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Computing the eigenvalues of the Laplace-Beltrami operator on the surface of a torus: a numerical approach. (English) Zbl 1152.65477

Glowinski, Roland (ed.) et al., Partial differential equations. Modelling and numerical simulation. Some papers based on the presentations at the international conference, Helsinki, Finland, fall 2005. Dordrecht: Springer (ISBN 978-1-4020-8757-8/hbk). Computational Methods in Applied Sciences (Springer) 16, 225-232 (2008).
Summary: We present a methodology for numerically computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on the surface of a torus. Beginning with a variational formulation, we derive an equivalent partial differential equation (PDE) formulation and then discretize the PDE using finite differences to obtain an algebraic generalized eigenvalue problem. This finite dimensional eigenvalue problem is solved numerically using the eigenfunction in Matlab which is based upon ARPACK. We show results for problems of order 16K variables where we computed lowest 15 modes. We also show a bifurcation study of eigenvalue trajectories as functions of aspect ration of the major to minor axis of the torus.
For the entire collection see [Zbl 1140.65007].

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs

Software:

ARPACK; Matlab; eigs
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