331 thus l0 d0lt has seven non trivial bands as shown

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Unformatted text preview: are mesh ( x = y = 1=4) with h h spacing by the 9-point approximation of the Laplacian 1 (U 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 0 Figure 9.3.7: The seven-band structure of L0 D0 LT . 0 + 2 (Uj+1 k + Uj;1 k + Uj k+1 + Uj k;1) ; 10 Ujk = 0: 3 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 10 28 16 18 16 20 57 28 25 23 40 112 52 34 32 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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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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