32 comparisons of results obtained using sor cg and

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Unformatted text preview: rior, but the SSOR-PCG method overtakes these methods for larger systems. One of the most successful preconditioning techniques utilizes incomplete factorization by Gaussian elimination. Thus, let M = L0D0 LT 0 (9.3.28) 9.3. Conjugate Gradient Methods 47 1=h SOR CG PCG 10 33 26 12 20 60 52 16 40 115 103 22 Table 9.3.2: Number of SOR, conjugate gradient, and SSOR-PCG iterations for Example 9.3.4. J J A= L 0= Figure 9.3.6: Nonzero structure of a discrete Poisson operator A (left) and an approximate lower triangular factor L0 (right). where L0 is determined to have a particular sparsity structure. Example 9.3.5. Consider a Dirichlet problem for Poisson's equation, which has the nonzero structure shown in Figure 9.3.6. Ordering the equations by rows, we have (Ax)n = ;xn + x(xn;1 + xn+1) + y (xn;J + xn+J ): (9.3.29a) If equation n corresponds to mesh point (j k), we may also write (9.3.29a) in the (doublesubscripted) mesh notation as (Ax)jk = ;xjk + x(xj;1 k + xj+1 k ) + y (xj k;1 + xj k+1): (9.3.29b) We insist that (L0 x)n = xn + bnxn;1 + cnxn;J : (9.3.30a) (LT x)n = xn + bn+1xn+1 + cn+J xn+J : 0 (9.3.30b) Taking a transpose 48 Solution Techniques for Elliptic Problems Likewise (D0x)n = dnxn : (9.3.30c) Multiplying (9.3.30b) and (9.3.30c) (D0LT x)n = dn(xn + bn+1 xn+1 + cn+J xn+J ): 0 Now, using (9.3.30a) (L0 D0LT x)n = (D0LT x)n + bn(D0 LT x)n;1 + cn(D0 LT x)n;J : 0 0 0 0 Combining the above two equations (L0D0 LT x)n = dn(xn + bn+1 xn+1 + cn+J xn+J ) + bndn;1(xn;1 + bnxn + cn+J ;1xn+J ;1 )+ 0 cndn;J (xn;J + bn+1;J xn+1;J + cnxn ): Regrouping terms (L0 D0LT x)n = (dn + dn;1b2 + dn;J c2 )xn + dn;1bn xn;1 + dnbn+1 xn+1+ 0 n n dncn+J xn+J + dn;J cnxn;J + bn+1;J dn;J cnxn+1;J + bn dn;1cn+J ;1xn+J ;1 : (9.3.31) Thus, L0 D0LT has seven non-trivial bands as shown in Figure 9.3.7. 0 Using incomplete factorization, we equate the coe cients of xn, xn+1 , and xn+J in (9.3.31) to those of (Ax)n in (9.3.29a). This yields dn = ;1 ; dn;1b2 ; dn;J c2 n n n = 1 2 : : : N: (9.3.32a) bn+1 = x=dn n = 1 2 : : : N ; 1: (9.3.32b) cn+J = y =dn n = 1 2 : : : N ; J: (9.3.32c) Example 9.3.6. ( 5], Section 14.5). Consider the solution of Laplace's equation on the unit square using a squ...
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