Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
226 R. Glowinski and D.C. Sorensen eigenvalue problem is solved to obtain the approximations. A visualization of our results show the expected Sturm–Liousville behavior of the eigenfunctions according to wave number. Eigenvalues are typically multiplicity one or two. However, we show that for certain ratios of the minor to major radii, it is possible to create eigenvalues of multiplicity three or four. This indicates an interesting bifurcation structure is associated with this ratio. A thorough discussion of the approximate solution of eigenvalue problems for elliptic operators is given by Babushka and Osborn [BO91]. 2 Variational Formulation of the Eigenvalue Problem Let Σ be the boundary of a three-dimensional torus defined by a great circle of radius R and a small circle of radius ρ (see Figure 1). Our goal here is to numerically approximate the eigenvalues and corresponding eigenfunctions of the Laplace–Beltrami operator associated with Σ. A variational formulation of this problem reads as follows: Find λ ∈ R, u ∈H 1 (Σ) such that ∫ ∫ ∇ Σ u ·∇ Σ vdΣ = λ uvdΣ, ∀v ∈H 1 (Σ). (1) Σ In the equation (1): (i) ∇ Σ is the tangential gradient on Σ, (ii) dΣ is the infinitesimal superficial (surfacic) measure, (iii) H 1 (Σ) ={v|v ∈L 2 (Σ), ∫ Σ |∇ Σv| 2 dΣ
Eigenvalues of the Laplace–Beltrami Operator on the Surface of a Torus 227 Find u ∈H 1 p(Ω 0 )andλ, such that ρ ∂u ∂v R + ρ cos θ ∂φ ∂φ + R + ρ cos θ ] ∂u ∂v dφdθ ρ ∂θ ∂θ ∫ = λ ρ(R + ρ cos θ)uvdφdθ, (2) Ω 0 for all v ∈Hp(Ω 1 0 ), with Ω 0 =(0, 2π) × (0, 2π) and with ∫Ω 0 [ Hp(Ω 1 0 )={v | v ∈H 1 (Ω 0 ), v(0,θ)=v(2π, θ), for a.e. θ ∈ (0, 2π), v(φ, 0) = v(φ, 2π), for a.e. φ ∈ (0, 2π)}, i.e., H 1 p(Ω 0 ) is a space of doubly periodic functions. In the following, keep in mind that 0
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226 R. Glowinski <strong>and</strong> D.C. Sorensen<br />
eigenvalue problem is solved to obtain the approximations. A visualization of<br />
our results show the expected Sturm–Liousville behavior of the eigenfunctions<br />
according to wave number. Eigenvalues are typically multiplicity one or two.<br />
However, we show that for certain ratios of the minor to major radii, it is<br />
possible to create eigenvalues of multiplicity three or four. This indicates an<br />
interesting bifurcation structure is associated with this ratio.<br />
A thorough discussion of the approximate solution of eigenvalue problems<br />
for elliptic operators is given by Babushka <strong>and</strong> Osborn [BO91].<br />
2 Variational Formulation of the Eigenvalue Problem<br />
Let Σ be the boundary of a three-dimensional torus defined by a great circle<br />
of radius R <strong>and</strong> a small circle of radius ρ (see Figure 1).<br />
Our goal here is to numerically approximate the eigenvalues <strong>and</strong> corresponding<br />
eigenfunctions of the Laplace–Beltrami operator associated with Σ.<br />
A variational formulation of this problem reads as follows:<br />
Find λ ∈ R, u ∈H 1 (Σ) such that<br />
∫<br />
∫<br />
∇ Σ u ·∇ Σ vdΣ = λ uvdΣ, ∀v ∈H 1 (Σ). (1)<br />
Σ<br />
In the equation (1):<br />
(i) ∇ Σ is the tangential gradient on Σ,<br />
(ii) dΣ is the infinitesimal superficial (surfacic) measure,<br />
(iii) H 1 (Σ) ={v|v ∈L 2 (Σ), ∫ Σ |∇ Σv| 2 dΣ