The structure of the linear systems obtained by using

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Unformatted text preview: he row-by-row 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 nested-dissection ordering has a larger bandwidth than the one with the row-by-row 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 row-by-row 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 ll-in 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 ll-in 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...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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