Ujk 1 1 ujk well explicitly eliminate the intermediate

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Unformatted text preview: .17e) with J = K = 3, the spectral radius of the Jacobi method is 1 (MJ ) = 1 ; sin2 6 ; sin2 6 = 2 thus, using (9.2.25a) p2 1:07: 1 + 1 ; 1=4 The SOR solution and errors after ve iterations are shown in Table 9.2.4. After ve iterations, the percentage errors at point (1 1) are 1.2, 0.29, and 0.014, respectively, for the Jacobi, Gauss-Seidel, and SOR methods (cf. Example 9.2.4). !OPT = 26 Solution Techniques for Elliptic Problems 0 0 0 0/1 0 0 0 0 0 0.24997 0.49999 1 0 0.00003 0.00001 0 0 0.49993 0.74998 1 0 0.00007 0.00002 0 0/1 1 1 1 0 0 0 0 Table 9.2.4: Solution of Example 9.2.8 after ve iterations ( = 4) using the SOR method with ! = 1:07 (left) and errors in this solution (right). Example 9.2.9. Let us examine the convergence rate of SOR a bit more closely for Laplace's equation. Using (9.2.17e), the spectral radius of Jacobi's method on a square is (MJ ) = 1 ; 2 sin2 = cos : 2J J Using (9.2.25a) 2 !OPT = p 2 2 = 1 + sin =J : 1 + 1 ; cos =J Now, using (9.2.25b) ; sin =J (M!OP T ) = !OPT ; 1 = 1 + sin =J 1 or, for large values of J , (M!OP T ) 1 ; 2 : J Recall (Example 9.2.5) that the spectral radius of Jacobi and Gauss-Seidel iteration under the same conditions is 1 ; O(1=J 2). Thus, SOR iteration is considerably better. We'll emphasize this by computing the convergence rate according to (9.2.7b). For the Jacobi method, we nd 2 R (MJ ) ; ln (MJ ) ; ln(1 ; 2J 2 ) Similarly, for the Gauss-Seidel method, we have R (MGS ) and for SOR, we have 2 : 2J 2 2 J2 2: J Thus, typically, the Jacobi or Gauss-Seidel methods would require O(J 2) iterations to obtain an answer having a speci ed accuracy while the SOR would obtain the same accuracy in only O(J ) iterations. R (M!OP T ) 9.2. Basic Iterative Solution Methods 27 The optimal relaxation parameter is not known for realistic elliptic problems because the eigenvalues of MJ are typically unavailable. Strikwerda 5], Section 13.3, describes a way of calculating approximate values of !OPT . The optimal relaxation parameter for many elliptic problems is close to 2 and may be approximated by an expression of the form !OP...
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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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