This preview shows page 1. Sign up to view the full content.
Unformatted text preview: he rowbyrow and nesteddissection orderings are shown in Figures 9.1.4 and 9.1.5, respectively, for the 4 4 mesh
of Figure 9.1.3. The matrix A obtained by the nesteddissection ordering has a larger
bandwidth than the one with the rowbyrow ordering. However, even with this simple
4 4 problem, we can see that the ll in is less with the nested dissection ordering than
with the rowbyrow ordering (Figures 9.1.4 and 9.1.5). 9.1. Direct Solution Methods 7 3 8 4 6 9 7 1 5 8 2 6 7
5 9 Figure 9.1.3: Nested dissection of a uniform 4 4 mesh. Unknowns at the nest level (left)
are eliminated rst to create the coarser mesh (center). These unknowns are eliminated
next to leave a single unknown (bottom).
A banded structure is not necessary for the e cient implementation of a Gaussian
elimination procedure. It is just about as simple to implement a \skyline" or \pro le"
elimination strategy where the local bandwidth is used. This requires an additional
vector indicating, e.g., the number of leading zeros in a row or column. Let lij = 0 0 j < mi (9.1.12a) and k = max(mi mj )
then the skyline form of the symmetric Cholesky decomposition (9.1.9) is
i;1
X2
di = aii ; dk lik
k=k (9.1.12b) (9.1.13a) 8 Solution Techniques for Elliptic Problems
2
6
6
6
6
6
6
6
6
6
6
6
6
4 3
7
7
7
7
7
7
7
7
7
7
7
7
5 2
6
6
6
6
6
6
6
6
6
6
6
6
4 3
7
7
7
7
7
7
7
7
7
7
7
7
5 Figure 9.1.4: Matrix A when the nite di erence equations for a 4 4 mesh are ordered
by rows (left) and the resulting llin of U (right).
2
6
6
6
6
6
6
6
6
6
6
6
6
4 3
7
7
7
7
7
7
7
7
7
7
7
7
5 2
6
6
6
6
6
6
6
6
6
6
6
6
4 3
7
7
7
7
7
7
7
7
7
7
7
7
5 Figure 9.1.5: Matrix A when the nite di erence equations have a nested dissection
ordering (left) and the resulting llin of U (right).
i;1
X
1
lji = d (aji ; dk ljk lik )
i k=k j = mi + 1 i + 2 : : : N i = 1 2 : : : N: (9.1.13b) This procedures ignores any zeros within the pro le of L. George 1] proved that the
pro le algorithm (9.1.12, 9.1.13) with the nested dissection ordering could solve a Dirichlet proble...
View
Full
Document
This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

Click to edit the document details