1549301742-The_Theory_of_Difference_Schemes__Samarskii

286 Difference Schen1es for Elliptic Equations

where E stands for the identity operator.

The self-adjointness of the operator A = A 1 + A 2 can be proved by

exactly the same reasoning as in Section 2 with further reference to some

properties such as

0

A~= A°'> 0, for y E S.h.

To make sure of it, it suffices to calculate the row sum with CY= 1 for fixed

i 2 = 1, 2, ... , N2 - 1. The outcome of this is

N1-l N1-l

L (A1 Y)i, V; 1 hi= - L (a1 YxJx,,i1 V; 1 h1

N1-l

L Y; 1 (a1 v:vJ.,,,,i, h1

i I= I

N1 -I

= L Y;, (A1 v);, h1.

i 1 =I

Multiplying this identity by h 2 and summing over i 2 = 1, 2, ... , N 2 - 1, we

establish (A 1 y, v) = (y, A 1 v) and, in a sin1ilar way, (A 2 y, v) = (y, A 2 v),

yielding

(Ay, v) = ((A 1 + A2) y, v) = (y, Av).

By definition, this means A* = A.

6. Equations with mixed derivatives. In this section we consider problem
   (23) involving the elliptic operator L with mixed derivatives
   2
   (30) Lu= L Laf3 'U,
   a,(3=1

assuming the ellipticity conditions

2 2 2

(31) c 1 L ~~ < L kL~μ(x) ~°' ~f3 < c 2 L ~~' x E G,

a,(3=1 a=!

to be valid, where c 1 > 0, c 2 > 0 are constants and e = (~ 1 , ~ 2 ) is an
arbitrary vector. Setting first ~ 1 = 1, ~ 2 = 0 and then ~ 1 = 0, ~ 2 = 1 it is
not difficult to check that

CY= 1,2.