A Two-Dimensional "Flea on the Elephant" Phenomenon and its Numerical Visualization

Localization phenomena (sometimes called “flea on the elephant'') for the operator $L^\varepsilon=-\varepsilon^2 \Delta u + p(\mathbf{x}) u$, $p(\mathbf{x})$ being an asymmetric double well potenti...

[1]  V. B. Andreev.  On the accuracy of grid approximations to nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square , 2006 .

[2]  Otared Kavian,et al.  Variational problems related to self-similar solutions of the heat equation , 1987 .

[3]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[4]  M. Rouleux,et al.  Effet tunnel entre puits degeneres , 1988 .

[5]  Jeffrey S. Ovall,et al.  Filtered Subspace Iteration for Selfadjoint Operators , 2017 .

[6]  The dirichlet problem in zygmund spaces in a square , 2007 .

[7]  Houde Han,et al.  Multiscale tailored finite point method for second order elliptic equations with rough or highly oscillatory coefficients , 2012 .

[8]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[9]  A. A. Dosiyev,et al.  New Properties of 9-Point Finite Difference Solution of the Laplace Equation , 2011 .

[10]  Dae-Yup Song.  Tunneling and energy splitting in an asymmetric double-well potential , 2008, 0803.3113.

[11]  Laurent Gosse.  Aliasing and two-dimensional well-balanced for drift-diffusion equations on square grids , 2020, Math. Comput..

[12]  Zhi-zhong Sun,et al.  A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation , 1995 .

[13]  Brandon Patterson,et al.  Localization of eigenfunctions of a one-dimensional elliptic operator , 2012 .

[14]  M. Selg.  Exactly Solvable Asymmetric Double-Well Potentials , 2000 .

[15]  Discrete Weighted Mean Approximation of a Model Convection-Diffusion Equation , 1982 .

[16]  Alexander Ostermann,et al.  Modification of dimension-splitting methods—overcoming the order reduction due to corner singularities , 2015 .

[17]  J. D. Val,et al.  On mean value solutions for the Helmholtz equation on square grids , 2002 .

[18]  R. B. Kellogg,et al.  Differentiability properties of solutions of the equation -ε 2 δ u + ru = f ( x,y ) in a square , 1990 .

[19]  V. Jelić,et al.  The double-well potential in quantum mechanics: a simple, numerically exact formulation , 2012, 1209.2521.

[20]  Barry Simon,et al.  Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima : asymptotic expansions , 1983 .

[21]  C. M. Elliott,et al.  Numerical Studies of the Cahn-Hilliard Equation for Phase Separation , 1987 .

[22]  D. Hua.  A regularity result for boundary value problems on Lipschitz domains , 1989 .

[23]  I. Kupka,et al.  Singular perturbation for the first eigenfunction and blow-up analysis , 2004, math-ph/0412088.

[24]  Zhongyi Huang,et al.  A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain , 2008, J. Sci. Comput..

[25]  G. Sweers,et al.  An asymptotic eigenvalue problem for a Schrödinger type equation on domains with boundaries , 2016 .

[26]  J. López-Gómez,et al.  Semiclassical analysis of general second order elliptic operators on bounded domains , 2000 .

[27]  Zhongyi Huang,et al.  Asymptotic Analysis and Numerical Method for Singularly Perturbed Eigenvalue Problems , 2018, SIAM J. Sci. Comput..

[28]  G. Jona-Lasinio,et al.  Tunnelling instability via perturbation theory , 1984 .

[29]  Laurent Gosse,et al.  A Truly Two-Dimensional Discretization of Drift-Diffusion Equations on Cartesian Grids , 2018, SIAM J. Numer. Anal..

[30]  A. Azzam.  On differentiability properties of solutions of elliptic differential equations , 1980 .

[31]  R. DeVore,et al.  Besov regularity for elliptic boundary value problems , 1997 .

[32]  David W. Zachmann,et al.  Schaum's outline of theory and problems of partial differential equations , 1986 .

[33]  J. V. Ralston,et al.  Approximate eigenfunctions of the Laplacian , 1977 .

[34]  L. Gosse.  Dirichlet-to-Neumann mappings and finite-differences for anisotropic diffusion , 2017 .

[35]  Bernard Helffer,et al.  Multiple wells in the semi-classical limit I , 1984 .

[36]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[37]  Lloyd N. Trefethen,et al.  Computed eigenmodes of planar regions , 2005 .

[38]  A. Konenkov.  Dirichlet and Neumann problems for Laplace and heat equations in domains with right angles , 2008 .

[39]  N. P. Landsman,et al.  A Flea on Schrödinger’s Cat , 2013 .

[40]  Y. Miyazaki.  Sobolev Trace Theorem and the Dirichlet Problem in the Unit Disk , 2014 .

[41]  L. Gosse.  ℒ-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations , 2018 .

[42]  James Ralston,et al.  On the construction of quasimodes associated with stable periodic orbits , 1979 .

[43]  B. Kellogg.  Some simple boundary value problems with corner singularities and boundary layers , 2006, Comput. Math. Appl..

[44]  L. M. Otto.  As Quantum Wells , 2012 .

[45]  Anita Mayo,et al.  Fast High Order Accurate Solution of Laplace’s Equation on Irregular Regions , 1985 .

[46]  Carmelo Clavero,et al.  A parameter robust numerical method for a two dimensional reaction-diffusion problem , 2005, Math. Comput..

[47]  F. Martinelli,et al.  New approach to the semiclassical limit of quantum mechanics , 1981 .

[48]  Garrett,et al.  Optimal Few-Point Discretizations of Linear Source Problems , 1974 .