Multiscale methods for problems with complex geometry

TL;DR

This paper introduces a multiscale computational method for elliptic problems on complex geometries, effectively capturing fine-scale features and singularities with improved convergence and stable matrix conditioning.

Contribution

It develops a new multiscale approach with corrected coarse spaces tailored for complex domains, offering an alternative to enrichment techniques like XFEM.

Findings

Achieves linear convergence rate in energy norm.

Matrix conditioning remains stable despite complex boundary cuts.

Numerical experiments verify analytical results.

Abstract

We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.