1549301742-The_Theory_of_Difference_Schemes__Samarskii

72 Basic Concepts of the Theory of Difference Schemes

implies that 111/J I le -l) < M h^2. This 111eans that the approxin1ation error is
of order 2 in the negative norn1 II · 11(-l)' Observe that the norn1 II · 11(-l)
is concordant with the norn1 11 u I lo = [f 01 dx (f 0 x u( ~) d~)
2
] 1^12 , so that
II uh 11(-l)--+ 11 u llo as h--+ 0.
Another conclusion can be drawn from this example. Of course, the
study of the local approxin1ation is unsufficient for detern1ination of the
order of the difference approxi111ation and proper evaluation of the quality
of a difference operator.
In the estin1ation of the approxin1ation error the well-founded choice
of the norm depends on the structure of an operator and needs investigation
in every particular case. A precise relationship between an operator and a
norn1 in the process of searching the error of approxin1ation will be estab-
lished in the general case in Section 4. Its concretization for the example
of interest leads naturally to the appearance of the negative norn1 II · 11(-l).
A similar obstacle occurs in trying to construct difference approxin1ations
of the operator Lu = ( k u')', where k( x) is a piecewise continuous function
(see Chapter 3).
Where searching a solution u(x, t) to a nonstationary equation (for
instance, to the heat conduction equation), it is not unreasonable to sepa-
rate the variable t (the tin1e). Some consensus of opinion is that the sought
function u( x, t) as a function of the argument x is an elen1ent of the space
Ho. Let wh be a grid in a domain G of the space {x = (x 1 , •.. , xP)} and let
wT be a grid on the segment 0 < t < t 0. The grid function y(x, t) = YhT(x, t)
is defined on the grid whT = wh X wT = { (x, t), x E wh, y E wT} and gives as
a function of the argun1ent x E wh a vector of the space Hh with the norn1
11 · 111 ,. In dealing with y( x, t) on the grid w hT n1any scientists generally
prefer either

(30)

or one of the alternative norn1s

(31) II y llhT = L T II y(t) llh'

tEwr

Let LhT vhT be a difference approximation of the operator Lu, where
u = u(x, t). The operator Lin assigns really the values to the grid functions
vhT ( x, t) defined on the grid w hT. If v( x, t) as a function of the argun1ent x
belongs to the space Ho, then vh(x,t) =Ph v(x,t) E Hh for any t E [O,t 0 ].
For any continuous int function v(x,t) and all t E wT, we put v 1 n(x,t) =