scholarly journals Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics

2020 ◽  
Author(s):  
Aihui Zhou ◽  
Bin Ying
Keyword(s):  
Finite Element ◽  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Unique Continuation ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Finite Element Approximations ◽  
Open Set ◽  
Unique Continuation Property ◽  
Nonlinear Eigenvalue

In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that the eigenfunction cannot be a polynomial on any open set, which may be reviewed as a refinement of the classic unique continuation property. Then we apply the non-polynomial behavior of eigenfunction to show that the adaptive finite element approximations are convergent even if the initial mesh is not fine enough. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing result.

Adaptive Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics

2011 ◽  
Vol 3 (4) ◽  
pp. 493-518 ◽  
Author(s):  
Huajie Chen ◽  
Xingao Gong ◽  
Lianhua He ◽  
Aihui Zhou
Keyword(s):  
Finite Element ◽  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Energy Functional ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Finite Element Approximations ◽  
Nonconvex Energy ◽  
Nonlinear Eigenvalue

AbstractIn this paper, we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional. We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.

An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

2016 ◽  
Vol 19 (2) ◽  
pp. 442-472
Author(s):  
Ye Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Singularly Perturbed ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

SYMMETRY REDUCTIONS AND A POSTERIORI FINITE ELEMENT ERROR ESTIMATORS FOR BIFURCATION PROBLEMS

2005 ◽  
Vol 15 (07) ◽  
pp. 2091-2107 ◽  
Author(s):  
C.-S. CHIEN ◽  
B.-W. JENG
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Continuation Method ◽  
Discretization Scheme ◽  
Element Discretization ◽  
Nonlinear Eigenvalue Problems ◽  
Error Estimators ◽  
Symmetry Reductions ◽  
Relaxation Scheme ◽  
Nonlinear Eigenvalue

We discuss efficient continuation algorithms for solving nonlinear eigenvalue problems. First, we exploit the idea of symmetry reductions and discretize the problem on a symmetry cell by the finite element method. Then we incorporate the multigrid V-cycle scheme in the context of continuation method to trace solution branches of the discrete problems, where the preconditioned Lanczos method is used as the relaxation scheme. Next, we apply the symmetry reduction technique to the two-grid finite element discretization scheme [Chien & Jeng, 2005] to solve some nonlinear eigenvalue problems in physical science. The two-grid centered difference discretization scheme described therein was also implemented for comparison. Sample numerical results are reported.

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