Nice! An analytic approach would be great.

]]>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!

]]>Apologies about the delay from my end- I’ve been writing up some notes to summarize the numerical strategy, include some validation experiments, and discuss the results so far.

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.]*

Yes, this is certainly one way to analyze the overlapping strategy: the partitions of unity will assure convergence of the Schwartz iteration in one step.

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.

]]>Here’s one possibility. You’re dividing the triangle into three sectors and a disk, and on each of these regions one can create an exact eigenfunction with Neumann conditions on the original boundary (and some garbage on the new boundaries). Now with some explicit C^2 partition of unity, one can splice together these exact eigenfunctions on the subregions into an approximate eigenfunction on the whole triangle, and the residual will be controlled by the H^1 error between the exact eigenfunctions on the intersection between the subregions.

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.

]]>I’ve updated http://www.math.sfu.ca/~nigam/polymath-figures/Schwarz.pdf to include the implementation details. As a numerical method, this is OK (not great because of conditioning issues!)

]]>The challenge with this is in how I compute the residual. Numerically, my strategy was to approximate $u_i$ (in the notes) by finite linear combinations of Fourier-Bessel functions. The trace of the approximations on the arcs can be written down readily; the application of the Laplacian on the sub-domains is also OK. However, to compute the L2 inner products, I used a quadrature. This is how I assemble the matrices to get the approximate eigenfunctions. Also, the conditioning of the eigenvalue problems wasn’t great. Since one is looking at minimizing the residual in $L^2$, the all-critical traces of $u_i^n \frac{\partial u_0^n}{\partial \nu}$ on the common interfaces play a role, but not as important as one may want. While I want to believe this method gives a good approximation by looking at the numerical residual, I am not 100% convinced.

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.

]]>Well, perhaps we don’t need a rigorous guarantee that the numerical algorithm converges, but instead go with a numerical recipe that in practice gives a numerical eigenfunction and numerical eigenvalue with very good residual , and then do some a posteriori analysis to rigorously conclude that the error is small. Indeed, if one has a demonstrable gap between the numerical eigenvalue and the true third eigenvalue , then some simpleplaying around with eigenvalue decomposition (computing the inner product of against other true eigenfunctions via integration by parts) shows that the residual controls the error in H^2 norm (and hence in L^infty norm, by the Sobolev inequality in my notes), at least if one can ensure that obeys the Neumann condition exactly.

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