TITLE:
A Poisson Solver Based on Iterations on a Sylvester System
AUTHORS:
Michael B. Franklin, Ali Nadim
KEYWORDS:
Poisson’s Equation, Sylvester System, Multigrid
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.6,
June
29,
2018
ABSTRACT: We present an iterative scheme for solving Poisson’s equation in 2D. Using finite
differences, we discretize the equation into a Sylvester system, AU +UB = F, involving tridiagonal matrices A and B. The iterations occur
on this Sylvester system directly after introducing a deflation-type parameter
that enables optimized convergence. Analytical bounds are obtained on the
spectral radii of the iteration matrices. Our method is comparable to Successive
Over-Relaxation (SOR) and amenable to compact programming via vector/array operations. It can also be implemented within a multigrid framework
with considerable improvement in performance as shown herein.