1549301742-The_Theory_of_Difference_Schemes__Samarskii

746

__ = 2_ ( Yi+1 - Yi _ Yi - Yi-1).

Yxx Ji. I h· 1.+l h· I '

A list of inner products and associated norrns on a grid:

N-l

(y, v) = ~ Yi V; h, llYll = J(y, y);

j\T

( Y, v] = ~ Yi vi h, llvll = j(y, y];

i=l

II~ 11(-l) = ( ~~

1

h ( kt 1 h~k)

2

)

112

;

W 7 ={tj=jr, r>O, j=O,l, ... }-agridintime;

T -· step of the grid W 7 ;

y = yJ = y(tj) - a function defined on the grid w 7 ;

y = yJ+1 = y(tj+1), jj = yj-1 = y(t1-1);

:Q 1 = (y - y)/r, Yt: = (y - y)/r, Y/ = (f;-y)/(2r);

Ytt = (f; - 2y + y)/r2;

Sy1nbols

;r = xi = ( :r\ii), ... , x~"), ... , x~,ip)) - a node of the p-dimensional

rectangular grid w h;

,i;\i,y) Lt' = h O' i O').

hex - step of the grid w h along the direction o•;

X~±la) = ( x(i1)' ... ';r~a) ±hex' ... ' X~ip));

Y = y(xi), y(±la) = y( x~±l"));

Yx 0 = (y(+la) _y)/hu;

Yx 0 = (v-y(-la))/hl,;

YXaXa = (y(+la) - 2y+ y(-la))/h~;
0
D - set of all grid functions given on the grid w,, and vanishing on its
boundary;
H - Hilbert space;
( y, v) - inner product of elcn1ents y, v E H, and associated norn1 11 y 11
J(y, y);
V(A) - domain of definition of an operator A;
R(A) - range of values of an operator A;
E - identity operator;
A : H 1--> H - operator A with the domain V(A) =Hand range R(A) CH;