An efficient algorithm for solving elliptic problems on percolation clusters

Abstract

We present an efficient algorithm to solve elliptic Dirichlet problems defined on the cluster of ${\mathbb{Z}}^d$ supercritical Bernoulli percolation, as a generalization of the iterative method proposed by S. Armstrong, A. Hannukainen, T. Kuusi and J.-C. Mourrat. We also explore the two-scale expansion on the infinite cluster of percolation, and use it to give a rigorous analysis of the algorithm.