Eliminating the pollution effect in Helmholtz problems by local subscale correction

TL;DR

This paper presents a multiscale Petrov-Galerkin method for Helmholtz problems that effectively eliminates pollution effects by using local subscale corrections, ensuring stability and accuracy for large wave numbers.

Contribution

The authors develop a novel multiscale approach with local corrections that removes pollution effects in Helmholtz problems with high wave numbers, improving stability and accuracy.

Findings

Method is stable under certain mesh and oversampling conditions.

Error is proportional to mesh size H, eliminating pollution effects.

Precomputation involves solving localized cell problems.

Abstract

We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number $\kappa$ in bounded domains in $\mathbb{R}^d$. The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale corrections in the spirit of numerical homogenization. The precomputation of the corrections involves the solution of coercive cell problems on localized subdomains of size $\ell H$; $H$ being the mesh size and $\ell$ being the oversampling parameter. If the mesh size and the oversampling parameter are such that $H\kappa$ and $\log(\kappa)/\ell$ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to $H$; pollution effects are eliminated in this regime.