1549301742-The_Theory_of_Difference_Schemes__Samarskii

54 Basic Concepts of the Theory of Difference Schemes

0

Figure 3.

the number N in the case of an equidistant grid. In dealing with various

non-equidistant grids w h we 1nean by step h the vector h = ( h 1 , h 2 , ... , h N)

with components h 1 , h 2 , .•. , hN.

The sa1ne observation remains valid with 1nultidirnensional domains

0. Here x = (x 1 , ... ,xp) and, on the same grounds, h = (h 1 ,h 2 ,. .. ,hp)
   if the grid wh is equidistant in each of the arguments x 1 , ... , xP'
   Throughout the entire chapter, the functions u( x) of the continuous
   argument x E 0 are the elements of son1e functional space H 0. The space
   Hh comprises all of the grid functions Yh(x), providing a possibility to
   replace within the framework of the finite difference method the space H 0
   by the space H h of grid functions Yh ( x). Recall that although the fixed
   notation II · II is usually adopted, there is a wide variety of possible choices
   of the functional form of II · II·
   In a common setting the set { Hh} of spaces of grid functions depending
   on the parameter h corresponds to the set of grids {wh}, making it possible
   to introduce in the vector space H1i the norm II · llh, which is a grid analog
   of the nonn II · llo of the initial space Ho. We give below two norms in the