The Dirichlet difference problem for Poisson's equation 245
x 2, '
2 ,'' '
( il hl 'i2h2)
" ," ,,
h 2
" , " " ' , " ' " ' " ,
xFigure 10.
- The Dirichlet difference problen1 in a rectangle. Let now
be a rectangle of sides I 1, 12 (see Fig .l 0) with boundary r.
Of our initial concern is the Dirichlet problem in the rectangle G 0
Go + r for the Poisson equation( l') 6.u= -f(x),
In order to form in Go the grid wh with steps h 1 = l 1 /N1 and h 2 = l 2 /N2,
where N 1 and N 2 are positive integers, we draw up two families of straight
lines such as
X(ii) 1 - - ; "1 h 1 ) "1^0 - -^0 J^1 ' ' ' ' ' N 1 ' X 2 (i^2 J = ; "2 h 2 ' ; "2 --^0 '^1 ' ' ' ' ) N 2 '
We call the points of intersection x = ( i 1 h 1 , i 2 h 2 ) of those straight lines
with the coordinates i 1 h 1 and i 2 h 2 nodes. If a nodal point x = ( i 1 h 1 , i 2 h 2 )
is inside the rectangle, that is, 0 < i 1 < N1 , 0 < i 2 < N2, it falls within
the collection of inner nodes. Let w h be the set of all inner nodes. The
total number of inner nodes is equal to ( N 1 - 1 )( N2 - 1).