1549301742-The_Theory_of_Difference_Schemes__Samarskii

124 Basic Concepts of the Theory of Difference Schemes

They are mutually adjoint, that is, (R 1 y, v) = (y, R 2 v) and

The useful relation

N-l

11 Rl y 112 = h -^2 L y~ h < 4 h -^4 11 y 112

i=l

provides the validity of the following estimates:

(17)

yielding

(18)

0

The difference operator R 3 y = (^1) i^12 yo h ~ ( R 1 - R 2 ) y, y E Qh , is obvi-
ously skew-syn1111etric due to the relationships
and, consequently, ( R 3 y, y) = 0. Its norm satisfies the estimate
It is not difficult to refine the estimate for 11 R 3 11 with the aid of relations
giving II R 3 II< 1 / h^2 •
Non-self-adjoint difference operators appear, for example, in the ap-
proximation of second-order elliptic operators with the first derivatives.
The operator Lu= v^11 (x)+b u'(x), x E [ 0, 1], b = const, is approximated