Partial Differential Equations - Modelling and ... - ResearchGate

Computing the Eigenvalues
of the Laplace–Beltrami Operator
on the Surface of a Torus: A Numerical
Approach
Roland Glowinski 1 and Danny C. Sorensen 2
1 University of Houston, Department of Mathematics, Houston, TX, 77004, USA
roland@math.uh.edu
2 Rice University, Department of Computational and Applied Mathematics,
Houston, TX, 77251-1892, USA sorensen@rice.edu
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.
1 Introduction
A large number of physical phenomena take place on surfaces. Many of these
are modeled by partial differential equations, a typical example being provided
by elastic shells. It is not surprising, therefore, that many questions
have arisen concerning the spectrum of some partial differential operators defined
on surfaces. This area of investigation is known as spectral geometry.
Among these operators defined on surfaces, a most important one is the so
called Beltrami Laplacian, also known as the Laplace–Beltrami operator. The
main goal of this chapter is to discuss the computation of the lowest eigenvalues
of the Laplace–Beltrami operator associated with the boundary of a torus
of R 3 . After a description of our methodology for the computation of these
eigenvalues and their corresponding eigenfunctions, we present selected results
from our numerical experiments. The methodology consists of obtaining
a finite difference discretization of a PDE that is equivalent to a more standard
variational formulation; then the resulting finite dimensional generalized