1549301742-The_Theory_of_Difference_Schemes__Samarskii

678 Methods for Solving Grid Equations

On account of the basic theoren1 proved in Section 1 of the present

chapter Seidel method converges if the operator A is self-adjoint and posi-

tive. !VI ore specifically, the sufficient stability condition (11) for the conver-

gence of iterations in scheme (3') with a non-self-adjoint operator B takes

the fonn

(8)

T

Bo - -A> 0.

2

Within the fra1nework of Seidel method we thus have B = A- + D, T = 1

and

Bo=~ [(A-+D)+(A- +D)*] =~(A- +A+ +2D),

since A+ = (A -)* and D* = D > 0 in light of the properties of the operator

A: A= A* > 0. With this in mind, condition (8) becomes

T 1 1

Bo - - A = B 0 - - A > 0 = - D > 0

2 2 2

thereby justifying that Seidel method converges with the rate of a geometric

progression.

2. The upper relaxation method. In order to accelerate the iteration
   process in view, we are forced to revise Seidel method by inserting in (5)
   the i tera ti on parameter w so that

(9) ( A-+wD l ) (y k+l -y)+AY=f. k k

This method falls within the category of relaxation methods and gives rise
to Seidel method in one particular case where w = 1. In the modern
literature the iteration process (9) with w > 1 is known as the upper
relaxation inethod.
By comparing of (9) with (6) it is easily seen that

1

B=A-+-D,

w

or, what amounts to the same,

B=wA-+D,

As far as a non-self-adjoint operator Bis concerned, the workable procedure
reduces to inversion of a lower triangle matrix.