A HIGHLY PARALLEL ALGORITHM FOR SOLVING POISSON EQUATION USING BLOCK DECOMPOSITIONS

摘要

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 for its simulations. In this paper, we consider solving the Poisson equation ▽~2u = f(x,y) in the Cartesian domain Ω = [-1, 1] (×) [-1, 1], subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to propose a two-level block decomposition scheme for decoupling the original linear system obtained from the discretization into independent subsystems. The first level of the block decomposition uses the eigenpairs of the second-order Chebyshev differentiation matrix in one space dimension and the second level of the decomposition exploits a special reflexivity property inherent in this differentiation matrix. The decomposition not only yields a more efficient algorithm but introduces high-degree coarse-grain parallelism. This approach can also be applied to Laplace eigenvalue problem discretized with the Chebyshev pseudo-spectral method as well, subject to the same boundary conditions.