Fast Solving the Cauchy Problems of Poisson Equation in an Arbitrary Three-Dimensional Domain

In this paper we propose a novel two-stage method to solve the three-dimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary. The solution is decomposed into a particular solution and a homogeneous solution. In the first stage a multiple-scale polynomial method (MSPM) is used to approximate the forcing term and then the formula of Tsai et al. [Tsai, Cheng, and Chen (2009)] is used to obtain the corresponding closed-form solution for each polynomial term. Then in the second stage we use a multiple/scale/direction Trefftz method (MSDTM) to find the solution of Laplace equation, of which the directions are uniformly distributed on a unit circle S-1, and the scales are determined a priori by the collocation points on boundary. Two examples of 3D data interpolation, and several numerical examples of direct and inverse Cauchy problems in complex domain confirm the efficiency of the MSPM and the MSDTM.