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Eigenvalues of the Laplace–Beltrami Operator on the Surface of a Torus 229
If i = I and j = I,
2[(Rρ −1 +1) −1 + Rρ −1 + cos(h/2)]u II
− (Rρ −1 +1) −1 (u 1I + u I−1I ) − (Rρ −1 + cos(h/2))u I1
− (Rρ −1 + cos(h/2))u II−1 = λρ 2 (Rρ −1 +1)h 2 u II . (11)
If 2 ≤ i ≤ I − 1andj =1,
2[(Rρ −1 + cos(h)) −1 + Rρ −1 + cos(h) cos(h/2)]u i1
− (Rρ −1 + cos(h)) −1 (u i+11 + u i−11 ) − (Rρ −1 +cos(3h/2))u i2
− (Rρ −1 + cos(h/2))u iI = λρ 2 (Rρ −1 + cos(h))h 2 u i1 . (12)
If 2 ≤ i ≤ I − 1andj = I,
2[(Rρ −1 +1) −1 + Rρ −1 + cos(h/2)]u iI
− (Rρ −1 +1) −1 (u i+1I + u i−1I ) − (Rρ −1 + cos(h/2))u i1
− (Rρ −1 + cos(h/2))u iI−1 = λρ 2 (Rρ −1 +1)h 2 u iI . (13)
If 2 ≤ i, j ≤ I − 1,
2[(Rρ −1 + cos θ j ) −1 + Rρ −1 + cos θ j cos(h/2)]u ij
− (Rρ −1 + cos θ j ) −1 (u i+1j + u i−1j ) − (Rρ −1 + cos(θ j + h/2))u ij+1
− (Rρ −1 + cos(θ j − h/2))u ij−1 = λρ 2 (Rρ −1 + cos θ j )h 2 u ij . (14)
These finite difference formulas generate an approximation to the problem
(3) in the form of a symmetric generalized eigenvalue problem
Ax = λDx, (15)
with A sparse and symmetric positive semi-definite and with D positive definite
and diagonal (independent of the ordering of the variables). We used
Matlab to solve the problem (15) to obtain approximations to eigenvalues
and corresponding eigenfunctions of (2).
5 Numerical Experiments
The Matlab function eigs which is based upon ARPACK [Sor92, LSY98] was
used to perform the numerical calculation of eigenvalues and corresponding
eigenvectors. In all cases, we computed the 15 lowest (algebraically smallest)
eigenvalues of the generalized eigenvalue problem (15) using the shift-invert
option with shift σ = −.0001. Since the eigenvalues are real and non-negative,
the eigenvalues closest to the origin are enhanced with this transformation
and thus easily computed with a Krylov method.