1549301742-The_Theory_of_Difference_Schemes__Samarskii

282 Difference Schernes for Elliptic Equations

is valid with

12

M - o

o-2 lfT'

V 'I '2

Proof Suppose that vk, k 2 ( x) and Ak, k) x) are the eigenfunctions and eigen-

values of the operator A that we have determined in Section 1 for ko: =

1, 2, ... , No: - 1 and CY = 1, 2. The expansion of the function y( x) with

respect to the orthonormal functions { vk, k 2 ( x)}

yields

y(x) = L ck,k2 vk,k2(x)

kl I k2

Ay = L ck,k2 Ak1A'2 vA>1h ,

7.:1) k'J

llYll2= L ci 1 k 2 '

k 1, k2

II Ay 11

2

= L ci,k2 >-Lk2 ·

k1 I k2

The function y( x) satisfies the inequality

I y(x) I< ( L I ck 1 k 2 I) max I vk,k 2 (x) I·

k k 1, 2 ki,k2

From formula (5) we get I vk,k 2 (x) I < 2/ .jl;r;. Applying the Cauchy-

Bunyakovski1 inequality we establish the chain of the relations

(21)

4

11 12 ( L I ck,l.:2 I Ak,k2 A,'.J'

ki J k2

4

2=

;,2 c2

2=

1

<-

(^11 12) ki J k2 k1k2 kik2 k 1 J k3 >,2 kik2
4
II Ay (^112) 2= ;,-2
II (^12) k,' k2 k, k2.