In this talk, a parallel local method is proposed to solve 3-D potential problems using deterministic boundary integral equation and Brownian random walk on sphere (WOS) methods. First, the Neumann data of the potential solution is expressed as a hyper-singular boundary integral equation (IE) using potential on a small hemisphere centered at the location of the sought-after Neumann data, then, the required potential on the hemisphere on selected Gauss quadrature points is obtained by the Brownian article WOS random walk samplings. Such a hybrid method results in an efficient localized method for the Neumann data on the domain boundary, as a result, potential in whole space can be obtained using an integral representation.

Compared with traditional multigrid solvers for finite element methods or Krylov iterative solver for integral equation methods accelerated with the fast multipole method, the proposed BIE_WOS does not require to solve any global matrix system, and it is highly parallel and applicable to complex domains with general Dirichlet data.

*Joint work with Yan Changhao and Zeng Xuan, Microelectronics, Fudan Unversity.