Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients

TL;DR

This paper improves the theoretical understanding of an iterative method for solving elliptic equations with rapidly oscillating stochastic coefficients, providing a uniform estimate for its contraction factor that scales logarithmically with domain size.

Contribution

The authors strengthen the contraction estimate for the iterative algorithm, making it uniform over initial functions and demonstrating logarithmic growth with domain size.

Findings

Contraction factor estimate is uniform over initial functions.

Estimate grows only logarithmically with domain size.

Improves theoretical bounds for the iterative method.

Abstract

We study the iterative algorithm proposed by S. Armstrong, A. Hannukainen, T. Kuusi, J.-C. Mourrat to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this iterative method is comparatively easy to compute. In this article, we strengthen the estimate for the contraction factor achieved by one iteration of the algorithm. We obtain an estimate that holds uniformly over the initial function in the iteration, and which grows only logarithmically with the size of the domain.