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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
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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

Information

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

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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