Contrast independent localization of multiscale problems

TL;DR

This paper introduces a new variational multiscale method with a novel interpolation operator that achieves contrast-independent localization error in multiscale problems with high contrast coefficients, improving accuracy.

Contribution

The paper develops a new interpolation operator for two-valued coefficients that ensures contrast-independent localization error, advancing multiscale method robustness.

Findings

Contrast-independent localization error achieved with the new operator

Numerical experiments confirm theoretical predictions

Applicable to geometries with inclusions and channels

Abstract

The accuracy of many multiscale methods based on localized computations suffers from high contrast coefficients since the localization error generally depends on the contrast. We study a class of methods based on the variational multiscale method, where the range and kernel of a quasi-interpolation operator defines the method. We present a novel interpolation operator for two-valued coefficients and prove that it yields contrast independent localization error under physically justified assumptions on the geometry of inclusions and channel structures in the coefficient. The idea developed in the paper can be transferred to more general operators and our numerical experiments show that the contrast independent localization property follows.