Unformatted text preview: on (right).
Example 9.2.5. We'll try to quantify the di erences in the convergence rates of Jacobi
and Gauss-Seidel iteration for Poisson's equation on a rectangle. Jacobi's method satis es
(9.2.12) and we let p be an eigenvector of MJ with corresponding eigenvalue thus, MJ p = p: (9.2.17a) Using (9.2.12), we see that the component form of this relation is pjk = x (pj;1 k + pj+1 k) + y (pj k;1 + pj k+1)
j = 1 2 ::: J ; 1
k = 1 2 ::: K ;1
where pjk is a component of p. (The double subscript notation for a vector component is
non-standard, but convenient in this case since it corresponds to a position in the nite
di erence mesh.) One may easily verify that pjk = sin mj sin nk
K (9.2.17c) 16 Solution Techniques for Elliptic Problems and
= 1 ; 4 x sin2 m ; 4 y sin2 n
2K m = 1 2 ::: J ; 1 n = 1 2 : : : K ; 1:
(9.2.17d) Remark 1. The (J ; 1)(K ; 1) eigenvectors and eigenvalues of MJ are indexed by m
Remark 2. The eigenvector p is the eigenfunction of the Laplacian sampled at the
mesh points j = 1 2 : : : J ; 1, k = 1 2 : : : K ; 1. The eigenvalue is, however, not
an eigenvalue of the Laplacian.
The largest eigenvalue and, hence, the spectral radius of MJ may be obtained by
setting m = n = 1 in (9.2.17d) to obtain (MJ ) = 1 ; 4 x sin2 2J ; 4 y sin2 2K :
In the special case of a square, x = y (9.2.17e) = 1=4 and J = K thus, (MJ ) = 1 ; 2 sin2 2J : For large values of J , we may approximate this as
(MJ ) 1 ; 2
2J 2 = 1 ; C x2 since J = a= x for an a a square region. Thus, the spectral radius approaches unity
and the convergence rate slows as J increases (or as x decreases).
In a similar manner, Let q and be an eigenvector-eigenvector pair of the GaussSeidel iteration matrix MGS for Poisson's equation on a rectangle (9.2.16). Thus, using
(9.2.15b) MGS q = (D ; L);1 Uq = q (9.2.18a) or, using (9.2.15b), in component form qjk ; ( xqj;1 k + y qj k;1) = ( xqj+1 k + y qj k+1)
j = 1 2 ::: J ;1
k = 1 2 : : : K ; 1:
(9.2.18b) 9.2. Basic Iterative Solution Methods 17 This problem appea...
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