1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular method 681

where xk = B^112 wk and C = B-^112 AB-^112.

Indeed, because the operator B is self-adjoint and positive: B = B* >

0, the square root B^112 does exist with the property

(B1f2)* =Bl/'-'-> 0.

By applying the operator B-^1!^2 to equation (13) and inserting wk =

B-^1!^2 x k we arrive at scheme ( 14). Arguing in reverse order we rnight

obtain the same result without any difficulties.

Lem1na 1 Given operators

A=A*>O,

the operator inequalities

B = B* > 0, C = B-1/2 AB-1;2

(15) fl > 0 J

are equivalent.

Proof A reasonable form of the functional at hand is

I= ((A -1 B) y, y) = (Ay, y) - f (By, y)

= (Ca:, a:) - f ( x, a:) = ( ( C - f E) :L', x) ,

J

where x = B^112 y is an arbitrary element of the space H due to the arbi-

trariness in the choice of the element y E H. Because of this, the equality

( 16) I = ( (A - f B) y, y) = ( ( C - f E) x, :e)

implies that the operators A - f B and C - f E are of the same sign. For

the sake of simplicity we may assume A-f 1 B > 0 and then insert f = fl in

(16), leaving us with the inequality l= ((C'-f 1 E)x,x) > 0. This means

that C > fl E, etc. Thus, the assertion of the lemma is completely proved.

From what has been said above another conclusion can be drawn in

this direction: the possible applications of the implicit scheme (6) in solving

the original equation Av = f are equivalent to the numerical solution of

the auxiliary equation Cu= <p through the use of the explicit scheme

( 17) -~--+ Xk+l - Xk C xk=<p, k=O,l, ... ,n, x 0 EHgiven,

Tk+l

if we accept C = B-^1 /^2 AB-^1!^2 and <p = B-^1 /^2 f. Under this agreement
all the results obtained in Section 2 for a particular implicit scheme can be
covered by the following statement concerning explicit schemes.