Numerical Experiments Using Dissection Methods to Solve n by n Grid Problems

TL;DR: This paper describes how these orderings for Gaussian elimination can be implemented In an efficient manner and provides numerical experiments which show that the execution times of these programs properly reflects the arithmetic operation counts.

Abstract: Recently the author has proposed two theoretically efficient orderings for Gaussian elimination when it is applied to systems of $n^2 $ linear equations arising in connection with the use of finite element methods on an n by n grid [6], [7]. These are efficient in the sense that if zeros are exploited, the amount of arithmetic required is $O(n^3 )$ or $O(n^{{7 / 2}} )$, compared to $O(n^4 )$ if the usual row by row numbering scheme is used. Similarly, the amount of fill suffered is $O(n^2 \log _2 n)$ and $O(n^{{5 / 2}} )$ compared to $O(n^3 )$. These comparisons ignored differences in the program and data structure complexity required to exploit the zeros for the different orderings. In this paper the author describes how these orderings can be implemented In an efficient manner and provides numerical experiments which show that the execution times of these programs properly reflects the arithmetic operation counts.