Numerical solution to the Neumann problem in a Lipschitz domain, based on random walks

TL;DR

This paper presents a probabilistic numerical scheme for solving linear elliptic equations with Neumann boundary conditions in Lipschitz domains, utilizing random walks and new numerical layer methods.

Contribution

It introduces a novel probabilistic numerical approach based on random walks and layer methods for Neumann problems in Lipschitz domains, extending previous techniques.

Findings

Effective numerical scheme for Neumann problems

Applicable to Lipschitz domains with complex boundaries

Demonstrates convergence and accuracy of the method

Abstract

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods.