Dear Math 116,

Just wanted to say, since you're probably in an hour of need doing HW3,
that HW2 was generally excellent... keep it up! This time I will grade HW3
very soon after Tues, so that we can get cracking on projects for the
final 2 wks.

Let's draw some useful conclusions:

i) We got exponential convergence in qu 1 (laplace) but not in qu 2
(helmholtz); this is because, given analytic boundary curve, the D(s,t)
kernel is analytic in the first case but merely continuous in the second
(its derivative has a log singularity, as mentioned in the Kress review as
to why *both* the single- and double-layer should be split up to handle
singularities explicitly). This is not something we'd discussed in class.
However you should have got about 5 digits accuracy in your scattered
field; some of you did. Therefore in HW3 qu 1, if you want exponential
convergence (& I will give bonus to those who show it), you need to split
both single and double, according to the formulae of Kress.

ii) Here's why many of your experiments to add a corner produced no worse
convergence than an analytic domain: you placed the corner at a line of
reflection symmetry (the x-axis), for both the field u (antisymmetric) and
the domain! This meant the exact tau=0 there (if your choice of boundary
points was symmetric too, then the numerical solution would also be
exactly symmetric - ouch!). So the corner was 'invisible', no density
there to cause trouble.

The Lesson: make everything as *generic* as possible. Notice how I made
you rotate your trefoil pi/10? Why was that? To prevent a reflection
symmetry which otherwise might lead to deceptively good results. Breaking
symmetries leads (usually) to worst-case scenarios, which a good
numerical analyst should be concerned about.

iii) I repeat a rule of programming: debug as you go, plotting everything
you can think of, no matter how simple, to test. Consider making simple
routines whose only purpose is to test another routine. Go back to simple
cases like the disk.

iv) Log plots are great to show errors, including spatially.

v) try to make your plots self-explanatory and beautiful - this means
labels, well-chosen axes scales or 3d viewpoint, colorscales. This is good
practise for scientific publication.

vi) When you test for accuracy, the usual thing is to compare N with 2N.
Comparisons of N=700 vs N=800 don't really tell you how many significant
digits you have, but 2N is usually much more accurate that N, so the
difference is a good estimate of the accuracy for the N case, which you
can then quote.

Have fun - remember you're writing codes which model acoustic,
electromagnetic, water-wave, scattering, which is happening all around
you!

	Alex

*-------------------------------------------------------------------~-^`^-,.-'
|\   Dr Alex Barnett   Rm 308, Bradley Hall, HB 6188, Department of Mathematics
| `                    Dartmouth College, Hanover, NH, 03755
|  http://math.dartmouth.edu/~ahb                   tel: (603) 646-3178