In this paper, we present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk on Spheres algorithm. The method combines a traditional Walk on Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an L-shaped insulated polyhedron.

翻译：本文提出一种随机方法，用于在平面多边形或圆弧边界区域上求解带混合边界条件的拉普拉斯方程，我们称之为"反射球面游走算法"。该方法将传统球面游走算法与诺伊曼边界反射机制相结合。通过该算法，我们实现了从特定四边形到对应正则区域的数值保角映射仿真，并计算出其保角模量。最后，我们展示了该方法在三维多面体域上的应用实例，并利用其模拟了L形绝热多面体的热流分布。