Computing ultra-precise eigenvalues of the Laplacian within polygons
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Computing ultra-precise eigenvalues of the Laplacian within polygons Robert Stephen Jones1
Received: 1 September 2016 / Accepted: 14 March 2017 © Springer Science+Business Media New York 2017
Abstract 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. Keywords Laplacian eigenvalue · Helmholtz equation · Method of particular solutions · Point-matching method · Polygon · Eigenvalue bound Communicated by: Alexander Barnett Robert Stephen Jones
[email protected] 1
Independent Researcher, Sunbury, Ohio, USA
R.S. Jones
Mathematics Subject Classification (2010) 65N35
1 Introduction The task is to calculate very precise eigenvalues of the Laplace operator within the depicted in Fig. 1, on which may be imposed either Dirichlet or polygon shapes Neumann boundary conditions. Specifically, we want values of λ = k 2 which satisfy r∈ (1) ( + λ) (k; r) = 0 (k; r) = 0 Dirichlet ∂ r ∈ ∂ (2) (k; r) = 0 Neumann ∂n is the boundary of and ∂/∂n is the outward-pointing normal derivative where ∂ on that boundary. For r, either polar (r, θ) or Cartesian (x, y) = (r cos θ, r sin θ) coordinates may be used. The Dirichlet and Neumann1 eigenvalues each form a non-accumulating, infinite “tower” of distinct real numbers, 0 < λ1 < λ 2 < λ 3 < · · ·
(3)
some of which may be multiple (also called degenerate). An “eigen-pair” shall be denoted, {λα , αi } (4) with i = 1, 2, 3, ..., gα αi βj da = δαβ δij (5)
where gα is the multiplicity of λα , and where the eigenfunctions shall be orthonormalized as indicated. The solution technique is substa
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