Some properties of difference elliptic operators 281The sides of rectangles, which constitute the domain Gare assumed to
be commensurable. All this enables us to place in the plane a grid with steps
h 1 and h 2 so that the boundary of the grid domain lies on the boundary of
the domain G. One trick we have encountered is to complete the domain
G to the rectangle and then denote it by G (see Fig. _!4). After that, we
construct in G a difference grid wh and extend it to G. The notation wh
will be used for the grid in the domain G.
Let v be a grid function defined on the grid w h and satisfying the
boundary condition vi /h = 0. Also, it will be sensible to introduce the
function
v(x) = { v(x),
0,
By definition,
II vllwh =II v llwh =II v II,
where
II v 112 = (v, v), (v, y) = L 1i(x) y(x) h 1 h 2 ,
xEwhllvll5h = L v^2 (x)h1h2,
xEwh
With these relations established, we deduce that estimate (19) holds for
any function v( x) defined on the grid w h in the domain G, providing the
values
o 0 =scz- 1 2 +1- 2 2 ) 'where 11 and 12 are the sides of the rectangle G.- The en1bedding theorem. Various a priori estimates for the equation
A y = <p can be obtained in the energy norms in light of the properties
of elliptic opei:ators. One might expect that the energy estimates imply a
uniform estimate, that is, an estimate in the norm
11 y 11 c = xEwh max^11 y( x) I.
The following grid theorem gives a definite answer to this hint.
0
The embedding theorem. For any function y( x) E rlh defined on
the grid wh = {x; = (i 1 h 1 , i 2 h 2 ), iex = 0, 1, ... , Nex, hex= lexf Nex, a= 1, 2}
and vanishing on the boundary (for xex = 0, lex; a= 1, 2) the estimate
(20) llYllc <Mo llAYll