  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.
