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Unformatted text preview: m for Poisson's equation on a J J square mesh in O(J 3) operations with
O(J 2 log2 J ) storage. These should be compared to the O(J 4) operations and O(J 3)
storage required for the banded algorithm of Table 9.1.1. George 1] additionally showed
that the nested ordering is optimal in the sense that all orderings of the mesh must yield
an operation count of at least O(J 3). 9.2. Basic Iterative Solution Methods 9 9.2 Basic Iterative Solution Methods
The direct methods of Section 9.1 require storage within the band or pro le which could
be signi cant for very large problems (e.g., in excess of 10,000 equations). Storage and,
perhaps, computer time can be reduced through the use of iterative techniques. As in
Section 9.1, we'll focus on techniques for N N linear system having the form (9.1.1).
Using a \ xed-point" strategy, we rewrite (9.1.1) in the form
x = Mx + b (9.2.1) and consider the iteration
x( +1) = Mx( ) + b = 0 1 ::: : (9.2.2) The iteration must be designed so that
lim x( ) = x:
Example 9.2.1. Write A=A+I;I where I is the N N identity matrix, and rewrite (9.1.1) as Ix = (I ; A)x + b:
This system has the form of (9.2.1) with M=I;A ^
b = b: Here are two fundamental convergence criteria. Theorem 9.2.1. The iteration (9.2.2) converges to a xed point x of (9.2.1) when
kMk < 1: (9.2.3) e( ) = x( ) ; x (9.2.4) Proof. De ne 10 Solution Techniques for Elliptic Problems and subtract (9.2.1) from (9.2.2) to obtain e( +1) = Me( ) :
Thus, e( +1) = Me( ) = M2 e( ;1) = : : : = M +1 e(0) or e( ) = M e(0) : Taking a norm ke( ) k = kM e(0) k kM kke(0) k kMk ke(0) k: (9.2.5) When kMk < 1, we see that ke( ) k ! 0 as ! 1 and, hence, the iteration converges. Theorem 9.2.2. The iteration (9.2.2) converges from any initial guess if and only if the
spectral radius (M) max j i(M)j < 1 1iN (9.2.6) where i , i = 1 2 : : : N , are eigenvalues of the N N matrix M.
Proof. From Lemma 3.3.1 we know (M) = (M ) kM k:
If the iteration (9.2.2) converges from any initial guess e...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14