Xi 1 1 xi i 1 2 n 9220b the acceleration

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Unformatted text preview: s-Seidel iteration when ! = 1, successive under relaxation when ! < 1, and successive over relaxation (SOR) when ! > 1. Over relaxation is the important method with elliptic problems. Using (9.2.8), we can write (9.2.20) in the vector form ^ Dx( +1) = Lx( +1) + Ux( ) + b (9.2.21a) ^ x( +1) = !x( +1) + (1 ; !)x( ) : (9.2.21b) We can further eliminate the provisional iterate and write (9.2.21a, 9.2.21b) in the form of (9.2.2) ^ x( +1) = M! x( ) + b! (9.2.21c) 9.2. Basic Iterative Solution Methods 19 with M! = (D ; !L);1 (1 ; !)D + !U] ^ b! = !(D ; !L);1b: (9.2.21d) Our goal is to nd the value of ! that minimizes (M! ) and, hence, maximizes the convergence rate. There is a wealth of theory on this subject and let us begin with some preliminary considerations. De nition 9.2.3. A matrix A is two cyclic if there is a permutation of its rows and columns that reduce it to the form where D1 and D2 are diagonal. D1 F G D2 De nition 9.2.4. A matrix A is weakly two cyclic if D1 and D2 are zero. Example 9.2.6. The matrix shown in Figure 9.2.2 is two interchange of its second and third rows and columns. 2 3 2 3 2 1c0 1c0 10 4a 1 c5 40 a 15 40 1 0a1 a1c ac cyclic as revealed by an 3 c a5 1 Figure 9.2.2: Matrix (left) whose second and third rows are interchanged (center) and whose second and third columns are interchanged to obtain a two-cyclic form (right). Example 9.2.7. Consider the Laplacian operator on a 4 4 grid as shown in Figure 9.2.3. Instead of ordering the equations and unknowns by rows, order them in \checkerboard" or \red-black" fashion by listing unknowns and equations at every other point. The resulting matrix has the two-cyclic form 2 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7: 6 7 6 7 6 7 6 7 6 7 4 5 20 Solution Techniques for Elliptic Problems 4 9 5 7 3 8 1 6 2 Figure 9.2.3: Red-black ordering of the Laplacian operator on a 4 4 square mesh De nition 9.2.5. A two-cyclic matrix of the form of (9.2.8) is consistently ordered if the eigenvalues of D;1 ( L + 1 U) are independent of for...
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