Summary
In this chapter, 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 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 eigs function 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.
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Glowinski, R., Sorensen, D.C. (2008). Computing the Eigenvalues of the Laplace-Beltrami Operator on the Surface of a Torus: A Numerical Approach. In: Glowinski, R., Neittaanmäki, P. (eds) Partial Differential Equations. Computational Methods in Applied Sciences, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8758-5_12
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DOI: https://doi.org/10.1007/978-1-4020-8758-5_12
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