Super-localization of elliptic multiscale problems

TL;DR

This paper introduces a novel localization method for elliptic multiscale problems that ensures super-exponential decay of basis functions, enabling efficient approximation with optimal convergence rates using smaller supports.

Contribution

It presents a new localization technique that achieves super-exponential decay of basis functions, reducing support size while maintaining optimal convergence in numerical homogenization.

Findings

Super-exponential decay of basis functions achieved.

Supports of size $ ext{O}(H| ext{log} H|^{(d-1)/d})$ suffice for optimal rates.

Compared to previous methods, the new technique reduces support size significantly.

Abstract

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter $H$. This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximate solution space. This paper presents a novel localization technique that enforces the super-exponential decay of the basis relative to $H$. This shows that basis functions with supports of width $\mathcal O(H|\log H|^{(d-1)/d})$ are sufficient to preserve the optimal algebraic rates of convergence in $H$ without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width $\mathcal O(H|\log H|)$.