1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 55

space Hh for the grids w h = {xi = ih} on the segment 0 < x < 1. These

norms are widely used in many applications to numerical analysis (here the

subscript h of Yh is omitted):

(1) the grid analog of the norm on the space C:

11 Y I le = xEwh m~x I Y( x) I or 11 Y 11 c = O<i<N max I Yi I ;

(2) the grid analog of the norm on the space L 2 :

(

N 1 )l/'2

11 Y 11 = i~ YT h or ( )

1/2

11 Y 11 = ,~ Y; h

In what follows we will use, as a rule, several norms associated with
inner products in the space H h (the grid analogs of the L 2 and W}-norms
are available in Chapter 1).
Given a solution u( x) of the original continuous problem, u E H 0 , and
a solution Yh of the appropriate approximate (difference) problem, Yh E H h,
the main goal of the possible theory of approxin1ate methods is the accurate
account of the proximity between Yh and u. It is worth noting here that in
a common setting the vectors Yh and u belong to different spaces. In this
context, two interesting possibilities reduce to the following ones:

• the grid function Yh defined at the nodes w h ( G) should be extended
  (for instance, via the linear interpolation) to all of the remaining
  points x of the domain G. As a final result we get a function fj( x, h)
  of the continuous argument x E G for which the difference f;( x, h )-
  u( x) belongs to the space Ho. The proxin1ity of Yh to u is well-
  characterized by the nun1ber II y(x, h) - u(x) 110 , where II · llo is the
  norm of the space Ho;


• the space H 0 is mapped onto the space H h, thereby putting every
  function u( x) E Ho in correspondence with a grid function uh ( x),
  x E wh, so that uh = Phu E Hh, where Ph is a linear operator
  from H 0 into Hh. It is possible to establish this correspondence in
  a number of different ways by approval of different operators Ph.
  If u( x) is a continuous function, we might accept uh ( x) = u( x) for
  x E wh. Son1etimes ·uh(J.:;) is determined at a node X; E wh as the
  integral mean value of u( x) over some neighborhood of this node (for
  instance, of diameter O(h)). In the sequel we will always assume
  that u(x) is a continuous function and keep uh(xi) = u(xi) for all
  xi E wh unless otherwise is explicitly stated.