My internet access is limited (I am on a public computer at the moment) but I/we will try to write a summary of our discussion after the conference!

The conjecture has been (numerically) examined and (numerically) verified on a fine, non-uniform, grid in parameter space away from the equilateral triangle. The grid spacing is chosen so that the variation of the eigenfunctions is controlled to 0.001. At each of these points, we have numerical upper and lower bounds on the second eigenvalue; these bounds provide an interval of width 1e-7 around the true eigenvalue. The eigenfunctions are computed so the Ritz residual is under 1e-11.

I have *something* coded up which uses the bounds near the equilateral triangle, but am not confident enough about these yet to present them.

[Done. - T.]

In the set-up I tried using numerically, the domains have non-trivial overlap. Solving boundary value problems this way would ensure nice convergence of the iteration. My misgiving came from the conditioning of the eigenvalue problems on the sub-domains; since the computations were in floating-point arithmetic, poor conditioning is worrying.

My thinking was that since the actual eigenfunction is C^2 in the interior, the non-standard eigenvalue problem for the disk will have smooth coefficients. My rationale for not using the partition of unity was that the approximation functions I used in each region satisfy $-\Delta u = \Lambda u$ exactly (but potentially not the boundary data). However, for the purpose of an analytical treatment, the partition of unity strategy may be easier to work with.

To illustrate what I mean by this, let us for simplicity assume that the triangle is covered into just two subregions instead of four. Let be exact eigenfunctions on respectively with the same eigenvalue , and obey the Neumann condition exactly on and respectively. We then glue these together to create a function on the entire triangle , where is a C^2 bump function that equals 1 outside of and equals 0 outside of . Then we may compute

.

Also u obeys the Neumann conditions exactly. Thus if u_1 and u_2 are close in H^1 norm on the common domain , the global residual will be small.

One advantage of this approach is that we don’t need to care too much about the boundary traces of u_1,u_2. But one does need a certain margin of overlap between the subregions so that the cutoffs lie in C^2 with reasonable bounds, it’s not enough for them to be adjacent.

One I got the approximate eigenfunction by this method, I still have to locate the extrema. I do this by interpolating the function by piecewise linears onto a mesh of the triangle, and then doing a search. This can be improved.

Let me add in some of the details of the implementation in the notes. Perhaps some collective trouble-shooting will help.

Using the finite element method, the quadratures are exact (since I use piecewise polynomials). The search proceeds as above. Since I’m using a quasi-regular discretization, both the Galerkin errors and the Lanczos errors are well-understood and the methods are provably convergent. This is a reliable, if not super-fast, work-horse.