A FAST POISSON SOLVER BY CHEBYSHEV PSEUDOSPECTRAL METHOD USING REFLEXIVE DECOMPOSITION

Abstract

Poisson equation is frequently encountered in mathematical modeling for scientific and engineering applications. Fast Poisson numerical solvers for 2D and 3D problems are, thus, highly requested. In this paper, we consider solving the Poisson equation $\nabla^{2}u = f(x,y)$ in the Cartesian domain $\Omega = [-1,1] \times [-1,1]$, subject to all types of boundary conditions, discretized with the Chebyshev pseudospectral method. The main purpose of this paper is to propose a reflexive decomposition scheme for orthogonally decoupling the linear system obtained from the discretization into independent subsystems via the exploration of a special reflexive property inherent in the second-order Chebyshev collocation derivative matrix. The decomposition will introduce coarse-grain parallelism suitable for parallel computations. This approach can be applied to more general linear elliptic problems discretized with the Chebyshev pseudospectral method, so long as the discretized problems possess reflexive property. Numerical examples with error analysis are presented to demonstrate the validity and advantage of the proposed approach.