5.7 Convergence Criteria and Iteration Errors
127
If the order of the initial error is known, it is possible to monitor the
norm of the difference between two iterates and compare it with the same
quantity a t the beginning of the iteration process. When the difference norm
has fallen three to four orders of magnitude, the error has usually fallen by
a comparable amount.
Both of these methods are only approximate; however, they are better
than the criterion based on the non-normalized difference between two successive iterates.
Error, exact
Error. estim
Difference
\
'
\
'
\
',
Error, exact
Error, estim
- - - - - - - - - Difference
\
0
50
100
50
100
150
Iter.
Iter.
Fig. 5.3. Variation of the norm of the exact iteration error, error estimate, residual
and difference between two iterations for the Laplace problem with the SOR solver
on a 20 x 20 CV grid: relaxation parameter smaller (left) and larger (right) than
the optimum
In order t o test the method of estimating iteration errors, we first present
the solution of a linear 2D problem using the SOR solver. The linear problem
is Laplace equation in the square domain (0 < x < 1; 0 < y < 1) with
Dirichlet boundary conditions chosen t o correspond to the solution +(x, y) =
100xy. The advantage of this choice is that the second order central difference
approximation t o the converged solution is exact on any grid so the actual
difference between the present iterate and the converged solution is easily
computed. The initial guess of the solution is zero everywhere within the
domain. We chose the SOR method as the iterative technique because, if the
relaxation parameter is greater than the optimum value, the eigenvalues are
complex.