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(9.3.21) with (cf. Problem 2 at the end of this section)
^
MSSOR = !(2 1 !) (D ; !L)D;1(D ; !U)
(9.3.26a)
;
1
1
^
NSSOR = ! I ; 2 ; ! (D ; !L)D;1] !U + (1 ; !)D]: (9.3.26b) ^
At the moment, the iteration matrix M of (9.3.21) and the preconditioning matrix M
of (9.3.15) are unrelated however, observe that the exact solution of (9.3.21a) satis es
^
^
Mx = Nx + b:
^
Multiplying by M;1 ^^
^
(I ; M;1 N)x = M;1 b: ^
Using (9.3.21b) to eliminate N)
^
^
M;1 Ax = M;1b:
^
This has the same form as the left preconditioning (9.3.15) thus, M serves as a preconditioner.
^
^
Examining (9.3.22  9.3.24), (9.3.26), however, we see that MGS and M! are not
symmetric. Thus, only the Jacobi and SSOR methods will furnish acceptable preconditionings and, of the two, we focus on the SSOR preconditioner (9.3.26a). 46 Solution Techniques for Elliptic Problems
At each PCG iteration (Figure 9.3.5) we must solve
^
MSSORz( ) = r( ) : (9.3.27a) If A is symmetric then U = LT and (9.3.26a) becomes
^
MSSOR = !(2 1 !) (D ; !L)D;1(D ; !LT )
(9.3.27b)
;
Thus, (9.3.27a) may be solved with a forward, diagonal, and backward substitution as
(D ; !L)z = !(2 ; !)r( ) (9.3.27c) D;1~ = z
z (9.3.27d) ~
(D ; !LT )z( ) = z: (9.3.27e) and
The choice of ! doesn't appear to be critical and may, e.g., be selected as unity.
Example 9.3.4 ( 5], Section 14.5). Consider the solution of Poisson's equation on a
square with uniform spacing x = y = h. Suppose that the forcing and boundary data
is such that the exact solution is u(x y) = cos x sin y:
The initial iterate was trivial inside the square and all other numerical parameters were
selected as for Example 9.3.2. Comparisons of results obtained using SOR, CG, and PCG
are presented in Table 9.3.2. The number of iterations of the SOR and CG algorithms is
increasing as 1=h while that of the PCG algorithm is increasing as 1=h1=2 . The work of
the conjugate gradient method is about twice that of the SOR method and that of the
PCG method is about four times the SOR method. Thus, for small systems, the SOR
and conjugate gradient method will be supe...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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