The Dirichlet difference proble111 for Poisson's equation 251a) b)
Figure 12. a) A disconnected grid. b) A connected grid.
Our purpose here is to construct a difference scheme for solving the
Dirichlet problem in the domain G = G + r, the complete posing of which
is to find an unknown solution to the equation
82 u 82 u
~ll = a ,2 +-a 2 = -f(x),
xi x2
which is continuous in the closed domain G = G + r and satisfies the
boundary condition ulr = μ(x).
At each of the inner nodes x E wh we approximate the difference
operatorby the three-point difference operator Aa.
If a node x E wh is regular with respect to xa, then the difference
operator Aa on the regular pattern (x(-l"J, x, x(+l"J) is similar to (11):
y( + l") - 2 y + y( -l" )
Aa y = Yx" "'" = h2
ry
But if a node x E w 1 ** i.) Q' , that is, a node is irregular with respect to x ~ '-" on
the irregular pattern, then the difference operator Aa can be rewritten as
- _l ( y( + l c.) - y - y - y( -l"))
(23a) Aa y - h (\' h O' h* O' for
- _l ( y( + l c.) - y - y - y( -l"))