Numerical Solution of the Robin Problem of Laplace Equations with a Feynman-Kac Formula and Reflecting Brownian Motions

TL;DR

This paper introduces a numerical approach combining Feynman-Kac formula and reflecting Brownian motions to solve Robin boundary problems for Laplace equations, demonstrating accuracy and efficiency.

Contribution

It develops a novel probabilistic numerical method using SRBM and local time simulation via WOS for Robin problems, enhancing solution accuracy.

Findings

Method accurately solves Robin boundary problems.

Simulation approach is computationally efficient.

Numerical results confirm method's effectiveness.

Abstract

In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman-Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorohod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres (WOS) method, approximation of the boundary local time is obtained and the Feynman-Kac formula is calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.