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

  • Ying Yang (a1) and Benzhuo Lu (a2)


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.


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[1]Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.
[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. 209223.
[3]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
[4]Babuška, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), pp. 207213.
[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. 631648.
[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. 8093.
[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. 22982320.
[8]Chen, Z. and Zhen, J., Finite element methods and their convergence for elliptic and parabolic interface problem, Numer. Math., 79 (1998), pp. 175202.
[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.
[10]Eisenberg, R. and Chen, D. P., Poisson-Nernst-Planck (PNP) theory of an open ionic channel, Biophys. J., 64(2) (1993), A22–A22.
[11]Elliott, C.M. and Larsson, S., A finite element model for the time-dependent joule heating problem, Math. Comput., 64 (1995), pp. 14331453.
[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. 565590.
[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. 642656.
[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. 69796994.
[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.
[16]Nernst, W., Die elektromotorische wirksamkeit der ionen, Z. Physik. Chem., 4 (1889), pp. 4129.
[17]Planck, M., Uber die erregung von electricität und wärme in electrolyten, Ann. Phys. Chem., (1890), pp. 39161.
[18]Schatz, A. H. and Wahlbin, L. B., Interior maximum norm estimates for finite element methods, Math. Comput., 31 (1977), pp. 414442.
[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. 15581566.
[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. 20172029.
[21]Xu, J. and Zhou, A., Local and parallel finite element algprithms based on two-grid discretizations, Math. Comput., 69 (2000), pp. 881909.
[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. 202212.
[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. 177193.
[24]Yue, X., Numerical analysis of nonstationary thermistor problem, J. Comput. Math., 12 (1994), pp. 213223.
[25]Zhou, A., Liem, C., Shih, T. and , L., Error analysis on bi-parameter finite element, Comput. Methods Appl. Mech. Eng., 158 (1998), pp. 329339.
[26]Zhu, Q. and Lin, Q., Superconvergence Theory of Finite Element Methods, Hunan Science Press, Changsha, 1989 (in Chinese).
[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. 270275.
[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. 13441351.



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