^{1}

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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.

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 condition number is of size

The MILU preconditioing introduced by Axelsson [

The numerical experiments [

We introduce a novel and heuristic proof for the conjecture in case of Poisson equation with Dirichlet boundary condition on rectangular domains. The MILU preconditioner is obtained from recursively calculating the row-sum equation at each grid node in the lexicographical ordering. In the case of the five-point method, it is well known [

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.

Consider the Poisson equation

at each grid node

where L, U, and D are its strictly lower and upper, and diagonal parts, respectively. MILU preconditioner is the matrix of the form

for

for

Here

the row sum property,

Lemma 1. Let

Then we have

Proof. Let

and this proves the lemma.

Theorem 1. Let

and, therefore,

Proof. We shall show that

Now, we are ready to estimate the condition number of the MILU preconditioned matrix

and

for the matrix

Using the inequality

Thus, we obtain the inequalities

In summary, we have the following.

Theorem 2. Let

Proof. Let

and applying these inequalities above into (2) and (3) gives

which shows the inequalites (6). On the other hand, the row sum property implies that 1 is an eigenvalue of

Corollary 1. The ratio of the maximum and minimum eigenvalues of the MILU preconditioned matrix is bounded by

Remark 1. Our analysis deals with the two dimensional Poisson equation. It naturally extends to the three dimensional equation in a dimension-by-dimension manner.

This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2009-0093827).