1549301742-The_Theory_of_Difference_Schemes__Samarskii

68 Basic Concepts of the Theory of Difference Schemes

Having stipulated the condition h± < 21i, the well-established expansions

of a sufficiently smooth function v( x) in a neighborhood of the node x such

as

h2 h3

v(x + h+) = v(x) + h+ v'(x) +

2

+ v^11 (x) +

6

+ v^111 (x) + O(h!),

I h

2

II ) h

3

v(x-h_)=v(x)-h_v(x)+ Ill^4

2

-v (x -5v (x)+O(h_)

lead to the following ones:

h h^2

vx = v'(x) + { v^11 (x) +

6

• v^111 (x) + O(h!),

h h^2

vx=v'(x)-

2

• v^11 (x)+f v^111 (x)+O(h~),

V - V- h^2 - h^2

Lh v = x n x = v^11 (x) + +

6

n - v^111 (x) + 0(1i^2 ).

With these relations in view, we derive the usefull expressions for 1/J( x)

(27) 1jJ = Lh v - Lv = h+ - 3 h_ v^111 + 0(1i^2 ) = O(n).

Thus, operator (26) provides the local approximation of order 1 on any
irregular pattern with h_ f- h+.

3. The error of approximation on a grid. So far we have considered the
   local difference approximation; meaning the approximation at a point. Just
   in this sense we spoke about the order of approximation in the preceding
   section. Usually some estimates of the difference approximation order on
   the whole grid are needed in various constructions.
   In preparation for this, let w h be a grid in a domain G of the Euclidean
   space { x = ( x 1 , ••• , xp)}, Hh be a vector space of grid functions defined on
   the grid wh and let the space Ho comprise all of the smooth functions v(x),
   whose norms are defined by II · llo and II · llh, respectively. In the sequel
   we take for granted the following assumptions:
   ( l) there exists an operator T\ such that T\ u = uh E H h for any
   u E Ho;
   (2) the norms II llo and II · llh are concordant, that is,

lim llPhH llh = llullo,

I h 1---+0

where I h I stands for the norm of a vector h.