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Unformatted text preview: sSeidel 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 twocyclic 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 \redblack" fashion by listing unknowns and equations at every other point.
The resulting matrix has the twocyclic form
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5 20 Solution Techniques for Elliptic Problems 4 9 5 7 3 8 1 6 2 Figure 9.2.3: Redblack ordering of the Laplacian operator on a 4 4 square mesh De nition 9.2.5. A twocyclic 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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 Spring '14
 JosephE.Flaherty

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