For this problem, I wrote a collection of routines that solves the 2-dimensional Laplace equation
for a specific domain with specific boundary. The domain is
. The boundary
is specified by the boundary values of the particular solution to
Laplace's equation:
One particular finite differencing of Laplace's equation leads to an iterative scheme that is successively applied to the interior points of the domain:
This iterative method is called Jacobi's method. I have implimented
this algorithm in the program LAPLACE.F90. The convergence
criterion is that the infinity norm between 
and 
be less than 0.000003. In FORTRAN 90, that is
do while ( MAXVAL( ABS(U - UOLD) ) > 0.000003)
.
.
.
JACOBI STEP
.
.
.
enddo
The program then returns the total number of iterations, as well as the
infinity norm between the final 
and the analytic solution from
wence came the boundary values. In FORTRAN 90, this is just
error = MAXVAL(ABS(U - U_ANALYTIC))
The code is described in more detail below.