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

• 作者: Chohong Min Gangjoon Yoon
• 来源期刊: 美国计算数学期刊(英文)
• 学科: 数学
• 关键词: Five-Point Finite DIFFERENCE Method Modified ILU PRECONDITIONING Condition Number
• 年，卷（期）: mgjssxqkyw,（2）
• 页码范围: 75-79
• 页数: 5页
• 分类号: O1

摘要

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.