Reduced-Basis approach for homogenization beyond the periodic setting

TL;DR

This paper introduces a reduced-basis method to efficiently compute homogenized coefficients in elliptic PDEs, significantly speeding up multiple similar calculations without sacrificing accuracy, especially beyond periodic settings.

Contribution

It develops a reduced-basis framework with sharp error estimates for homogenization problems beyond periodic assumptions, enabling faster computations.

Findings

Speeds up homogenization computations for multiple parameters

Maintains high accuracy with a posteriori error bounds

Decouples offline and online computation stages

Abstract

We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the macroscopic scale is required at the microscopic scale. This is a framework very much adapted to model order reduction attempts. The purpose of this work is to show how the reduced-basis approach allows to speed up the computation of a large number of cell problems without any loss of precision. The essential components of this reduced-basis approach are the {\it a posteriori} error estimation, which provides sharp error bounds for the outputs of interest, and an approximation process divided into offline and online stages, which decouples the generation of the approximation space and its use for Galerkin projections.