Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions |
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Authors: | Beth A Wingate Mark A Taylor |
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Affiliation: | (1) Los Alamos National Laboratory, MS D413 Computer, Computational and Statistical Sciences Division and Center for Nonlinear Studies, Los Alamos, NM 87544, USA;(2) Sandia National Laboratories, MS318 Exploratory Simulation Technologies, Albuquerque, NM 87185, USA |
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Abstract: | The spatial resolution of eigenfunctions of Sturm–Liouville equations in one-dimension is frequently measured by examining
the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster
like O(1/N
2) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional
eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated
than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In
this work an alternative measure, ℓ, is proposed for estimating the characteristic length scale of eigenfunctions of Sturm–Liouville
equations that does not rely on knowledge of the roots. It is first shown that ℓ is a reasonable measure for evaluating the
eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown
that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional
result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing
the minimum distance between Fekete points to ℓ shows that ℓ is a reasonably good measure of the characteristic length scale
in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle
proposed by Taylor MA, Wingate BA (2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some
of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement. |
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Keywords: | Length scale Eigenfunctions Sturm– Liouville |
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