Approximations and Bounds for Eigenvalues of Elliptic Operators

Fox, L.; Henrici, Péter; Moler, Cleve B.

doi:10.1137/0704008

Cited by 201 publications

(152 citation statements)

References 8 publications

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“…The boundary coordinate q ͓0,L͔ parametrizes ⌫; its arc length is L =3͑1+ / 4͒. This Fourier-Bessel basis originates with Fox, Henrici, and Moler 39 for the L-shaped domain; we believe it is new in quantum physics. In Ref.…”

Section: B Choice Of Basis Functionsmentioning

confidence: 95%

Quantum mushroom billiards

Barnett

Betcke

2007

Chaos

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We report the first large-scale statistical study of very high-lying eigenmodes ͑quantum states͒ of the mushroom billiard proposed by L. A. Bunimovich ͓Chaos 11, 802 ͑2001͔͒. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy ͑1.7%͒. We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E −1/3 ͑E is the eigenvalue͒. We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16 000 odd modes ͑an order of magnitude more than any existing study͒. We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers ͑ϳ10 5 ͒ orders of magnitude faster than with competing methods. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2816946͔ Quantum chaos is the study of the quantum (wave) properties of Hamiltonian systems whose classical (ray) dynamics is chaotic. Billiards are some of the simplest and most studied examples; physically their wave analogs are vibrating membranes, quantum, electromagnetic, or acoustic cavities. They continue to provide a wealth of theoretical challenges. In particular "mixed" systems, where ray phase space has both regular and chaotic regions (the generic case), are difficult to analyze. Six years ago Bunimovich described 1 a mushroom billiard with simple mixed dynamics free of the usual island hierarchies of Kolmogorov-Arnold-Moser (KAM). He concluded by anticipating the growth of "quantum mushrooms;" it is this gardening task that we achieve here, by developing advanced numerical methods to collect an unprecedented large number n of eigenmodes (much higher than competing numerics 2 or microwave studies 3 ). Since uncertainties scale as n −1Õ2 , a large n is vital for accurate spectral statistics and for studying the semiclassical (high eigenvalue) limit. We address three main issues: (i) The conjecture of Percival 4 that semiclassically modes live exclusively in invariant (regular or chaotic) regions, and occur in proportion to the phase space volumes. (ii) The mechanism for dynamical tunneling, or quantum coupling between classically isolated phase space regions. (iii) The distribution of spacings of nearest-neighbor eigenvalues, about which recent questions have been raised. 3 We show many pictures of modes, including the boundary phase space (the so-called Husimi function).

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Section: B Choice Of Basis Functionsmentioning

confidence: 95%

Quantum mushroom billiards

Barnett

Betcke

2007

Chaos

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“…Recent works have indicated that highly accurate results may be obtained with meshless methods, as compared to grid-based methods [1,2]. Over the last years, several meshless methods have been proposed, as the Smoothed Particle Hydrodynamics (SPH) [3], the Diffuse Element Method (DEM) [4], the Element Free Galerkin method (EFG) [5], the Reproducing Kernel Particle Method (RKPM) [6,7], the Partition of Unity Finite Element method (PUFEM) [8], the h-p Clouds [9], the Moving Least-Square Reproducing Kernel method (MLSRK) [10], the meshless Local Boundary Integral Equation method (LBIE) [11], the Meshless Local Petrov-Galerkin method (MLPG) [12], meshless point collocation methods using reproducing kernel approximations [13], the Merhod of Fundamental Solutions (MFS) [14], the Method of Particular Solutions (MPS) [15] and more. In the present work we imply the MPC method for the solution of equations that describe the MHD flow.…”

Section: Introductionmentioning

confidence: 99%

An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

Bourantas

Skouras

Loukopoulos

et al. 2009

Journal of Computational Physics

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a b s t r a c tThe aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to Dirichlet, Neumann and Robin boundary conditions. Toward this, the meshless point collocation method (MPCM) is used for MHD flow problems in channels with fully insulating or partially insulating and partially conducting walls, having rectangular, circular, elliptical or even arbitrary cross sections. MPC is a truly meshless and computationally efficient method. The maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently, and the convergence proof for the numerical solution of the Poisson problem with Dirichlet boundary conditions have been attained also. Additionally, in the present work convergence is attained for Neumann and Robin boundary conditions, accordingly. The shape functions are constructed using the Moving Least Squares (MLS) approximation. The refinement procedure with meshless methods is obtained with an easily handled and fully automated manner. We present results for Hartmann number up to 10 5 . The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional FEM and BEM, for the Dirichlet, Neumann and Robin boundary conditions of interior problems with simple or complex boundaries.

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“…The method of particular solutions (MPS), introduced by Fox, Henrici, and Moler in [3] and revived by Betcke and Trefethen in [4], can be used to compute eigenvalues of the Laplacian by solving an equation such as (2.8). Our approach is similar to the MPS in that both algorithms locate an approximate eigenvalue by minimizing a function f ( ) whose value is the smallest singular value of some matrix A( ).…”

Section: Comparison With the Methods Of Particular Solutionsmentioning

confidence: 99%

“…The other, known as the method of particular solutions (MPS), uses special function series and seeks the singular values of matrices obtained by imposing boundary conditions to their discretization. It was first introduced in [3] and has recently been revived by [4], where the authors resolve a major shortcoming of the method that had been already observed in the original [3]. In its current form as formulated by [4], we shall refer to it as the modified MPS method.…”

Section: P Guidotti and J V Lambersmentioning

confidence: 99%