1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes

Indeed, putting x = A-^1 y we obtain from (55) the inequality

(y, y) < 6.(A-1y, y),

meaning E < 6.A -l or A -l > E / 6.. By virtue of the inequalities

(Ax, x)^2 <II Ax 112 · 11x11^2 < 6.(Ax, x) 11x112

419

we deduce that (Ax, x) < 6.ll x 112, giving A< 6.E. Lemma 5 implies that

B = A-^1 + O"TE > (1/6.+ O'T) E, B - 0.5rA> 0 for O' > 0' 0 ,

where A= E, 0' 0 = ~ - 1/( r6.).

Lemma 6 Let A be a positive definite operator and inequality (55) hold.

Then

(57)

(58)

( 59)

11 (E + O"rA)-^1 (E - (1 - O') rA) II< 1

II (E + O'TA)-^1 II< 1

1

II (E + O'TA)-^1 II< -

c

1 1

for O' > i5' = -- -

-^0 2 r6.'

for O' > 0,

1 1 - c

for O' > i5' = - - --

• c 2 r6. '

O<s<l.
Proof 1) Since B > ~TA for O' > 0' 0 , by applying Theorem 1 to scheme
( 48) we deduce that for a solution of problem ( 48) the estimate is valid for
any Yn E H and <p = 0:
( 60) 11 Yn + 1 11 < 11 Yn 11 ·
As can readily be observed, scheme (47) for <p = 0 can be rewritten as
Yn+1 =Syn, S = (E + O"rA)-^1 (E - (1 - O') rA).
From here and relation (60) we obtain estimate (57).
2) In order to estimate the norm II B-^1 II, where B = E + O'TA, it
suffices to establish an inequality of the fonn B > O' E, O' > 0. Then

bllxll^2 < (Bx,x) <II Bx II· llxll, II Bx II> bllxll,

and, consequently, 'II B-^1 II< l/b. If O' > 0, then B > E and 11 B-^1 II < 1.
If O' > 0' 0 , then
1-s
B > E + 0' 0 r A= E + 0.5 r A - 6. A.

In conformity with (56), A< 6.E, making it possible to conclude that

B > E + 0.5 rA-(1 - s) E > c E + 0.5 rA > s E

and, therefore, II B-^1 II < 1/s. Observe that estimate (58) holds true also
for any non-self-adjoint operator A > 0, thereby completing the proof of
the lemma.