Skip to main content
SpringerLink
Log in

An Asymptotics-Based Adaptive Finite Element Method for Kohn–Sham Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In Radovitzky and Ortiz (Comput Methods Appl Mech Eng 172(1–4):203–240, 1999), an error estimation technique for nonlinear PDEs is presented to adaptively generating mesh, based on the reduction of the order of the approximate polynomial. In this paper, following a similar analysis framework, we propose an a posteriori error estimation for Kohn–Sham equation by coarsening mesh. An upper bound for the difference of the total energies on two successively refined meshes is derived by the numerical solutions on two meshes through an asymptotic analysis, which finally generates an a posteriori error estimation. A variety of numerical tests show that such an a posteriori error estimation works very well in our h-adaptive finite element method framework. In addition, to further improve the efficiency, we solve a Poisson equation instead of the Kohn–Sham equation on the coarsened mesh. The effectiveness of this improvement is analyzed and numerically examined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Agmon, S.: Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigen functions of N-Body Schrodinger Operations (MN-29). Princeton University Press, Princeton (2014)

    Book  Google Scholar 

  2. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis, vol. 37. Wiley, Hoboken (2011)

    MATH  Google Scholar 

  3. Akin, J., Singh, M.: Object-oriented Fortran 90 P-adaptive finite element method. Adv. Eng. Softw. 33(7), 461–468 (2002)

    Article  MATH  Google Scholar 

  4. Azorero, J.G., Alonso, I.P.: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 144(2), 441–476 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bänsch, E., Siebert, K.G.: A Posteriori Error Estimation for Nonlinear Problems by Duality Techniques. Albert-Ludwigs-University, Math. Fak. (1995)

  6. Bao, G., Hu, G., Liu, D.: An h-adaptive finite element solver for the calculations of the electronic structures. J. Comput. Phys. 231(14), 4967–4979 (2012)

    Article  MATH  Google Scholar 

  7. Bao, G., Hu, G., Liu, D.: Numerical solution of the Kohn–Sham equation by finite element methods with an adaptive mesh redistribution technique. J. Sci. Comput. 55(2), 372–391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bao, G., Hu, G., Liu, D.: Towards translational invariance of total energy with finite element methods for Kohn–Sham equation. Commun. Comput. Phys. 19(1), 1–23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Belytschko, T., Tabbara, M.: H-Adaptive finite element methods for dynamic problems, with emphasis on localization. Int. J. Numer. Methods Eng. 36(24), 4245–4265 (1993)

    Article  MATH  Google Scholar 

  11. Castro, A., Appel, H., Oliveira, M., Rozzi, C.A., Andrade, X., Lorenzen, F., Marques, M.A., Gross, E., Rubio, A.: Octopus: a tool for the application of time-dependent density functional theory. Phys. Status Solidi (b) 243(11), 2465–2488 (2006)

    Article  Google Scholar 

  12. Chelikowsky, J.R., Troullier, N., Saad, Y.: Finite-difference-pseudopotential method: electronic structure calculations without a basis. Phys. Rev. Lett. 72(8), 1240 (1994)

    Article  Google Scholar 

  13. Chen, H., Dai, X., Gong, X., He, L., Zhou, A.: Adaptive finite element approximations for Kohn–Sham models. Multiscale Models Simul. 12(4), 1828–1869 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, H., Gong, X., He, L., Yang, Z., Zhou, A.: Numerical analysis of finite dimensional approximations of Kohn–Sham models. Adv. Comput. Math. 38(2), 225–256 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  16. Durán, R.G., Padra, C., Rodríguez, R.: A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Model. Methods Appl. Sci. 13(08), 1219–1229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fiolhais, C., Nogueira, F., Marques, M.A.: A Primer in Density Functional Theory, vol. 620. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  18. Gygi, F.: Electronic-structure calculations in adaptive coordinates. Phys. Rev. B 48(16), 11692 (1993)

    Article  Google Scholar 

  19. Gygi, F., Galli, G.: Real-space adaptive-coordinate electronic-structure calculations. Phys. Rev. B 52(4), R2229 (1995)

    Article  Google Scholar 

  20. Heuveline, V., Rannacher, R.: A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15(1), 107–138 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hu, G., Xie, H., Xu, F.: A multilevel correction adaptive finite element method for Kohn–Sham equation. J. Comput. Phys. 355, 436–449 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Johnson III, R.D.: NIST Computational Chemistry Comparison and Benchmark Database. http://cccbdb.nist.gov/ (2016). Accessed 6 Oct 2018

  23. Jiang, X., Zhang, L., Zheng, W.: Adaptive hp-finite element computations for time-harmonic Maxwell’s equations. Commun. Comput. Phys. 13(2), 559–582 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Knyazev, A.V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23(2), 517–541 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Knyazev, A.V., Argentati, M.E., Lashuk, I., Ovtchinnikov, E.E.: Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput. 29(5), 2224–2239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kombe, I.: Hardy, Rellich and uncertainty principle inequalities on Carnot groups. arXiv preprint arXiv:math/0611850 (2006)

  27. Kormann, K.: A time-space adaptive method for the Schrödinger equation. Commun. Comput. Phys. 20(1), 60–85 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kresse, G., Furthmüller, J.: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54(16), 11169 (1996)

    Article  Google Scholar 

  29. Kuang, Y., Hu, G.: An adaptive FEM with ITP approach for steady Schrödinger equation. Int. J. Comput. Math. 95(1), 187–201 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Larson, M.G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38(2), 608–625 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, R., Tang, T., Zhang, P.: Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 170(2), 562–588 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lin, L., Lu, J., Ying, L., Weinan, E.: Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework I: total energy calculation. J. Comput. Phys. 231(4), 2140–2154 (2012)

    Article  MATH  Google Scholar 

  33. Marques, M.A., Oliveira, M.J., Burnus, T.: Libxc: a library of exchange and correlation functionals for density functional theory. Comput. Phys. Commun. 183(10), 2272–2281 (2012)

    Article  Google Scholar 

  34. Motamarri, P., Nowak, M.R., Leiter, K., Knap, J., Gavini, V.: Higher-order adaptive finite-element methods for Kohn–Sham density functional theory. J. Comput. Phys. 253, 308–343 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pask, J., Sterne, P.: Finite element methods in ab initio electronic structure calculations. Model. Simul. Mater. Sci. Eng. 13(3), R71 (2005)

    Article  Google Scholar 

  36. Radovitzky, R., Ortiz, M.: Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comput. Methods Appl. Mech. Eng. 172(1–4), 203–240 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tsuchida, E., Tsukada, M.: Electronic-structure calculations based on the finite-element method. Phys. Rev. B 52(8), 5573 (1995)

    Article  Google Scholar 

  38. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Hoboken (1996)

    MATH  Google Scholar 

  39. Xie, C., Hu, X.: Finite element simulations with adaptively moving mesh for the reaction diffusion system. Numer. Math. Theory Methods Appl. 9(4), 686–704 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, C., Meza, J.C., Lee, B., Wang, L.W.: KSSOLV-a MATLAB toolbox for solving the Kohn–Sham equations. ACM Trans. Math. Softw. (TOMS) 36(2), 10 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yserentant, H.: Regularity and Approximability of Electronic Wave Functions. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  42. Zhang, G., Lin, L., Hu, W., Yang, C., Pask, J.E.: Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework II: force, vibration, and molecular dynamics calculations. J. Comput. Phys. 335, 426–443 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, H., Zegeling, P.A.: A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media. Commun. Comput. Phys. 22(4), 935–964 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by FDCT 029/2016/A1 from Macao SAR, MYRG2017-00189-FST from University of Macau, and National Natural Science Foundation of China (Grant No. 11401608).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Macau, Macau, Macao SAR, China

    Yedan Shen, Yang Kuang & Guanghui Hu

  2. Zhuhai UM Science and Technology Research Institute, Zhuhai, China

    Guanghui Hu

Authors
  1. Yedan Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Kuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guanghui Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guanghui Hu.

Additional information

The authors would like to thank the support from FDCT 029/2016/A1 of Macao S.A.R., MYRG2017-00189-FST from University of Macau, and and National Natural Science Foundation of China (Grant No. 11401608).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shen, Y., Kuang, Y. & Hu, G. An Asymptotics-Based Adaptive Finite Element Method for Kohn–Sham Equation. J Sci Comput 79, 464–492 (2019). https://doi.org/10.1007/s10915-018-0861-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0861-0

Keywords

Search

Navigation