# An efficient algorithm for solving elliptic problems on percolation clusters

## TL;DR

This paper introduces a new efficient algorithm for solving elliptic Dirichlet problems on supercritical Bernoulli percolation clusters, extending previous iterative methods and providing a rigorous analysis via two-scale expansion.

## Contribution

It generalizes an existing iterative method to percolation clusters and rigorously analyzes its effectiveness using two-scale expansion techniques.

## Key findings

- Algorithm significantly improves computational efficiency.
- Provides rigorous theoretical analysis of the method.
- Demonstrates applicability to infinite 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.