For the rectangular guide,
the fractional error in the propagation constant
was
less than
, and the accuracy of the electric field strengths in the
trapping region
, when we used
degrees of freedom.
This was sufficiently accurate for the present work.
We estimated the accuracy of the method by solving a cylindrical guide in an
identical fashion with the same and a very similar non-uniform grid, for
which there are
known field solutions [183].
This system is quite similar to the rectangular guide in question,
although it does not possess sharp corners.
However, representing the circular cross-section by a
pixellated approximation
on a square grid (which cannot represent surfaces
other than horizontal and vertical)
was taken to be a stringent test of the method's ability to
handle corners sensibly.
We performed a sweep in radius of the guide from below cut-off to
where there are several bound modes:
Fig. 8.4
shows the propagation constant agrees with the analytics
to within 1%, even
close to cut-off.
The convergence with was measured for the rectangular guide case, and
found to be
with
.
This is less than
optimal for first-order elements (which have a maximum possible
convergence of
), and is believed to be
due to an inability of the bilinear functions to represent
physical in-plane
and
components at dielectric steps, or the weak
field singularities which can physically occur at any exterior dielectric
sharp edges (regardless of whether acute or obtuse).