A Wachspress-Habetler extension to the HSS iteration method in Rn×n

Abstract

In the study of implicit iterations for the two dimensional heat and Helmhotz equations, one constructs a splitting of the form $A=U+V$ where U and V are the difference approximations parallel to the x and y axis, respectively. In the commutative case for U and V, several investigations have taken place. For the noncommutative case, a symmetric positive definite matrix F is found, such that $UFV=VFU$ and then, the investigations use this to address the non-commutative nature of U with V. The purpose of this paper is to study the same problem type for the commutativity of H and S in the $HSS$ splitting of $A=H+S$, where H and S are the symmetric and skew-symmetric parts of A, respectively. Although throughout the literature in $C^{n\times n}$, the H of $HSS$ stands for hermitian, we use it in the symmetric matrix case. One then applies this result to the $HSS$ iteration method. Since it is well known that the commutativity of U and V plays an important role in the analysis of ADI methods; especially for the solution of the Helmhotz Equation, one hopes that this commutativity will improve performance of the $HSS$ method. This extension is similar to that of Wachspress and Habetler's variation of the Peaceman-Rachford method. Along the way, suitable conditions are found for A, which yield a symmetric non-zero matrix $P=\sqrt{F}$ for which $N=PAP$ is a normal matrix.