1549301742-The_Theory_of_Difference_Schemes__Samarskii

234 Homogeneous Difference Schemes

where D = A-^1!^2 JIA-^1!^2 , and, after this, pass to D^112 y = z:

1 = ( 1 + ex) ( n-^1 z, z) - 11 z 112

= (1 +ex) (A^1 l^2 JI-l A^1 l^2 z, z) - II z 112

= (1 +ex) (A-^1 v, v) - (A-^1 v, v) = A^1 l^2 z.

As far as J > 0, we have (1 +cx)JI-^1 > A-^1 , thereby justifying that

for any operators A = A* > 0 and A = A* > 0 the inequality A-^1 <

(1 + cx)A-^1 follows from the inequality A< (1 + cx)A. Moreover, we claim

that inequalities (20) are equivalent to

(1 - ex) E < C < (1 +ex) E,

Indeed,

( 1 + CY) (A -^1 x' x) - (A - l x, x) = ( 1 + CY) 11 y 112 - ( A^1 I^2 A -l A^1 I^2 y' y)

= (l+cx)llYll^2 -(Cy,y) > 0.

Thus, (18) implies that -cxE < E-C < cxE and C = A^112 A-^1 A^112.

By the definition of the norm of self-adjoint operator,

Substituting this estimate into (17) we deduce from (16) that

or, what amounts to the same,

ll'u - u ll;t <II j - f ll;t-^1 +CY II f ll;t-^1 •

If we are in possession of ~ an operator Ao = A 0 of ~ rather simplified

structure than the operator A satisfying the condition A > c 1 Ao, c 1 > 0,

then

provided the in verse Ai)^1 exists.
theorem.

1

II f ll;t-^1 < - II f ll;t-^1 ,

Fi 0

Thus, we have proved the comparison