Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods

Yuan, Quan; He, Zhiqing

doi:10.1016/j.cam.2009.08.114

Cited by 17 publications

(15 citation statements)

References 13 publications

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“…1 has a simple structure and there are many previous works (e.g. [3], [13], [14]) on computing eigenvalues and eigenvectors of the L shape domain in the literature, we indeed observed that (41) on a uniform grid of size 21 × 21 computed rough estimates of eigenvalues and eigenfunctions fast.…”

Section: The Gradient Descent Methodssupporting

confidence: 63%

An unconstrained global optimization framework for real symmetric eigenvalue problems

Kim

2019

Applied Numerical Mathematics

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Section: The Gradient Descent Methodssupporting

confidence: 63%

An unconstrained global optimization framework for real symmetric eigenvalue problems

Kim

2019

Applied Numerical Mathematics

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“…It is an exercise in geometry to figure out all of the specific shapes with dihedral-symmetry that also yield exponentially-convergent eigenfunction expansions (per the current procedure). 21 Of course the division of eigenfunctions according to dihedral symmetry is hardly new. For example, Cureton and Kuttler [7] identify the symmetry classes for the regular hexagon, which can be lined up using A = S0, Be = S1, Co = S2, S ′ = S3, S = C0, Bo = C1, Ce = C2, and A ′ = C3; where the S0-C3 are class names in that reference.…”

Section: Some Numerical Considerationsmentioning

confidence: 99%

Computing ultra-precise eigenvalues of the Laplacian within polygons

Jones¹

2017

Adv Comput Math

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The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such eigenvalue calculations can be extended to unprecedented precision, very often to well over a hundred digits, and sometimes to over a thousand digits.A curious observation is that as one increases the number of terms in the eigenfunction expansion, the approximate eigenvalue may be made to alternate above and below the exact eigenvalue. This alternation provides a new method to bound eigenvalues, by inspection.Symmetry must be exploited to simplify the geometry, reduce the number of non-analytic vertices and disentangle degeneracies. The symmetry-reduced polygons considered here have at most one non-analytic vertex from which all edges can be seen. Dirichlet, Neumann, and periodic-type edge conditions, are independently imposed on each polygon edge.The full shapes include the regular polygons and some with re-entrant angles (cut-square, L-shape, 5-point star). Thousand-digit results are obtained for the lowest Dirichlet eigenvalue of the L-shape, and regular pentagon and hexagon.1 Very recently, P. Amore, et al. [1] have independently calculated several highly precise eigenvalues (up to dozens of digits) for various shapes, including the L-shape and the cut-square. They also used the FHM method for some problems, but their focus was on a highorder Richardson extrapolation method with finite elements, which, among other things, can handle more varied shapes. Fortunately, their calculations provide (partial) independent validation of my results.2004, when they demonstrated the numerical advantages of the so-called "generalized singular-value decomposition" (GSVD) method, a relative to the point-matching method.Fortunately, the answer to make the point-matching method work is short and simple: One must both (a) select adequate matching points (both number and distribution) and (b) keep enough precision in the intermediate calculations. My empirical observations are that Chebyshev nodes chosen as matching points (as suggested by Betcke and Trefethen [5]) often work well, even where equally-spaced points do not, and the precision of the intermediate calculations must be significantly higher, often several times, than the precision of the eigenvalue. (a) DC-27 (even and odd) λ ≈ 1044.72 (b) DS-65: λ ≈ 4293.91 (c) DS-1396: λ ≈ 100001.28 FIG. 3. Several representative Dirichlet eigenfunctions within the unit-edged regular pentagon. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. (a) The degenerate pair just below DS-18 (see Fig. 2d, LEFT). (b) Another example that strongly reveals the pentagram nodal pattern. (c) The first eigenfunction with eigenvalue above 100,000, corresponding to the 8139 th overall eigenvalue, and having about fifty wavelengths along a pentagon edge (Λ ≈ 0.020). Notice how the boundary shape is revealed within the sea of apparent chaos (...

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“…The eigenvalue problem over an L-shaped domain has been well investigated by many researchers, e.g., Fox, Henrici, and Moler [11] and Yuan and He [45]. In Table 5.5, we list the first six approximate eigenvalues given by [11] and the lower and upper bounds given by our proposed method using a uniform mesh.…”

Section: Domain With Reentrant Corner: L-shaped Onementioning

confidence: 99%

“…Yuan and He [45] adopted the Lehmann-Goerisch method to produce high-precision eigenvalue bounds for an L-shaped domain, for example, 9.6397238404 ≤ λ 1 ≤ 9.6397238444. However, the a priori estimation λ 11 < ν = 71 < λ 12 used in [45] is based on the eigenvalue bound from [36], which is not a verified result and is available only for special domains of symmetry.…”

Section: Domain With Reentrant Corner: L-shaped Onementioning

confidence: 99%

“…However, the a priori estimation λ 11 < ν = 71 < λ 12 used in [45] is based on the eigenvalue bound from [36], which is not a verified result and is available only for special domains of symmetry. This implies that the combination of the Lehmann-Goerisch method and our algorithm can give high-precision eigenvalue bounds, even for a domain of general shape.…”

Section: Domain With Reentrant Corner: L-shaped Onementioning

confidence: 99%

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Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape

Li¹,

Oishi²

2013

SIAM J. Numer. Anal.

103

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Abstract. The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct a computable a priori error estimation for the FEM solution of Poisson's problem, which holds even for nonconvex domains with reentrant corners. Second, new computable lower bounds are developed for the eigenvalues. Because the interval arithmetic is implemented throughout the computation, the desired eigenvalue bounds are expected to be mathematically correct. We illustrate several computation examples, such as the cases of an L-shaped domain and a crack domain, to demonstrate the efficiency and flexibility of the proposed method. 1. Introduction. The eigenvalue problem for the Laplacian, −Δu = λu, has been well investigated in history, mainly in the study of acoustic theory and problems of vibrating elastic membranes and electromagnetic waveguides. There is considerable literature on approximating the eigenvalues by numerical methods; see, e.g., the review papers of Boffi [5] and Kuttler and Sigillito [19] and the lecture book of Weinberger [41].In this paper, we aim to provide concrete lower and upper bounds for eigenvalues of the Laplacian over a polygonal domain. The need for such bounds arises in many practical problems, for example, the estimation of the Poincaré constant in the embedding theorem of Sobolev spaces (see, e.g., [16]) and the investigation of the spectral distribution of the differential operator when exploring the existence of solutions of nonlinear differential equations (see, e.g., [30,27]). All these problems are closely related to the elliptic eigenvalue problem. Moreover, to apply the computational results to a mathematical proof, strictly correct eigenvalue bounds are desired.It is well known that the upper bound for eigenvalues can be easily computed using the Rayleigh-Ritz method, while the lower bound cannot be determined easily. Many theories propose a qualitative analysis of eigenvalue approximation using various numerical methods, but a concrete lower bound is usually unavailable because of the appearance of unknown constants in the estimation formula. In the following, we present a short review of methods that can provide explicit bounds:

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Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods

Cited by 17 publications

References 13 publications

An unconstrained global optimization framework for real symmetric eigenvalue problems

An unconstrained global optimization framework for real symmetric eigenvalue problems

Computing ultra-precise eigenvalues of the Laplacian within polygons

Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape

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