A posteriori error estimator for the eigenvalue problem associated to the Schrödinger operator with magnetic field

Summary.The ground state for the Neumann realization of the Schrödinger operator for constant and sufficiently large magnetic field presents a localization in the boundary of the domain and particularly in the corners where the angle is minimum. As the solution decreases exponentially fast away of the corner, it is rather difficult to catch it numerically. A natural idea is to try using a mesh refinement method coupled to a posteriori error estimates. The purpose of this paper is to provide such an estimator adapted to the problem.

[1]  Xingbin Pan.  Upper critical field for superconductors with edges and corners , 2002 .

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

[3]  I Babuska,et al.  Feedback, Adaptivity and A-Posterior Estimates in Finite Elements: Aims, Theory and Experience. , 1984 .

[4]  The boundary integral method for magnetic billiards. , 1999, chao-dyn/9912022.

[5]  Mats G. Larson,et al.  A Posteriori and a Priori Error Analysis for Finite Element Approximations of Self-Adjoint Elliptic Eigenvalue Problems , 2000, SIAM J. Numer. Anal..

[6]  Y. Maday,et al.  Numerical analysis of the adiabatic variable method for the approximation of the nuclear Hamiltonian , 2001 .

[7]  Superconductivity in a wedge: analytical variational results☆ , 1999 .

[8]  Virginie Bonnaillie.  On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners , 2003 .

[9]  Christine Bernardi,et al.  Indicateurs d'erreur pour l'équation de la chaleur , 2000 .

[10]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[11]  B. Helffer,et al.  Magnetic Bottles in Connection with Superconductivity , 2001 .

[12]  Rolf Rannacher,et al.  A posteriori error control for finite element approximations of elliptic eigenvalue problems , 2001, Adv. Comput. Math..

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

[14]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[15]  Xing-Bin Pan,et al.  Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity , 1999 .

[16]  Analyse numérique de la supraconductivité , 2003 .

[17]  H. Jadallah.  The onset of superconductivity in a domain with a corner , 2001 .

[18]  Anthony T. Patera,et al.  A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem* , 1999 .

[19]  Virginie Bonnaillie.  On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners , 2005 .

[20]  Andrew J. Bernoff,et al.  Onset of superconductivity in decreasing fields for general domains , 1998 .

[21]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .