Hearing Ellipsoids: A Benchmark of Krylov, Spectral Galerkin, and Filter Diagonalization Eigensolvers

We present a computational benchmark for large Dirichlet eigenvalue spectra on triaxial ellipsoids, comparing finite-element discretizations combined with ARPACK, inertia-based spectrum slicing (SLEPc/MUMPS), a stretched spectral Galerkin (SSG) method, and a multi-probe block-Hankel filter diagonalization that we extend here from a landmark detector into a gap-free complete-spectrum solver. Rather than ranking methods by runtime alone, we ask whether the resulting spectra preserve reliable geometric information. The unit ball serves as a quality-assurance case, since its eigenvalues are known in closed form; for the triaxial case a=1, b=1.5, c=2.3 we add a reverse-geometry validation based on the three-term Weyl law, reconstructing volume, surface area, and integrated mean curvature from finite spectral segments. As the eigenvalue index grows, the error is increasingly dominated by the mesh rather than by the eigensolver. SSG gives the strongest geometric consistency in our tests, reconstructing the volume to about 0.025%. The block-Hankel method extended by a Weyl-equal-count tiling, an analytic one-sided bandpass, and coprime decimation voting produces a complete ordered spectrum of 2034 levels including multiplicities; spectrum slicing returns the same 2034 levels on the identical matrices in 4.9 core-hours against roughly 69,400. For this task, the established method is therefore superior by four orders of magnitude, and we report that plainly. What the comparison establishes instead is agreement; three unrelated solvers on one discretization coincide to a median relative deviation of 3×10−11, with Sylvester inertia counts certifying the level count exactly. On the unit ball, where degeneracies reach 2ℓ+1=51, the certified spectrum reproduces the exact level count without deviation up to λ=600 and resolves degeneracies up to 35. There, the same certificate does more than confirm—it repairs, adding 110 level groups the extraction had lost, without reference to any other solver.