A Parallel Method for Solving Laplace Equations with Dirichlet Data Using Local Boundary Integral Equations and Random Walks

In this paper, a hybrid approach for solving the Laplace equation in general threedimensional (3-D) domains is presented. The approach is based on a local method for the Dirichletto-Neumann (DtN) mapping of a Laplace equation by combining a deterministic (local) boundary integral equation (BIE) method and the probabilistic Feynman–Kac formula for solutions of elliptic partial differential equations. This hybridization produces a parallel algorithm where the bulk of the computation has no need for data communication between processors. Given Dirichlet data of the solution on a domain boundary, a local BIE is established over the boundary of a local region formed by a hemisphere superimposed on the domain boundary. By using a homogeneous Dirichlet Green’s function for the whole sphere, the resulting BIE involves only the Dirichlet data (the solution value) over the surface of the hemisphere, while over the patch of the domain boundary intersected by the hemisphere, both Dirichlet and Neumann data are used. Then, first, the solution value on the surface of the hemisphere is computed by the Feynman–Kac formula, which is implemented by a Monte Carlo walk-on-spheres algorithm. Second, a boundary collocation method is employed to solve the integral equation on the aforementioned local patch of the domain boundary to yield the required Neumann data there. As a result, a local method of finding the DtN mapping is obtained, which can be used to find all Neumann data on the whole domain boundary in a parallel manner. Finally, the potential solution in the whole space can be computed by an integral representation using both the Dirichlet and Neumann data over the domain boundary.