Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-04-30T19:12:55.822Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 June 2015

Huajie Chen ,
Xingao Gong ,
Lianhua He  and
Aihui Zhou
Huajie Chen*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/ Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Xingao Gong*
Affiliation:
Department of Physics, Fudan University, Shanghai 200433, China
Lianhua He*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/ Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Aihui Zhou*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/ Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Email: hjchen@lsec.cc.ac.cn
Email: xggong@fudan.edu.cn
Email: helh@lsec.cc.ac.cn
Corresponding author. Email: azhou@lsec.cc.ac.cn
Get access

Abstract

In 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.

Keywords

35Q5565N1565N2565N3081Q05Adaptive finite elementconvergencemicro-structurenonlinear eigenvalue
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 4 , August 2011 , pp. 493 - 518
Copyright
Copyright © Global-Science Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
[2] Arnold, D., Mukherjee, A. and Pouly, L., Locally adapted tetrahedral meshes using bisection, SIAM J. Sci. Comput., 22 (2000), pp. 431–448.Google Scholar
[3] Babuska, I. and Rheinboldt, W. C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736–754.CrossRefGoogle Scholar
[4] Bao, W. and Du, Q., Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697.CrossRefGoogle Scholar
[5] Beck, T. L., Real-space mesh techniques in density-function theory, Rev. Mod. Phys., 72 (2000), pp. 1041–1080.Google Scholar
[6] Binev, P., Dahmen, W. and Devore, R., Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), pp. 219–268.CrossRefGoogle Scholar
[7] Blanc, X. and Cancès, E., Nonlinear instability of density-independent orbital-free kinetic energy functionals, Chem. Phys., 122 (2005), pp. 214106–214120.Google ScholarPubMed
[8] Cascon, J. M., Kreuzer, C., Nochetto, R. H. and Siebert, K. C., Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524–2550.Google Scholar
[9] Cancés, E., Chakir, R. and Maday, Y., Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput., 45 (2010), pp. 90–117.Google Scholar
[10] Cancés, E., Chakir, R. and Maday, Y., Numerical analysis of the planewave discretization of orbital-free and Kohn-Sham models, arXiv: 1003.1612, 2010.Google Scholar
[11] Carstensen, C., Convergence of an adaptive FEM for a class of degenerate convex minimization problems, IMA J. Numer. Anal., 3 (2008), pp. 423–439.Google Scholar
[12] Carstensen, C. and Gedicke, J., An adaptive finite element eigenvalue solver of quasioptimal computational complexity, Preprint 662, DFG Research Center MATHEON, , 2009.Google Scholar
[13] Chen, H., Gong, X. and Zhou, A., Approximations of a nonlinear eigenvalue problem and applications to a density functional model, Math. Meth. Appl. Sci., 33 (2010), pp. 1723–1742.Google Scholar
[14] Chen, H., He, L. and Zhou, A., Finite element approximations of nonlinear eigenvalue problems in quantum physics, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 1846–1865.Google Scholar
[15] Chen, H. and Zhou, A., Orbital-free density functional theory for molecular structure calculations, Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 1–28.Google Scholar
[16] Chen, L., Holst, M. J. and Xu, J., The finite element approximation of the nonlinear Poisson-Boltzmann equation, SIAM J. Numer. Anal., 45 (2007), pp. 2298–2320.CrossRefGoogle Scholar
[17] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, 1978.Google Scholar
[18] Dai, X., Xu, J. and Zhou, A., Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math., 110 (2008), pp. 313–355.Google Scholar
[19] Dalfovo, F., Giorgini, S., Pitaevskii, L. P. and Stringari, S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), pp. 463–512.Google Scholar
[20] Dörfler, W., A robust adaptive strategy for the non-linear Poisson’s equation, Comput., 55 (1995), pp. 289–304.CrossRefGoogle Scholar
[21] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.Google Scholar
[22] Garau, E. M. and Morin, P., Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA J. Numer. Anal., DOI: 10.1093/imanum/drp055.Google Scholar
[23] Garau, E. M., Morin, P. and Zuppa, C., Convergence of adaptive finite element methods for eigenvalue problems, M3AS., 19 (2009), pp. 721–747.Google Scholar
[24] Giani, S. and Graham, I. G., A convergent adaptive method for elliptic eigenvalue problems, SIAM J. Numer. Anal., 47 (2009), pp. 1067–1091.CrossRefGoogle Scholar
[25] Goodwin, L., Needs, R. J. and Heine, V., A pseudopotential total energy study of impurity-promoted intergranular embrettlement, J. Phys. Condens. Mat., 2 (1990), pp. 351–365.Google Scholar
[26] Harrison, R., Moroz, I. and Tod, K. P., A numerical study of the Schrödinger-Newton equations, Nonlinear., 16 (2003), pp. 101–122.Google Scholar
[27] He, L. and Zhou, A., Convergence and optimal complexity of adaptive finite element methods, Int. J. Numer. Anal. Model., to appear.Google Scholar
[28] Holst, M., Tsogtgerel, G. and Zhu, Y., Local convergence of adaptive methods for nonlinear partial differential equations, arXiv: 1001.1382, 2010.Google Scholar
[29] Langwallner, B., Ortner, C. and Suli, E., Existence and convergence results for the Galerkin approximation of an electronic density functional, M3AS, 20 (2010), pp. 2237–2265.Google Scholar
[30] Le Bris, C., Handbook of Numerical Analysis, Vol. X. Special Issue: Computational Chemistry, North-Holland, Amsterdam, 2003.Google Scholar
[31] Lieb, E. H., Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53 (1981), pp. 603–641.Google Scholar
[32] Mekchay, K. and Nochetto, R. H., Convergence of adaptive finite element methods for general second order linear elliplic PDEs, SIAM J. Numer. Anal., 43 (2005), pp. 1803–1827.CrossRefGoogle Scholar
[33] Morin, P., Nochetto, R. H. and Siebert, K., Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466–488.CrossRefGoogle Scholar
[34] Morin, P., Nochetto, R. H. and Siebert, K., Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), pp. 631–658.Google Scholar
[35] Morin, P., Siebert, K. G. and Veeser, A., A basic convergence result for conforming adaptive finite elements, Math. Models. Methods. Appl. Sci., 18 (2008), pp. 707–737.Google Scholar
[36] Penrose, R., On gravity’s role in quantum state reduction, Gen. Rel. Grav., 28 (1996), pp. 581–600.Google Scholar
[37] Siebert, K. G., A convergence proof for adaptive finite elements without lower bound, IMA J. Numer. Anal., DOI: 10.1093/imanum/drq001.Google Scholar
[38] Stevenson, R., Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245–269.Google Scholar
[39] Veeser, A., Convergent adaptive finite elements for the nonlinear Laplacian, Numer. Math., 92 (2002), pp. 743–770.Google Scholar
[40] Verfürth, R., A Review of A Posteriore Error Estimation and Adaptive Mesh Refinement Techniques, Teubner, B. G., 1996.Google Scholar
[41] Wang, Y. A. and Carter, E. A., Orbital-free kinetic-energy density functional theory, in: Theoretical Methods in Condensed Phase Chemistry (Schwartz, S.D., ed.), Kluwer, Dordrecht, 2000, pp. 117–184.Google Scholar
[42] Zhou, A., An analysis of finite-dimensional approximations for the ground state solution of Bose-Einstein condensates, Nonlinear., 17 (2004), pp. 541–550.Google Scholar
[43] Zhou, A., Finite dimensional approximations for the electronic ground state solution of a molecular system, Math. Meth. Appl. Sci., 30 (2007), pp. 429–447.Google Scholar