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

Guidotti, Patrick; Lambers, James V.

doi:10.1080/01630560802099233

Cited by 9 publications

(15 citation statements)

References 10 publications

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“…First of all the calculation of the descriptor would greatly benefit from advanced numerical methods [5,9,13,14] that could solve the Helmholtz equation with an higher degree of accuracy and possibly faster (it is known that finite difference schemes may introduce spurious modes and that there exists a dependence between the grid resolution and the largest index of the eigenpair that can be computed). We also believe in the importance of quantitatively characterizing the influence that the perturbations on the boundaries of the curves have on the coefficients of the descriptors or equivalently the sensitivity of the HD with respect to morphological perturbations.…”

Section: Discussionmentioning

confidence: 99%

Drums, curve descriptors and affine invariant region matching

Zuliani

Bertelli

Kenney

et al. 2008

Image and Vision Computing

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Section: Discussionmentioning

confidence: 99%

Drums, curve descriptors and affine invariant region matching

Zuliani

Bertelli

Kenney

et al. 2008

Image and Vision Computing

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“…Hence to the difference of analytic functions whose derivatives are again analytic, in general the (F, G)-derivatives of pseudoanalytic functions are solutions of another Vekua equation with the coefficients given by (23). Obviously, this process of construction of new Vekua equations associated with the previous ones via relations (23) can be continued and we arrive at the following definition.…”

Section: Definition 5 a Pair Of Solutions F And G Of A Vekua Equationmentioning

confidence: 99%

“…Obviously, this process of construction of new Vekua equations associated with the previous ones via relations (23) can be continued and we arrive at the following definition.…”

Section: Definition 5 a Pair Of Solutions F And G Of A Vekua Equationmentioning

confidence: 99%

“…As another example, let us consider the eigenvalue problem in the so-called star-shaped domain [23] whose boundary is defined by the expression in the polar coordinates…”

Section: Complex Variables and Elliptic Equations 821mentioning

confidence: 99%

“…We computed the eigenvalues ¼ ffiffiffi ffi p and compared them with those reported in [23] (Table 6). …”

Section: Complex Variables and Elliptic Equations 821mentioning

confidence: 99%

See 2 more Smart Citations

Construction and application of Bergman-type reproducing kernels for boundary and eigenvalue problems in the plane

Campos

Castillo-Pérez

Kravchenko

2012

Complex Variables and Elliptic Equations

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We show how the Bergman-type reproducing kernels for the elliptic operator D ¼ div p grad þ q with variable coefficients defined in a bounded domain in the plane can be constructed using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q and with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be obtained following a simple algorithm consisting in recursive integration. Then the complete system of solutions is used for constructing the corresponding reproducing kernel. We study theoretical and numerical aspects of the method and apply it to solve boundary value and eigenvalue problems for the stationary Schro¨dinger operator in bounded domains.

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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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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

Cited by 9 publications

References 10 publications

Drums, curve descriptors and affine invariant region matching

Drums, curve descriptors and affine invariant region matching

Construction and application of Bergman-type reproducing kernels for boundary and eigenvalue problems in the plane

Computing ultra-precise eigenvalues of the Laplacian within polygons

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