Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

Abstract

In this paper, we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Laplacian − ▵ on a general domain Ω ⊂ ℝ² subject to homogeneous Dirichlet or Neumann boundary conditions. The basic idea is to look for eigenfunctions as the superposition of generalized eigenfunctions of the corresponding free space operator, in the spirit of the classical method of particular solutions (MPS). The main novelties of the proposed approach are the possibility of targeting each eigenvalue independently without the need for extensive scanning of the positive real axis and the use of small matrices. This is made possible by iterative inclusion of more basis functions in the expansions and a projection idea that transforms the minimization problem associated with MPS and its variants into a relatively simple zero-finding problem, even for expansions with very few basis functions.

Keywords:

• Dirichlet
• Inverse spectral problem
• Laplacian eigenvalue problem
• Neumann

AMS Subject Classification:

• 35P15
• 65N25

Notes

∗Approximations to the ten smallest eigenvalues of −Δ on the star-shaped domain with boundary given by ρ(θ) = 1 + 0.05sin(2θ), 0 ≤ θ < 2π. The value M _max indicates the largest value of M used. The number of iterations refers to the number of inner iterations.

∗Approximations to the ten smallest eigenvalues of −Δ on the star-shaped domain with boundary given by ρ(θ) = 1 + 0.25sin(4θ), 0 ≤ θ < 2π. The value M _max indicates the largest value of M used. The number of iterations refers to the number of inner iterations.

∗Approximations to the ten smallest eigenvalues of −Δ on the unit square for Dirichlet boundary conditions. The value M _max indicates the largest value of M used.

∗Approximations to the ten smallest nonzero eigenvalues of −Δ on the unit square for Neumann boundary conditions. The value M _max indicates the largest value of M used.

∗Approximations to the ten smallest simple eigenvalues of −Δ on the 128-sided regular polygon. These approximations are compared with those computed in [9, Ch. 8], denoted by λ _k , k = 1, 2, …, 10.

∗Smallest two singular values σ₁ and σ₂ of the matrix for the domain [0,1] × [0,1 + ∊], with m ₀ = 0 and M = 20. For ∊ = 0, the Laplacian has a double eigenvalue at 49.34802200544679, and for ∊ = 10⁻⁶, there are two simple eigenvalues at 49.34794304873002 and 49.34800226626760.

∗Approximations to the three smallest eigenvalues of −Δ on an L-shaped domain using Fourier–Bessel functions of order 2m/3 for each positive integer m, centered at the singular corner. These approximations are compared with those computed in [9, Ch. 8], denoted by λ _k , k = 1, 2, 3.

∗Approximations to the three smallest eigenvalues of −Δ on an L-shaped domain using free space eigenfunctions centered in the interior, with M = 40. These approximations are compared with those computed in [9, Ch. 8], denoted by λ _k , k = 1, 2, 3.

∗Approximations to the three smallest eigenvalues of −Δ on the GWW1 drum, with M = 40. These approximations are compared with those computed in [9, Ch. 8], denoted by λ _k , k = 1, 2, 3.