Adaptive finite element approximations for Kohn-Sham models

Multiscale Modeling and Simulation

Volumn 12, Issue 4, 2014, Pages 1828-1869

  Chen, Huajie ^a   Dai, Xiaoying ^b   Gong, Xingao ^c   He, Lianhua ^b   Zhou, Aihui ^b  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^b LSEC   (China)

^c FUDAN UNIVERSITY   (China)

Author keywords

Adaptive finite element approximation; Complexity; Convergence; Kohn Sham density functional theory; Nonlinear eigenvalue problem

Indexed keywords

APPROXIMATION ALGORITHMS; DENSITY FUNCTIONAL THEORY; EIGENVALUES AND EIGENFUNCTIONS; ELECTRONIC STRUCTURE; ERROR ANALYSIS; GROUND STATE;

ADAPTIVE FINITE ELEMENT; COMPLEXITY; CONVERGENCE; KOHN SHAMS; NONLINEAR EIGENVALUE PROBLEM;

FINITE ELEMENT METHOD;

EID: 84920034827     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/130916096     Document Type: Article

References (65)

1. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
2. S. Agmon, Lectures on the Exponential Decay of Solutions of Second-Order Elliptic Operators, Princeton University Press, Princeton, NJ, 1981.
3. C.O. Almbladh and U. von Barth, Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues, Phys. Rev. B, 31 (1985), pp. 3231-3244.
4. A. Anantharaman and E. Cancès, Existence of minimizers for Kohn-Sham models in quantum chemistry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), pp. 2425-2455.
5. I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of onedimensional boundary value problems, Numer. Math., 44 (1984), pp. 75-102.
6. G. Bao, G. Hu, and D. Liu, Numerical solution of the Kohn-Sham equation by finite element methods with an adaptive mesh redistribution technique, J. Sci. Comput., 55 (2013), pp. 372-391.
7. T.L. Beck, Real-space mesh techniques in density-function theory, Rev. Mod. Phys., 72 (2000), pp. 1041-1080.
8. R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1-102.
9. P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), pp. 219-268.
10. E.J. Bylaska, M. Holst, and J.H. Weare, Adaptive finite element method for solving the exact Kohn-Sham equation of density functional theory, J. Chem. Theory Comput., 5 (2009), pp. 937-948.
11. E. Cancès, R. Chakir, and Y. Maday, Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models, M2AN Math. Model. Numer. Anal., 46 (2012), pp. 341-388.
12. J.M. Cascon, C. Kreuzer, R.H. Nochetto, and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524-2550.
13. H. Chen, X. Gong, L. He, Z. Yang, and A. Zhou, Numerical analysis of finite dimensional approximations of Kohn-Sham equations, Adv. Comput. Math., 38 (2013), pp. 225-256.
14. H. Chen, X. Gong, L. He, and A. Zhou, Adaptive finite element approximations for a class of nonlinear eigenvalue problems in quantum physics, Adv. Appl. Math. Mech., 3 (2011), pp. 493-518.
15. H. Chen, L. He, and A. Zhou, Finite element approximations of nonlinear eigenvalue problems in quantum physics, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 1846-1865.
16. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
17. X. Dai, Adaptive and Localization Based Finite Element Discretizations for the First- Principles Electronic Structure Calculations, IMA J. Appl. Math., to appear.
18. X. Dai, X. Gong, Z. Yang, D, Zhang, and A. Zhou, Finite volume discretizations for eigenvalue problems with applications to electronic structure calculations, Multiscale Model. Simul., 9 (2011), pp. 208-240.
19. X. Dai, L. He, and A. Zhou, Convergence Rate and Quasi-Optimal Complexity of Adaptive Finite Element Computations for Multiple Eigenvalues, preprint, arXiv:1210.1846, 2012.
20. X. Dai, L. Shen, and A. Zhou, A local computational scheme for higher order finite element eigenvalue approximations, Internat. J. Numer. Anal. Model., 5 (2008), pp. 570-589.
21. X. Dai, J. Xu, and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math., 110 (2008), pp. 313-355.
22. X. Dai and A. Zhou, Three-scale finite element discretizations for quantum eigenvalue problems, SIAM J. Numer. Anal., 46 (2008), pp. 295-324.
23. W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124.
24. R.G. Durán, C. Padra, and R. Rodríguez, A posteriori error estimates for the finite element approximation of eigenvalue problems, Math. Models Methods Appl. Sci., 13 (2003), pp. 1219-1229.
25. A. Edelman, T.A. Arias, and S.T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
26. J. Fang, X. Gao, and A. Zhou, A Kohn-Sham equation solver based on hexahedral finite elements, J. Comput. Phys., 231 (2012), pp. 3166-3180.
27. J.L. Fattebert, R.D. Hornung, and A.M. Wissink, Finite element approach for density functional theory calculations on locally refined meshes, J. Comput. Phys., 223 (2007), pp. 759-773.
28. E.M. Garau and P. Morin, Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA J. Numer. Anal., 31 (2011), pp. 914-946.
29. E.M. Garau, P. Morin, and C. Zuppa, Convergence of adaptive finite element methods for eigenvalue problems, Math. Models Methods Appl. Sci., 19 (2009), pp. 721-747.
30. S. Giani and I.G. Graham, A convergent adaptive method for elliptic eigenvalue problems, SIAM J. Numer. Anal., 47 (2009), pp. 1067-1091.
31. X. Gong, L. Shen, D. Zhang, and A. Zhou, Finite element approximations for Schrödinger equations with applications to electronic structure computations, J. Comput. Math., 23 (2008), pp. 310-327.
32. L. He and A. Zhou, Convergence and complexity of adaptive finite element methods for elliptic partial differential equations, Internat. J. Numer. Anal. Model., 8 (2011), pp. 615-640.
33. V. Heuveline and R. Rannacher, A posteriori error control for finite element approximations of ellipic eigenvalue problems, Adv. Comput. Math., 15 (2001), pp. 107-138.
34. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. B (2), 136 (1964), pp. 864-871.
35. W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. A (2), 140 (1965), pp. 1133-1138.
36. M.G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal., 38 (2000), pp. 608-625.
37. D. Mao, L. Shen, and A. Zhou, Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates, Adv. Comput. Math., 25 (2006), pp. 135-160.
38. R.M. Martin, Electronic Structure: Basic Theory and Practical Method, Cambridge University Press, Cambridge, 2004.
39. K. Mekchay and R.H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliplic PDEs, SIAM J. Numer. Anal., 43 (2005), pp. 1803-1827.
40. P. Morin, R.H. Nochetto, and K.G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), pp. 631-658.
41. P. Morin, K.G. Siebert, and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 18 (2008), pp. 707-737.
42. P. Motamarri, M.R. Nowak, K. Leiter, J. Knap, and V. Gavini, Higher-order adaptive finite-element methods for Kohn-Sham density functional theory, J. Comput. Phys., 253 (2013), pp. 308-343.
43. J.E. Pask, B.M. Klein, P.A. Sterne, and C.Y. Fong, Finite-element methods in electronicstructure theory, Comput. Phys. Comm., 135 (2001), pp. 1-34.
44. J. Pask and P. Sterne, Finite element methods in ab initio electronic structure calculations, Modelling Simul. Mater. Sci. Eng., 13 (2005), pp. R71-R96.
45. P. Pulay, Convergence acceleration of iterative sequences. The case of scf iteration, Chem. Phys. Lett., 73 (1980), pp. 393-398.
46. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2, Academic Press, New York, 1975.
47. R. Schneider, T. Rohwedder, A. Neelov, and J. Blauert, Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure, J. Comput. Math., 27 (2009), pp. 360-387.
48. L. Shen, Parallel Adaptive Finite Element Algorithms for Electronic Structure Computing Based on Density Functional Theory, Ph.D. thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 2005.
49. L. Shen and A. Zhou, A defect correction scheme for finite element eigenvalues with applications to quantum chemistry, SIAM J. Sci. Comput., 28 (2006), pp. 321-338.
50. R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245-269.
51. R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comput., 77 (2008), pp. 227-241.
52. P. Suryanarayana, V. Gavini, T. Blesgen, K. Bhattacharya, and M. Ortiz, Non-periodic finite-element formulation of Kohn-Sham density functional theory, J. Mech. Phys. Solids, 58 (2010), pp. 256-280.
53. T. Torsti, T. Eirola, J. Enkovaara, T. Hakala, P. Havu, V. Havu, T. Hoynalanmaa, J. Ignatius, M. Lyly, I. Makkonen, T.T. Rantala, J. Ruokolainen, K. Ruotsalainen, E. Rasanen, H. Saarikoski, and M.J. Puska, Three real-space discretization techniques in electronic structure calculations, Phys. Status Solidi B, 243 (2006), pp. 1016-1053.
54. E. Tsuchida and M. Tsukada, Electronic-structure calculations based on the finite-element method, Phys. Rev. B (3), 52 (1995), pp. 5573-5578.
55. E. Tsuchida and M. Tsukada, Adaptive finite-element method for electronic-structure calculations, Phys. Rev. B (3), 54 (1996), pp. 7602-7605.
56. E. Tsuchida and M. Tsukada, Large-scale electronic-structure calculations based on the adaptive finite-element method, J. Phys. Soc. Japan, 67 (1998), pp. 3844-3858.
57. R. Verfürth, A Review of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.
58. Z. Yang, Finite Volume Discretization Based First-Principles Electronic Structure Calculations, Ph.D. thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 2011.
59. H. Yserentant, Regularity and Approximability of Electronic Wave Functions, Lecture Notes in Math. 2000, Springer-Verlag, Berlin, 2010.
60. D. Zhang, Applications of Finite Element Methods in Electronic Structure Calculations, Ph.D. thesis, Fudan University, Shangahi, 2007.
61. D. Zhang, L. Shen, A. Zhou, and X. Gong, Finite element method for solving Kohn-Sham equations based on self-adaptive terahedral mesh, Phy. Lett. A, 372 (2008), pp. 5071-5076.
62. D. Zhang, A. Zhou, and X. Gong, Parallel mesh refinement of higher order finite elements for electronic structure calculations, Comm. Comput. Phys., 4 (2008), pp. 1086-1105.
63. A. Zhou, An analysis for finite dimensional approximations for the ground state solution of Bose-Einstein condensates, Nonlinearity, 17 (2004), pp. 541-550.
64. A. Zhou, Finite dimensional approximations for the electronic ground state solution of a molecular system, Math. Methods Appl. Sci., 30 (2007), pp. 429-447.
65. PHG, http://lsec.cc.ac.cn/phg/.