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：本文提出了一种随机方法，用于模拟多边形或圆弧有界平面区域中具有混合边界条件的拉普拉斯方程。我们将此方法称为反射球面行走算法。该方法将传统球面行走算法与诺伊曼边界处的反射操作相结合。我们应用该算法模拟从特定四边形到对应典型区域的数值共形映射，并计算其共形模量。最后，我们展示了该方法在三维多面体区域的应用示例，并用于模拟L形绝缘多面体上的热流过程。