Computing ultra-precise eigenvalues of the Laplacian within polygons
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The classic eigenvalue problem of the Laplace operator inside a variety of polygons is numerically solved by using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper. It is demonstrated that such eigenvalue calculations can be extended to unprecedented precision, often to well over a hundred digits, or even thousands of digits. To work well, geometric symmetry must be exploited. The de-symmetrized fundamental domains (usually triangular) considered here have at most one non-analytic vertex. Dirichlet, Neumann, and periodic-type edge conditions are independently imposed on each symmetry-reduced polygon edge. The method of particular solutions is used whereby an eigenfunction is expanded in an N-term Fourier-Bessel series about the non-analytic vertex and made to match at a set of N points on the boundary. Under the right conditions, the so-called point-matching determinant has roots that approximate eigenvalues. A key observation is that by increasing the number of terms in the expansion, the approximate eigenvalue may be made to alternate above and below, while approaching what is presumed to be the exact eigenvalue. This alternation effectively provides a new method to bound eigenvalues, by inspection. Specific examples include Dirichlet and Neumann eigenvalues within polygons with re-entrant angles (L-shape, cut-square, 5-point star) and the regular polygons. Thousand-digit results are reported for the lowest Dirichlet eigenvalues of the L-shape, and regular pentagon and hexagon.
KeywordsLaplacian eigenvalue Helmholtz equation Method of particular solutions Point-matching method Polygon Eigenvalue bound
Mathematics Subject Classification (2010)65N35
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I wish to thank Alex Barnett for making the specific suggestion to expand about the non-analytic vertex using fractional-order Bessel functions, i.e., non-integral m-values (private communication, December 2014). Indeed, after sharing some regular pentagon results with him, he suggested that instead of expanding about the center of the regular pentagon, I should expand about one of its vertices. That one simple, and—in hind-sight—obvious suggestion, immediately turned my eight-digit calculations into multi-hundred-digit calculations because of the exponential convergence. I also wish to thank James Kuttler and Nick Trefethen for suggestions and encouragement. I am also encouraged by the recent independent efforts of Mark Boady and Paolo Amore, et al., and wish to thank them for interesting dialogs. Of course, this project was made possible by free software, most specifically GMP  and the GP/PARI  calculator.
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