1549301742-The_Theory_of_Difference_Schemes__Samarskii

398 Stability Theory of Difference Schemes

When the triangle inequality

II Y ll(ll <II 'fJ ll(ll +II fJ ll(ll

is put together with estimates (2), (3) and (4), we derive the a priori esti-

mates

(5)

and

The forthcoming energy norms will be taken as the norm II · 11(1) in

addition to the approved basic norm 11y11 = ~:

(7)

(8)

llYllA = V(Ay, Y)

llYlls = V(By, Y)

for A= A*> 0,

for B = B* > 0.

Schen1e ( 1) is said to be stable in the space HA (or in H B) if estimate

(5) is- valid in the norm II · 11(1) =II · llA (or II · 111 = II · 113 ).

2. A primary family of schemes. We will discover stability in a certain
   primary family of difference schemes. Before going further, we regard op-
   erators A and B to be bounded linear operators defined on the entire space
   Hh, 'D(A) = 'D(B) = Hh. In what follows the difference problem (1) is
   presupposed to be solvable for any input data Yo and ip(t), that is, there
   exists a bounded operator B-^1 with the domain 'D(B-^1 ) = Hh. For the
   sake of simplicity, we take for granted in the detailed account below that

(1) the operators A and B are independent oft, that is, are constant
operators;
(2) Bis a posieive operator: B > O;
(3) A is a self-adjoint positive operator: A= A* > 0.
Conditions (1)-(3) in combination with the solvability requirement
single out a fa1nily of admissible sche1nes known as a primary family from
the set of all possible schemes (1). Observe that condition 1) can be weak-
ened in a number of different ways. Sometimes we will deal with operators
A and B, which are dependent on t, that is, withA = A(t) and B = B(t).
In the weighted sche1ne for the heat conduction equation in Example
1 of Section 3 the operators A and B are specified by A = -A and B = E +