A Super-Localized Generalized Finite Element Method

TL;DR

This paper introduces a new multi-scale finite element method for elliptic PDEs with complex coefficients, achieving high accuracy and efficiency through localized basis functions and rigorous error analysis.

Contribution

It develops a super-localized generalized finite element method with uniform approximation rates and higher-order extensions, advancing numerical homogenization techniques.

Findings

Method achieves super-exponential localization of basis functions.

Numerical experiments confirm high accuracy for high-contrast coefficients.

Error analysis supports theoretical approximation guarantees.

Abstract

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients.