Unformatted text preview: are mesh ( x = y = 1=4) with h h spacing by the 9-point
approximation of the Laplacian
6 j+1 k+1 + Uj+1 k;1 + Uj;1 k+1 + Uj;1 k;1) 9.3. Conjugate Gradient Methods 49 J-1
J L D0LT =
0 Figure 9.3.7: The seven-band structure of L0 D0 LT .
+ 2 (Uj+1 k + Uj;1 k + Uj k+1 + Uj k;1) ; 10 Ujk = 0:
3 The 9-point approximation of the Laplacian is accurate to O(h4) compared to the O(h2)
accuracy of the 5-point formula. The Dirichlet boundary conditions are established so
that the exact solution is
u(x y) = e3x sin 3y:
The problem is solved using the PCG method with the following preconditioners:
None. No preconditioning,
SSOR-5. SSOR iteration using the 5-point approximation of the Laplacian,
SSOR-9. SSOR iteration using the 9-point approximation of the Laplacian, and
ILU(0)-5. Incomplete Cholesky factorization of the 5-point Laplacian with no ll
in. Each method was started with a trivial solution in the interior and one that satis ed
the exact solution on the boundary. The iteration was terminated when the di erence
between successive iterates in the L2 norm was less than 10;10. Each SSOR method used
! = 2=(1 + h). The number of iterations to reach convergence are recorded in Table
9.3.2. 50 Solution Techniques for Elliptic Problems 1=h None ILU(0)-5 SSOR-5 SSOR-9
Table 9.3.3: Number of iterations to convergence for the PCG solution of Example 9.3.6
using the indicated preconditioners.
The number of iterations for the two SSOR preconditioners increases as O(h;1=2).
The more accurate SSOR-9 iterative preconditioning di ers little from the SSOR-5 preconditioning. The number of iterations for the incomplete factorization preconditioner is
increasing slower than O(1=h) but faster than O(h;1=2 ). All preconditioners in the table
are better than using the conjugate gradient method without preconditioning.
Thus far, we have used incomplete factorization without allowing any ll in. In
Example 9.3.6, we referred to this preconditioning as ILU(0). The I...
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