Analysis of multiscale methods for the two-dimensional Helmholtz equation with highly heterogeneous coefficient. Part II. Two-scale Localized Orthogonal Decomposition
Mario Ohlberger, Barbara Verfürth · arXiv (Cornell University) · 2016
In this paper, we suggest a two-scale Localized Orthogonal Decomposition (LOD) method in Petrov-Galerkin formulation for the Helmholtz equation with highly heterogeneous coefficient. The method is based on the homogenized (two-scale) formulation and approximates the numerical solution of the related Heterogeneous Multiscale Method introduced in the first part (Analysis of multiscale methods for the two-dimensional Helmholtz equation with highly heterogeneous coefficient. Part I. Homogenization and Heterogeneous Multiscale Method, preprint, University M\unster, 2016). There, a severe resolution condition on the mesh sizes has been observed that is nevertheless optimal in that setting and known as effect in the finite element literature. The LOD in Petrov-Galerkin formulation, following the ideas of Gallistl and Peterseim (Comput. Methods Appl. Mech. Engrg. 295:1-17, 2015), overcomes this pollution effect. Standard finite element functions are used for the trial space, whereas the test functions are enriched by solutions of subscsale 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, a reasonable resolution of a few degrees of freedom per wave length, is sufficient for the LOD to be stable and quasi-optimal.