A Simple Proof of Gustafsson’s Conjecture in Case of Poisson Equation on Rectangular Domains

Gangjoon Yoon¹, Chohong Min²

¹Institute of Mathematical Sciences, Ewha Womans University, Seoul, South Korea.
²Department of Mathematics, Ewha Womans University, Seoul, South Korea.

DOI: 10.4236/ajcm.2015.52005                

Abstract

We consider the standard five-point finite difference method for solving the Poisson equation with the Dirichlet boundary condition. Its associated matrix is a typical ill-conditioned matrix whose size of the condition number is as big as . Among ILU, SGS, modified ILU (MILU) and other ILU-type preconditioners, Gustafson shows that only MILU achieves an enhancement of the condition number in different order as . His seminal work, however, is not for the MILU but for a perturbed version of MILU and he observes that without the perurbation, it seems to reach the same result in practice. In this work, we give a simple proof of Gustafsson's conjecture on the unnecessity of perturbation in case of Poisson equation on rectangular domains. Using the Cuthill-Mckee ordering, we simplify the recursive equation in two dimensional grid nodes into a recursive one in the level that is one-dimensional. Due to the simplification, our proof is easy to follow and very short.

Keywords

Five-Point Finite Difference Method, Modified ILU Preconditioning, Condition Number

Conflicts of Interest

The authors declare no conflicts of interest.

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