Numerical justification of fundamental solutions and the quasi-Monte Carlo method for Poisson-type equations

Abstract

The numerical method developed in this paper is a generalization of the work for solving Poisson-type equations for arbitrary shaped domains in two dimensional (2D) and three dimensional (3D) cases. Regardless of the geometric shape of the boundary and without generating complex computational grids in the domain, we compute a particular solution by domain embedding and domain transformation. For numerical integration, we adopt the quasi-Monte Carlo method to avoid the difficulty of singularity and domain discretization. Three numerical examples in 2D and 3D were given to demonstrate the simplicity and effectiveness of the proposed method.