Computable Error Estimates for Ground State Solution of Bose-Einstein Condensates

In this paper, we propose a computable error estimate of the Gross-Pitaevskii equation for ground state solution of Bose-Einstein condensates by general conforming finite element methods on general meshes. Based on the proposed error estimate, asymptotic lower bounds of the smallest eigenvalue and ground state energy can be obtained. Several numerical examples are presented to validate our theoretical results in this paper.

[1]  Paulsamy Muruganandam,et al.  Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation , 2002 .

[2]  B. I. Schnieder,et al.  Numerical approach to the ground and excited states of a Bose-Einstein condensed gas confined in a completely anisotropic trap , 1999 .

[3]  Daniel Peterseim,et al.  Two-Level Discretization Techniques for Ground State Computations of Bose-Einstein Condensates , 2013, SIAM J. Numer. Anal..

[4]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[5]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[6]  Hehu Xie,et al.  A multi-level correction scheme for eigenvalue problems , 2011, Math. Comput..

[7]  Lin Qun,et al.  Stokes Eigenvalue Approximations from Below with Nonconforming Mixed Finite Element Methods , 2010 .

[8]  Elliott H. Lieb,et al.  Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional , 1999, math-ph/9908027.

[9]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[10]  Zachary Dutton,et al.  Near-Resonant Spatial Images of Confined Bose-Einstein Condensates in a 4-Dee Magnetic Bottle , 1998, cond-mat/9804278.

[11]  W. Bao,et al.  Mathematical Models and Numerical Methods for Bose-Einstein Condensation , 2012, 1212.5341.

[12]  Tomaÿs Vejchodsky,et al.  COMPUTING UPPER BOUNDS ON FRIEDRICHS' CONSTANT , 2012 .

[13]  Qiang Du,et al.  Computing the Ground State Solution of Bose-Einstein Condensates by a Normalized Gradient Flow , 2003, SIAM J. Sci. Comput..

[14]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[15]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[16]  Carsten Carstensen,et al.  Guaranteed lower bounds for eigenvalues , 2014, Math. Comput..

[17]  Succi,et al.  Numerical solution of the gross-pitaevskii equation using an explicit finite-difference scheme: An application to trapped bose-einstein condensates , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  Hehu Xie,et al.  A full multigrid method for eigenvalue problems , 2014, J. Comput. Phys..

[19]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[20]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[21]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[22]  C.-S. Chien,et al.  A Two-Grid Discretization Scheme for Semilinear Elliptic Eigenvalue Problems , 2005, SIAM J. Sci. Comput..

[23]  S. Repin.  A Posteriori Estimates for Partial Differential Equations , 2008 .

[24]  Xuefeng Liu.  A framework of verified eigenvalue bounds for self-adjoint differential operators , 2015, Appl. Math. Comput..

[25]  Tomás Vejchodský,et al.  Two-Sided Bounds for Eigenvalues of Differential Operators with Applications to Friedrichs, Poincaré, Trace, and Similar Constants , 2013, SIAM J. Numer. Anal..

[26]  Hehu Xie,et al.  A posterior error estimator and lower bound of a nonconforming finite element method , 2014, J. Comput. Appl. Math..

[27]  Eric Cornell,et al.  Very Cold Indeed: The Nanokelvin Physics of Bose-Einstein Condensation , 1996, Journal of research of the National Institute of Standards and Technology.

[28]  Pekka Neittaanmäki,et al.  Reliable Methods for Computer Simulation: Error Control and a Posteriori Estimates , 2004 .

[29]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .

[30]  Carsten Carstensen,et al.  Guaranteed lower eigenvalue bounds for the biharmonic equation , 2014, Numerische Mathematik.

[31]  Ivan Hlaváček,et al.  Convergence of a finite element method based on the dual variational formulation , 1976 .

[32]  Zhimin Zhang,et al.  Eigenvalue approximation from below using non-conforming finite elements , 2010 .

[33]  Xie He-hu.  The Asymptotic Lower Bounds of Eigenvalue Problems by Nonconforming Finite Element Methods , 2012 .

[34]  Hehu Xie,et al.  Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods , 2013 .

[35]  Weizhu Bao,et al.  Ground-state solution of Bose--Einstein condensate by directly minimizing the energy functional , 2003 .

[36]  R. J. Dodd,et al.  Approximate Solutions of the Nonlinear Schrödinger Equation for Ground and Excited States of Bose-Einstein Condensates , 1996, Journal of research of the National Institute of Standards and Technology.

[37]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[38]  Edwards,et al.  Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[39]  Xuefeng Liu,et al.  Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape , 2012, SIAM J. Numer. Anal..

[40]  C.-S. Chien,et al.  Two-grid discretization schemes for nonlinear Schrödinger equations , 2008 .

[41]  C. Wieman,et al.  Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments , 2002 .

[42]  W. Ketterle.  Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser* , 2002 .

[43]  Succi,et al.  Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[44]  Hehu Xie,et al.  A type of multilevel method for the Steklov eigenvalue problem , 2014 .

[45]  C. Wieman,et al.  Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor , 1995, Science.

[46]  W. Bao,et al.  MATHEMATICAL THEORY AND NUMERICAL METHODS FOR , 2012 .

[47]  Wolfgang Ketterle,et al.  Bose–Einstein condensation of atomic gases , 2002, Nature.

[48]  Tomás Vejchodský,et al.  Complementarity based a posteriori error estimates and their properties , 2012, Math. Comput. Simul..

[49]  J. Pasciak,et al.  New convergence estimates for multigrid algorithms , 1987 .

[50]  Jun Hu,et al.  The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods , 2011, 1112.1145.

[51]  Yvon Maday,et al.  Numerical Analysis of Nonlinear Eigenvalue Problems , 2009, J. Sci. Comput..

[52]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[53]  Hehu Xie,et al.  Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods , 2011, 1109.5977.

[54]  Shi Jin,et al.  Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrödinger Equations in the Semiclassical Regimes , 2003, SIAM J. Sci. Comput..

[55]  Aihui Zhou.  An analysis of finite-dimensional approximations for the ground state solution of Bose?Einstein condensates , 2004 .