1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 113

5. The method of energy inequalities. One of the general and very effective
   ways of constructing a priori estin1ates is the method of energy inequalities.
   We bring several examples illustrating how to use this method in deriving
   a priori estimates in difference problems and find, for instance, the rate of
   convergence of a difference scheme on the basis of these estimates.
   In this section we consider the simplest model problem

(43) u^11 (x)+f(x)=O, O<x<l, u(O) = u(l) = 0.

Example 1 Assuming that an equidistant grid wh is given on the segment

[ 0, 1], we now consider the difference approximation of problem (43)

( 44) Y-xx + f(x) = 0, Yo= YN = 0 ·

Multipling equation ( 44) by hy and summing up the resulting equality over

the grid nodes of w 1 " we eventually get

N-l N-l

( 45) L (Y-xx )i Yi h + L f; Yi h = 0.

i=l i=l

We now rewrite (45) in terms of inner products, whose use permits us to

reduce it to the following one:

( 46) (y_ xx 'y) + (f, y) = 0'

Via transform of the first summand in ( 46) by the Green difference formula

(8^11 ) we find that

( 47) or II Yxll^2 = (f, Y) ·

The estimation of the inner product (f, y) can be done using the Cauchy-
Bunyakowskil inequality (12): I (f, y) I< II f II · II y II· By Lemma 3,

Putting these together with ( 47) we clecluce in agreen1ent with Lemma 1
that

We finally get one possible a priori esti1nate for the solution of problem
( 44):

( 48) llYllc < llJll/(4J2).