1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some properties of difference elliptic operators 289

Let us investigate the properties of the operator Ay = -Ay for any

0

y E rlh = H. The operator Ay = ~ (A-y + A+y) with the members

is self-adjoint: A = A* for k 12 = k 21 and admits both estimates (28) and

(29).

The operators A- and A+ will be the subject of separate investiga-

tions. For arbitrary elements y, v E H the summation by parts formula

yields

2 2

(34) (A+y, v) = - 2= ((kexf3 Yx~)xa 'v) 2= (kexf3 Yx~ 'vxJ ex'

ex, f3=1 ex, f3=1

2 2

(35) (A-y, v) = - 2= ((ka/3 Yx~)xa, v) = 2= [kex,6 Yx~ 'Vx 0 ) ex·

ex,(3=1 ex,(3=1

The interchange of y and v followed by that of CY and f3 results in the

relations

2 2

(A+v,y) = """' L__, (k exf3 v x~' y Xa ] -ex - L (kf3ex Yx~ , vxJ f3.

ex, f3=1 a,(3=1

Putting these together with (34) we verify that (A+)* = A+ only if k 12 =

k 21. In a similar manner it is plain to show that in this case (A-)* = A-

and, hence, A* = A. In this regard, it is necessary to take into account

that Yx, = 0 for X 2 = 0 and Yx, = 0 for x 2 = 12 , while Yx 2 = 0 for x 1 = 0

and Yx 2 = 0 for x 1 =1 1. Passing now to the expression (A+y,y), which is

always representable by

2 N2-l

(A+y, Y) = L (kexf3 Yx~ 'Yx 0 ) + L (k11 (Yx,)

2

)ii=Ni hl h2

ex,(3=1

Ni-I

+ L (k22 (YxJ2)i2=N2 h1 h2,

ii= 1

we carry the su111 over CY, f3 under the sign of the inner product in the first
summand, leading by (31) to

2 ( 2 ) 2

Cl f; ((Yx,J

2

, 1) < ex~! kexf3 Yx~, Yx,, < C2f;((Yx 0 )

2

,1) ·