Multiscale Finite Element approach for "weakly" random problems and related issues

TL;DR

This paper introduces a perturbation-based multiscale finite element method for efficiently solving elliptic problems with weakly random coefficients, achieving comparable accuracy to direct methods with reduced computational effort.

Contribution

It develops a modified multiscale finite element basis tailored for weakly random problems, extending deterministic multiscale methods to stochastic settings.

Findings

Significant computational speed-up demonstrated.

Approach maintains approximation properties of deterministic multiscale basis.

Complete theoretical analysis provided for the stochastic extension.

Abstract

We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale Finite Element Method. Using numerical experiments, we demonstrate the efficiency of our approach and the computational speed-up with respect to a more standard approach. We provide a complete analysis of the approach, extending that available for the deterministic setting.