Skip to main content
SpringerLink
Log in

Two-scale finite element Green’s function approximations with applications to electrostatic potential computation

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

In this paper, a two-scale finite element approach is proposed and analyzed for approximations of Green’s function in three-dimensions. This approach is based on a two-scale finite element space defined, respectively, on the whole domain with size H and on some subdomain containing singular points with size h (h ≪ H). It is shown that this two-scale discretization approach is very efficient. In particular, the two-scale discretization approach is applied to solve Poisson-Boltzmann equations successfully.

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

Similar content being viewed by others

References

  1. N. Baker, M. Holst, and F. Wang, Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II. Refinement at solvent-accessible surfaces in biomolecular systems, J. Comput. Chem., 2000, 21: 1343–1352.

    Article  Google Scholar 

  2. F. Fogolari, A. Brigo, and H. Molinari, The Poisson-Boltzmann equation for biomolecular electrostatics: A tool for structural biology, J. Mol. Recognit., 2002, 15: 377–392.

    Article  Google Scholar 

  3. M. Holst, The Poisson-Boltzmann Equation, Analysis and Multilever Numerical Solution, Thesis, UCSD, 1994.

  4. M. Holst, N. Baker, and F. Wang, Adaptive multilevel finite element solution of the Posson-Boltzmann equation I. Algorithms and examples, J. Comput. Chem., 2000, 21: 1319–1342.

    Article  Google Scholar 

  5. K. A. Sharp, Incorporating solvent and ion screeing into molecular dynamics using the finite-difference Poisson-Boltzman method, J. Comput. Chem., 1991, 12: 454–468.

    Article  Google Scholar 

  6. Y. Yang and A. Zhou, A finite element recovery approach to Green’s function approximations with applications to electrostatic potential computation, J. Comput. Applied Math., 2009, 225: 202–212.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. E. Davis and J. A. McCammon, Electrostatics in biomolecular structure and dynamics, Chem. Rev., 1990, 90: 509–521.

    Article  Google Scholar 

  8. K. A. Sharp and B. Honig, Calculating total electrostatic energies with the non-linear Poisson-Boltzmann equation, J. Phys. Chem., 1990, 94: 7684–7692.

    Article  Google Scholar 

  9. K. A. Sharp and B. Honig, Electrostatic interactions in macomolecules: Theory and applications, Annu. Rev. Biophys. Biophys. Chem., 1990, 19: 301–332.

    Article  Google Scholar 

  10. T. Simonson, Macomolecular electrostatics: Continuum models and their growing pains, Curr. Opin. Struct. Biol., 2001, 11: 243–252.

    Article  Google Scholar 

  11. A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comput., 1977, 31: 414–442.

    Article  MATH  MathSciNet  Google Scholar 

  12. Z. Cai and S. F. McCormick, On the accuracy of the finite volume method for diffusion equations on composite grids, SIAM J. Numer. Anal., 1990, 27: 636–655.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. G. Ciarlet and J. L. Lions, Handbook of Numerical Analysis, Vol. II, Finite Element Method, Part I, North-Holland, Amsterdam, 1991.

    Google Scholar 

  14. R. E. Ewing, R. D. Lazarov, and P. S. Vassilevski, Locla refinement technique for elliptic problem on cell-centered grids, I: Error analysis, Math. Comput., 1991, 56: 437–462.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. S. Vassilevski, S. I. Petrova, and R. D. Lazarov, Finite difference scheme on triangular celcentered grids with local refinement, SIAM J. Sci. Stat. Comput., 1992, 13: 1287–1313.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Zhou, C. Liem, T. Shih, and L. Lü, Error analysis on bi-parameter finite element, Comput. Methods Appl. Mech. Engrg., 1998, 158: 329–339.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comput., 1982, 38: 437–445.

    Article  MATH  MathSciNet  Google Scholar 

  18. Q. Zhu and Q. Lin, Superconvergence Theory of Finite Element Methods (in Chinese), Hunan Science Press, Changsha, 1989.

    Google Scholar 

  19. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

    MATH  Google Scholar 

  20. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.

    MATH  Google Scholar 

  21. J. Nitsche, L ∞ -Error Analysis for Finite Elements, in: Whiteman, Jr., ed., The Mathematics of Finite Elements and Applications, Academic Press, New York, 1979, 173–186.

    Google Scholar 

  22. J. Xu and A. Zhou, Local and parallel finite element algprithms based on two-grid discretizations, Math. Comput., 2000, 69: 881–909.

    MATH  MathSciNet  Google Scholar 

  23. C. M. Cortis and R. A. Friesner, Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes, J. Comput. Chem., 1997, 18: 1591–1608.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computational Science and Mathematics, Guilin University of Electronic Technology, Guilin, 541004, China

    Ying Yang

  2. The State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Aihui Zhou

Authors
  1. Ying Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aihui Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was partially supported by the National Science Foundation of China under Grant Nos. 10425105 and 10871198, and the National Basic Research Program under Grant No. 2005CB321704.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, Y., Zhou, A. Two-scale finite element Green’s function approximations with applications to electrostatic potential computation. J Syst Sci Complex 23, 177–193 (2010). https://doi.org/10.1007/s11424-010-9274-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-010-9274-3

Key words

Search

Navigation