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5. Solution of Linear Equation Systems
This is the desired estimate of the error. Due to the oscillations in the solution,
the estimate may not be accurate on any particular iteration, but, as we shall
show below, it is quite good on the average.
In order to remove some of the effects of the oscillation, the eigenvalue
estimates should be averaged over a range of iterations. Depending on the
problem and the number of anticipated iterations, the averaging range may
vary from 1 to 50 (typically 1% of the expected number of iterations).
Finally, we want a method that can treat both real and complex eigenvalues. The error estimator for the complex case (5.93) gives low estimates if
the principal eigenvalue (XI) is real. Also, the contribution of XI to zn drops
out in this case so the eigenvalue estimate is quite poor. However, this fact
can be used to determine whether XI is real or complex. If the ratio
is small, the eigenvalue is probably real; if r is large, the eigenvalue is probably complex. For real eigenvalues, r tends to be smaller than 10W2 and, for
complex eigenvalues, r % 1. One can therefore adopt a value of r = 0.1 as
an indicator of type of eigenvalue and use the appropriate expression for the
error estimator.
A compromise is to use the reduction of the residual as a stopping criterion. Iteration is stopped when the residual norm has been reduced to some
fraction of its original size (usually by three or four orders of magnitude). As
we have shown, the iteration error is related to the residual via Eq. (5.15) so
reduction of the residual is accompanied by reduction of the iteration error. If
the iteration is started from zero initial values, than the initial error is equal
to the solution itself. When the residual level has fallen say three to four
orders of magnitude below the initial level, the error is likely to have fallen
by a comparable amount, i.e. it is of the order of 0.1% of the solution. The
residual and the error usually do not fall in the same way at the beginning
of iteration process; caution is also needed because, if the matrix is poorly
conditioned, the error may be large even when the residual is small.
Many iterative solvers require calculation of the residual. The above approach is then attractive as it requires no additional computation. The norm
of residual prior to the first inner iteration provides a reference for checking
the convergence of inner iterations. At the same time it provides a measure
of the convergence of the outer iterations. Experience shows that inner iterations can be stopped when the residual has fallen by one to two orders of
magnitude. Outer iterations should not be stopped before the residual has
been reduced by three to five orders of magnitude, depending on the desired
accuracy. The sum of absolute residuals (the L1 norm) can be used instead
of the r.m.s. (L2) norm. The convergence criterion should be more stringent
on refined grids, because the discretization errors are smaller on them than
on coarse grids.
5. Solution of Linear Equation Systems
This is the desired estimate of the error. Due to the oscillations in the solution,
the estimate may not be accurate on any particular iteration, but, as we shall
show below, it is quite good on the average.
In order to remove some of the effects of the oscillation, the eigenvalue
estimates should be averaged over a range of iterations. Depending on the
problem and the number of anticipated iterations, the averaging range may
vary from 1 to 50 (typically 1% of the expected number of iterations).
Finally, we want a method that can treat both real and complex eigenvalues. The error estimator for the complex case (5.93) gives low estimates if
the principal eigenvalue (XI) is real. Also, the contribution of XI to zn drops
out in this case so the eigenvalue estimate is quite poor. However, this fact
can be used to determine whether XI is real or complex. If the ratio
is small, the eigenvalue is probably real; if r is large, the eigenvalue is probably complex. For real eigenvalues, r tends to be smaller than 10W2 and, for
complex eigenvalues, r % 1. One can therefore adopt a value of r = 0.1 as
an indicator of type of eigenvalue and use the appropriate expression for the
error estimator.
A compromise is to use the reduction of the residual as a stopping criterion. Iteration is stopped when the residual norm has been reduced to some
fraction of its original size (usually by three or four orders of magnitude). As
we have shown, the iteration error is related to the residual via Eq. (5.15) so
reduction of the residual is accompanied by reduction of the iteration error. If
the iteration is started from zero initial values, than the initial error is equal
to the solution itself. When the residual level has fallen say three to four
orders of magnitude below the initial level, the error is likely to have fallen
by a comparable amount, i.e. it is of the order of 0.1% of the solution. The
residual and the error usually do not fall in the same way at the beginning
of iteration process; caution is also needed because, if the matrix is poorly
conditioned, the error may be large even when the residual is small.
Many iterative solvers require calculation of the residual. The above approach is then attractive as it requires no additional computation. The norm
of residual prior to the first inner iteration provides a reference for checking
the convergence of inner iterations. At the same time it provides a measure
of the convergence of the outer iterations. Experience shows that inner iterations can be stopped when the residual has fallen by one to two orders of
magnitude. Outer iterations should not be stopped before the residual has
been reduced by three to five orders of magnitude, depending on the desired
accuracy. The sum of absolute residuals (the L1 norm) can be used instead
of the r.m.s. (L2) norm. The convergence criterion should be more stringent
on refined grids, because the discretization errors are smaller on them than
on coarse grids.