1549301742-The_Theory_of_Difference_Schemes__Samarskii

112 Basic Concepts of the Theory of Difference Schemes

Lern1na 3 Any function y( x) defined on the equidistant grid

wh = { X; = ih, i = 0, 1, ... ' N, Xo = 0, XN = l}

and vanishing at the points x = 0 and x = l admits the estimates

( 41) h

(^2 2 2)! (^2 2)
4 llv,,ll < llvll < 8 llv'"ll.
Proof We have occasion to use the expansion of y( x) with respect to the
eigenfunctions of problem (14):
N-1
y(x)= L ckμ(k)(x), ck= (y(x), μ(k)(x)),
k=l
By the first Green formula (8) we thus have
( 42) (-Ay, Y) =II Yxll^2 ,
where Ay = Y-xx ,
By definition, A μ(k) = ->.k μ(k) and, therefore,
N-1
-Ay= L ck>.kμ(l')(x).
k=I
Let us substitute this expression into ( 42) and take into account that {μCk)}
is an orthonormal system. As a final result we obtain
N-1
11Yxl1
2
= -(A Y, Y) = L >.kc% ·
k=l
We deduce from here that >.^2
1 llvll2 < llY.rll < >.N_ 1 llvll2, where
4. 2 7rh
A1 = h2 sm 2 / '
4? 7rh

.N-1 = h2 cos~ 2/.
The next goal of our studies is to construct a lower bound for >. 1 with
respect to O'. = 7rh / (2/):
. = 7r2 ( sin O'. ) 2
1 l2 ()'.
Since h < 0.5!, the quantity O'. is varying on the half-interval (0, 7r /4].
It is easy to check that the minimum of the function for O'. E ( 0, 7r / 4] is
attained at the point O'. = 7r / 4, that is, >. 1 (h) attains its minimum in the
case where h = 0.5 l. This serves to motivate that >. 1 > 8 / 12. Taking into
account also that >. N-l < 4 / h^2 , we come to ( 41) and finish the proof of
the lemma.