I just checked in eigenvalue_ex3. It's an example I made last semester 
when I was teaching about eigenvalue problems, and I thought it'd make a 
good libMesh example. We compute eigenvalues of two "isospectral drums": 
they have different shapes, but their entire spectrum is exactly the 
same (so, in principle, these two drums would sound the same). This is a 
well-known result, e.g.
http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum

Also, it's a good benchmark problem, because the eigenvalues have been 
tabulated to high precision in the literature (as indicated in the 
example comments).

Dave

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