Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T05:29:45.095Z Has data issue: false hasContentIssue false
  • English
  • Français

An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations

Published online by Cambridge University Press:  03 June 2015

Ying Yang  and
Benzhuo Lu
Ying Yang*
Affiliation:
Department of Computational Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, Guangxi, China
Benzhuo Lu*
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, the National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Corresponding author. Email: yangying@lsec.cc.ac.cn
Email: bzlu@lsec.cc.ac.cn
Get access

Abstract

Poisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.

Keywords

65N3092C40Poisson-Nernst-Planck equationsfinite element methoderror bounds
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 1 , February 2013 , pp. 113 - 130
Copyright
Copyright © Global-Science Press 2013

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]Allegretto, W., Lin, Y. and Zhou, A., A box scheme for coupled systems resulting from microsensor thermistor problems, Dynam. Contin. Discrete Impuls. Systems, 5 (1999), pp. 209–223.Google Scholar
[3]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
[4]Babuška, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), pp. 207–213.Google Scholar
[5]Barcilon, V., Chen, D. P., Eisenberg, R. S. and Jerome, J. W., Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study, SIAM J. Appl. Math., 3 (1997), pp. 631–648.Google Scholar
[6]Cardenas, A. E., Coalson, R. D. and Kurnikova, M. G., Three-dimensional Poisson-Nernst-Planck theory studies: influence of membrane electrostatics on gramicidin a channel conductance, Biophys. J., 79(1) (2000), pp. 80–93.Google Scholar
[7]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.Google Scholar
[8]Chen, Z. and Zhen, J., Finite element methods and their convergence for elliptic and parabolic interface problem, Numer. Math., 79 (1998), pp. 175–202.CrossRefGoogle Scholar
[9]Bolintineanu, D. S., Sayyed-Ahmad, A., Davis, H. T. and Kaznessis, Y. N., Poisson-Nernst-Planck models of nonequilibrium ion electrodiffusion through a protegrin transmembrane pore, PLoS Comput. Biol., 5 (2009), e1000277.Google Scholar
[10]Eisenberg, R. and Chen, D. P., Poisson-Nernst-Planck (PNP) theory of an open ionic channel, Biophys. J., 64(2) (1993), A22–A22.Google Scholar
[11]Elliott, C.M. and Larsson, S., A finite element model for the time-dependent joule heating problem, Math. Comput., 64 (1995), pp. 1433–1453.Google Scholar
[12]Jerome, J. W., Consistency of semiconductor modeling: an existence/ stability analysis for the stationary van Boosbroeck system, SIAM J. Appl. Math., 45 (1985), pp. 565–590.Google Scholar
[13]Kurnikova, M. G., Coalson, R. D., Graf, P. and Nitzan, A., A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transpoort through the gramicidin a channel, Biophys. J., 76(2) (1999), pp. 642–656.Google Scholar
[14]Lu, B. Z., Holst, M. J., McCammo, J. A. and Zhou, Y. C., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: finite element solutions, J. Comput. Phys., 229 (2010), pp. 6979–6994.Google Scholar
[15]Lu, B. Z., Zhou, Y. C., Huber, G. A., Bond, S. D., Holst, M. J. and McCammon, J. A., Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution, J. Chem. Phys., 127 (2007), 135102.Google Scholar
[16]Nernst, W., Die elektromotorische wirksamkeit der ionen, Z. Physik. Chem., 4 (1889), pp. 4–129.Google Scholar
[17]Planck, M., Uber die erregung von electricität und wärme in electrolyten, Ann. Phys. Chem., (1890), pp. 39–161.Google Scholar
[18]Schatz, A. H. and Wahlbin, L. B., Interior maximum norm estimates for finite element methods, Math. Comput., 31 (1977), pp. 414–442.CrossRefGoogle Scholar
[19]Song, Y. H., Zhang, Y. J., Bajaj, C. L. and Baker, N. A., Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis, Biophys. J., 3 (2004), pp. 1558–1566.Google Scholar
[20]Song, Y. H., Zhang, Y. J., Shen, T. Y., Bajaj, C. L., McCammon, J. A. and Baker, N. A., Finite element solution of the steady-state Smoluchowski equation for rate constant calculations, Biophys. J., 4 (2004), pp. 2017–2029.Google Scholar
[21]Xu, J. and Zhou, A., Local and parallel finite element algprithms based on two-grid discretizations, Math. Comput., 69 (2000), pp. 881–909.CrossRefGoogle Scholar
[22]Yang, Y. and Zhou, A., A finite element recovery approach to Green’s function approximations with applications to electrostatic potential computation, J. Comput. Appl. Math., 225 (2009), pp. 202–212.Google Scholar
[23]Yang, Y. and Zhou, A., Two-scale finite element Green’s function approximations with applications to electrostatic potential computation, J. Syst. Sci. Complex., 23 (2010), pp. 177–193.Google Scholar
[24]Yue, X., Numerical analysis of nonstationary thermistor problem, J. Comput. Math., 12 (1994), pp. 213–223.Google Scholar
[25]Zhou, A., Liem, C., Shih, T. and Lü, L., Error analysis on bi-parameter finite element, Comput. Methods Appl. Mech. Eng., 158 (1998), pp. 329–339.Google Scholar
[26]Zhu, Q. and Lin, Q., Superconvergence Theory of Finite Element Methods, Hunan Science Press, Changsha, 1989 (in Chinese).Google Scholar
[27]Zhou, Y. C., Lu, B. Z., Huber, G. A., Holst, M. J. and McCammon, J. A., Continuum simulations of acetylcholine consumption by acetylcholinesterase-a Poisson-Nernst-Planck approach, J. Phys. Chem. B, 112(2) (2008), pp. 270–275.Google Scholar
[28]Zhou, Z., Payne, P., Vasquez, M., Kuhn, N. and Levitt, M., Finite-difference solution of the Poisson-Boltzmann equation: complete elimination of self-energy, J. Comput. Chem., 11 (1996), pp. 1344–1351.Google Scholar