An Efficient Online-Offline Method for Elliptic Homogenization Problems

TL;DR

This paper introduces an online-offline numerical method for elliptic homogenization that reconstructs the effective matrix offline to enable efficient online computations, significantly reducing costs while maintaining accuracy.

Contribution

The paper proposes a novel online-offline approach that reconstructs the effective matrix via local least-squares offline, improving efficiency in solving elliptic homogenization problems.

Findings

Numerical tests confirm the method's efficiency in 2D and 3D.

The method reduces computational cost without sacrificing accuracy.

Refined estimates support the theoretical analysis of the reconstruction operator.

Abstract

We present a new numerical method for solving the elliptic homogenization problem. The main idea is that the missing effective matrix is reconstructed by solving the local least-squares in an offline stage, which shall be served as the input data for the online computation. The accuracy of the proposed method are analyzed with the aid of the refined estimates of the reconstruction operator. Two dimensional and three dimensional numerical tests confirm the efficiency of the proposed method, and illustrate that this online-offline strategy may significantly reduce the cost without loss of accuracy.