Offline-online approximation of multiscale eigenvalue problems with random defects

TL;DR

This paper introduces an efficient offline-online multiscale method for approximating eigenvalues and eigenfunctions of elliptic problems with random, multiscale coefficients, improving computational speed and accuracy for multiple realizations.

Contribution

It develops a novel multiscale approximation method combining Localized Orthogonal Decomposition with an offline-online strategy for random defect problems.

Findings

The method provides rigorous a priori error estimates.

Numerical results show improved performance with the online modification.

The approach is effective for moderate and high defect probabilities.

Abstract

In this paper, we consider an elliptic eigenvalue problem with multiscale, randomly perturbed coefficients. For an efficient and accurate approximation of the solutions for many different realizations of the coefficient, we propose a computational multiscale method in the spirit of the Localized Orthogonal Decomposition (LOD) method together with an offline-online strategy similar to [M{\aa}lqvist, Verf\"urth, ESIAM Math. Model. Numer. Anal., 56(1):237-260, 2022]. The offline phase computes and stores local contributions to the LOD stiffness matrix for selected defect configurations. Given any perturbed coefficient, the online phase combines the pre-computed quantities in an efficient manner. We further propose a modification in the online phase, for which numerical results indicate enhanced performances for moderate and high defect probabilities. We show rigorous a priori error estimates for eigenfunctions as well as eigenvalues.