1549301742-The_Theory_of_Difference_Schemes__Samarskii

712 Methods for Solving Grid Equations

homogeneous equations for the error zi +^1 = yi +^1 - u:

zi+l/2 _ zi

(1)

Tj+l

zi+l _ z)+l/2

(2)

Tj+l

zo =YD - u.

-0 - '

zi+l I = 0

-Yh '

Here we accept, on the same grounds as before, zi+^1!^2 = yi+^112 - u.

In this view, it see111s reasonable to turn to operator-difference schemes

and then follow the usual practice: the operators A 1 y = -A 1 y and A 2 y =

0

-A 2 y are introduced for any y from the space H = Q of all grid functions

defined on the grid wh and vanishing on the boundary /1i under the inner

product structure

(y,v) = L y(x)v(x)h 1 h 2

.rEw7 1

N1-l N2-1

=I: I: y(i1h1,i2h2)v(i 1 h 1 ,i 2 h 2 )hJ1 2.

i1=l i2;;;:l

It is well known that the operators so defined possess some remarkable

properties:

(3)

C\'=l,2.

What is more, the relation A 1 A2 = A 2 A1 takes places in various rectangles
only.

2. The general fran1ework of ADM. The object of subsequent discussions
   is an operator equation

(4) Au= f,

where A : H 1-+ H, His a finite-din1ensional Euclidean space with an inner
product (y, v) and associated norm 11y11 = J(Y:Y}.