An iterative method for elliptic problems with rapidly oscillating coefficients

TL;DR

This paper presents a new iterative method for solving elliptic equations with rapidly oscillating random coefficients, combining homogenization techniques with multiscale iteration to improve convergence independent of domain size.

Contribution

The authors introduce a novel iterative approach that integrates homogenized equations into multiscale computations, outperforming standard multigrid methods in large-scale separation regimes.

Findings

Explicit contraction factor estimate independent of domain size

Numerical experiments confirm method effectiveness

Open-source code available for implementation

Abstract

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method, with openly available source code.