistl, Peterseim. Comput. Methods Appl. Mech. Engrg. 295:1-17, 2015), we use standard finite element functions for the trial space, whereas the test functions are enriched by solutions of subscale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter $m$, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.