An improved high-order method for elliptic multiscale problems

TL;DR

This paper introduces a high-order multiscale method for elliptic problems with oscillatory coefficients, achieving optimal convergence without restrictive assumptions, and improves localization to enhance accuracy on smaller subdomains.

Contribution

It presents a novel high-order multiscale method with an improved localization strategy based on the Localized Orthogonal Decomposition framework, enhancing accuracy for elliptic problems with rough coefficients.

Findings

Achieves higher-order convergence rates based on RHS regularity.

Improves localization to reduce errors on small subdomains.

Demonstrates effectiveness through numerical experiments.

Abstract

In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only. Hence, no restrictive assumptions on the coefficient, the domain, or the exact solution are required. In the spirit of the Localized Orthogonal Decomposition, the method constructs coarse problem-adapted ansatz spaces by solving auxiliary problems on local subdomains. More precisely, our approach is based on the strategy presented by Maier [SIAM J. Numer. Anal. 59(2), 2021]. The unique selling point of the proposed method is an improved localization strategy curing the effect of deteriorating errors with respect to the mesh size when the local subdomains are not large enough. We present a rigorous a priori error analysis and demonstrate the performance of the method in a series of numerical experiments.