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Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

A. H. Barnett
2009 SIAM Journal on Numerical Analysis  
In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes.  ...  The Dirichlet eigenvalue or "drum" problem in a domain Ω ⊂ R 2 becomes numerically challenging at high eigenvalue (frequency) E.  ...  Application to accurate inclusion of high eigenvalues.  ... 
doi:10.1137/080724022 fatcat:3jnpdqbv6veqdp3mml5r3nb22u

Range of the first three eigenvalues of the planar Dirichlet Laplacian [article]

Michael Levitin, Rustem Yagudin
2002 arXiv   pre-print
We conduct extensive numerical experiments aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian.  ...  The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio λ_3/λ_1.  ...  Numerical analysis of random domains To the best of our knowledge, there have been no large scale numerical experiments on low eigenvalues of the Dirichlet Laplacian for planar domain.  ... 
arXiv:math/0203231v1 fatcat:frbvi6isrbf37dag3i3ffecax4

Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues [article]

A. H. Barnett, Andrew Hassell
2010 arXiv   pre-print
One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions.  ...  We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain Ω⊂^n with piecewise smooth boundary.  ...  An important application is to solving (1.1)-(1.3) via global approximation methods, including the method of particular solutions (MPS) [9, 4] .  ... 
arXiv:1006.3592v1 fatcat:sykwejspvbgedjypyef2yejfhi

Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues

A. H. Barnett, A. Hassell
2011 SIAM Journal on Numerical Analysis  
One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions.  ...  We study eigenfunctions φ j and eigenvalues E j of the Dirichlet Laplacian on a bounded domain Ω ⊂ R n with piecewise smooth boundary.  ...  The work of AHB was supported by NSF grant DMS-0811005, and a Visiting Fellowship to ANU in February 2009 as part of the ANU  ... 
doi:10.1137/100796637 fatcat:fxe3okl2lzbdnpccuebanjlodm

Metric deformation and boundary value problems in 3D

S. Panda, S. P. Khastgir
2014 Progress of Theoretical and Experimental Physics  
The method seems to work quite well for these shapes for both, Dirichlet as well as Neumann boundary conditions.  ...  of the perturbative series for the energy.  ...  We thank the referees and the editor for bringing many critical points to our notice and several constructive suggestions for improving the text.  ... 
doi:10.1093/ptep/ptu051 fatcat:obns4gxigvanpirm74ywqwgrcu

Range of the First Three Eigenvalues of the Planar Dirichlet Laplacian

Michael Levitin, Rustem Yagudin
2003 LMS Journal of Computation and Mathematics  
AbstractExtensive numerical experiments have been conducted by the authors, aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian.  ...  The results improve the previously known theoretical estimates of M. Ashbaugh and R. Benguria. Some properties of a maximizer of the ratio λ3/λ1are also proved in the paper.  ...  It is the authors, nevertheless, who take full responsibility for the realization of this idea, and any criticism for possible shortcomings of this realization should be addressed to them.  ... 
doi:10.1112/s1461157000000346 fatcat:5zoovuiplvghphun3zq3wu7xfe

A non-iterative sampling approach using noise subspace projection for EIT

Cédric Bellis, Andrei Constantinescu, Thomas Coquet, Thomas Jaravel, Armin Lechleiter
2012 Inverse Problems  
For a subsequent implementation in a discrete setting, the quality of classical finite-dimensional approximations of the measurement operator is discussed.  ...  An introductory overlook to the forward and inverse conductivity problems is followed by the exposition of the so-called Picard criterion as a characterization of the range of the relative Neumann-to-Dirichlet  ...  It has been pursued by Vincent Choquet and Julien Alaterre who are kindly acknowledged for providing us with the experimental results of Section 5.4.  ... 
doi:10.1088/0266-5611/28/7/075015 fatcat:apluusr3uzbfzj3ueupx4eonfi

Geometrical structure of Laplacian eigenfunctions [article]

Denis S. Grebenkov, Binh-Thanh Nguyen
2013 arXiv   pre-print
We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition.  ...  The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.  ...  The authors thank H. Obuse and A. Barnett for providing their images of eigenfunctions and allowing us to reproduce them. We acknowledge fruitful discussions with Y. G. Sinai and A. L. Delitsyn.  ... 
arXiv:1206.1278v2 fatcat:pmeiikacdrhjxjqjpoh44tvaii

Geometric Aspects of Spectral Theory

Michiel van den Berg, Daniel Grieser, Thomas Hoffmann-Ostenhof, Iosif Polterovich
2012 Oberwolfach Reports  
The workshop "Geometric Aspects of Spectral Theory" brought together leading researchers working in various areas of this vast field of mathematics.  ...  The meeting featured presentations on some of the most fascinating recent developments in the subject, including five survey talks given by top experts, as well as reports on the progress made by graduate  ...  formulae for the concerning the behaviour of solutions (ε, u) under perturbation of ∂Ω a Questions.  ... 
doi:10.4171/owr/2012/33 fatcat:4u2lgrvogfbjldyun42obedgau

Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces [article]

Alexander Strohmaier
2016 arXiv   pre-print
The aim of the lecture was to explain the mathematical theory behind computations of eigenvalues and spectral determinants in geometrically non-trivial contexts.  ...  These are lecture notes from a series of three lectures given at the summer school "Geometric and Computational Spectral Theory" in Montreal in June 2015.  ...  I would like to thank the organizers of the summer school for the perfect organization and the hospitality.  ... 
arXiv:1604.02722v2 fatcat:vva5jdnlkzbb5ij7ym7edm6oda

Convergence and Error Analysis of FE-HMM/FE^2 for Energetically Consistent Micro-Coupling Conditions in Linear Elastic Solids [article]

Andreas Fischer, Bernhard Eidel
2018 arXiv   pre-print
All results of the present work are valid for both the Finite Element Heterogeneous Multiscale Method FE-HMM and for FE^2.  ...  The analysis addresses aspects of (i) regularity and how its loss affects the convergence behavior on both scales compared with the a priori estimates, of (ii) error propagation from micro to macro and  ...  Coupling Direct solution method Lagrange multiplier method For a uniform micro mesh in 2D with N nodes per edge Tab. 1 displays the number of degrees of freedom for both methods in the cases of Dirichlet  ... 
arXiv:1805.03077v2 fatcat:xifsixbw75fnblc4zovnibqe64

Membranes with thin and heavy inclusions: asymptotics of spectra [article]

Yuriy Golovaty
2021 arXiv   pre-print
We study the asymptotic behaviour of eigenvalues and eigenfunctions of 2D vibrating systems with mass density perturbed in a vicinity of closed curves.  ...  The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces.  ...  Using the method of quasimodes we describe the asymptotic behaviour of eigenvalues of (1.2) as perturbation of σ(P).  ... 
arXiv:2101.10059v1 fatcat:nw34uiwp4nd6bh3mssumzar474

A general method for central potentials in quantum mechanics [article]

Amlan K. Roy
2019 arXiv   pre-print
This offers very high-quality results for both ground and higher lying states for arbitrary values of potential parameters (covering both weak and strong coupling) with equal ease and efficacy.  ...  Essentially this allows optimal, nonuniform spatial discretization of the pertinent single-particle Schrodinger equation satisfying Dirichlet boundary condition leading to standard diagonalization of symmetric  ...  for some high-lying states; ν = 48, 49 and l = 0 − 9.  ... 
arXiv:1904.08719v1 fatcat:cr25vetdnbf7jfarmzaqoc3oiu

Localised modes due to defects in high contrast periodic media via two-scale homogenization [article]

I.V. Kamotski, V. P. Smyshlyaev
2018 arXiv   pre-print
Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with square  ...  Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V.V.  ...  The present manuscript represent a substantially refined and updated version of the one initially published in the BICS preprints series (2006).  ... 
arXiv:1801.03372v1 fatcat:7mcgojk5ujfpfhkqbc3k3e7k2q

Trapped modes in finite quantum waveguides

A. L. Delitsyn, B. T. Nguyen, D. S. Grebenkov
2012 European Physical Journal B : Condensed Matter Physics  
The Laplace operator in infinite quantum waveguides (e.g., a bent strip or a twisted tube) often has a point-like eigenvalue below the essential spectrum that corresponds to a trapped eigenmode of finite  ...  For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode.  ...  We study the eigenvalue problem for the Laplace operator with Dirichlet boundary con- dition −ΔU = λU, U | ∂D = 0. (1) Solution in rectangular branches Let u i (x, y) denote the restriction of the solution  ... 
doi:10.1140/epjb/e2012-21038-y fatcat:o73te6hcmfg7joecufabglszwy
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