Abstract
In this paper, a linearized structure-preserving Galerkin finite element method is investigated for Poisson-Nernst-Planck (PNP) equations. By making full use of the high accuracy estimation of the bilinear element, the mean value technique and rigorously dealing with the coupled nonlinear term, not only the unconditionally optimal error estimate in \(L^2\)-norm but also the unconditionally superclose error estimate in \(H^1\)-norm for the related variables are obtained. Then, the unconditionally global superconvergence error estimate in \(H^1\)-norm is derived by a simple and efficient interpolation post-processing approach, without any coupling restriction condition between the time step size and the space mesh width. Finally, numerical results are provided to confirm the theoretical findings. The numerical scheme preserves the global mass conservation and the electric energy decay, and this work has a great improvement of the error estimate results given in Prohl and Schmuck (Numer. Math. 111, 591–630 2009) and Gao and He (J. Sci. Comput. 72, 1269–1289 2017).
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References
Wei, G., Zheng, Q., Chen, Z., Xia, K.: Variational multiscale models for charge transport. SIAM Rev. 54, 699–754 (2012)
Xu, S., Chen, M., Majd, S., Yue, X., Liu, C.: Modeling and simulating asymmetrical conductance changes in Gramicidin pores. Mol. Based Math. Biol. 2, 509–523 (2014)
Brezzi, F., Marini, L., Pietra, P.: Numerical simulation of semiconductor devices. Comput. Methods Appl. Mech. Eng. 75, 493–514 (1989)
Gajewski, H., Groger, K.: On the basic equations for carrier transport in semiconductors. J. Math. Anal. Appl. 113, 12–35 (1986)
Singer, A., Norbury, J.: A Poisson-Nernst-Planck model for biological ion channels-an asymptotic analysis in a three-dimensional narrow funnel. SIAM J. Appl. Math. 70(3), 949–968 (2009)
Flavell, A., Machen, M., Eisenberg, R., Kabre, J., Liu, C., Li, X.: A conservative finite difference scheme for Poisson-Nernst-Planck equations. J. Comput. Electron. 15, 1–15 (2013)
Flavell, A., Kabre, J., Li, X.: An energy-preserving discretization for the Poisson-Nernst-Planck equations. J. Comput. Electron. 16, 431–441 (2017)
He, D., Pan, K.: An energy preserving finite difference scheme for the Poisson-Nernst-Planck system. Appl. Math. Comput. 287–288, 214–223 (2016)
Liu, H., Wang, Z.: A free energy satisfying finite difference method for Poisson-Nernst-Planck equations. J. Comput. Phys. 268, 363–376 (2014)
Mirzadeh, M., Gibou, F.: A conservative discretization of the Poisson-Nernst-Planck equations on adaptive Cartesian grids. J. Comput. Phys. 274, 633–653 (2014)
Hu, J.W., Huang, X.D.: A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson-Nernst-Planck equations. Numer. Math. 145, 77–115 (2020)
He, D.D., Pan, K.J., Yue, X.Q.: A positivity preserving and free energy dissipative difference scheme for the Poisson-Nernst-Planck system. J. Sci. Comput. 81, 436–458 (2019)
Sun, Y.Z., Sun, P.T., Zheng, B., Lin, G.: Error analysis of finite element method for Poisson-Nernst-Planck equations. J. Comput. Appl. Math. 301, 28–43 (2016)
Prohl, A., Schmuck, M.: Convergent discretizations for the Nernst-Planck-Poisson system. Numer. Math. 111, 591–630 (2009)
Gao, H.D., He, D.D.: Linearized conservative finite element methods for the Nernst-Planck-Poisson equations. J. Sci. Comput. 72, 1269–1289 (2017)
Gao, H.D., Sun, P.T.: A linearized local conservative mixed finite element method for Poisson-Nernst-Planck equations. J. Sci. Comput. 77, 793–817 (2018)
Shi, D.Y., Yang, H.J.: Superconvergence analysis of finite element method for Poisson-Nernst-Planck equations. Numer. Methods Partial Differ. Equ. 35, 1206–1223 (2019)
Shi, X.Y., Lu, L.Z.: Superconvergent estimate of a Galerkin finite element method for nonlinear Poisson-Nernst-Planck equations. Appl. Math. Lett. 104, 106253 (2020)
Yang, Y., Tang, M., Liu, C., Lu, B.Z., Zhong, L.Q.: Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem, Adv. Comput. Math. 46,(78) (2020)
Shi, X.Y., Lu, L.Z.: Nonconforming finite element method for coupled Poisson-Nernst-Planck equations. Numer. Methods Partial Differ. Equ. 37(3), 2714–2729 (2021)
He, M.Y., Sun, P.T.: Error analysis of mixed finite element method for Poisson-Nernst-Planck system. Numer. Methods Partial Differ. Equ. 33(6), 1924–1948 (2017)
Lu, B., Holst, M. J., Mccammon, A., Zhou,C.: Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions, J. Comput. Phys. 229,(19) pp. 6979-6994 (2010)
Jerome, J.W., Kerkhoven, T.: A finite element approximation theory for the drift diffusion semiconductor model. SIAM J. Numer. Anal. 28(2), 403–422 (1991)
Yang, Y., Lu, B.: An error analysis for the finite element approximation to the steady-state Poisson-Nernst-Planck equations. Adv. Appl. Math. Mech. 5(1), 113–130 (2013)
Liu, X.L., Xu, C.J.: Efficient time-stepping/spectral methods for the Navier-Stokes-Nernst-Planck-Poisson equations. Commun. Comput. Phys. 21(5), 1408–1428 (2017)
Hollerbach, U., Chen, D.P., Eisenberg, R.S.: Two- and three-dimensional Poisson-Nernst-Planck simulations of current flow through gramicidin A. J. Sci. Comput. 16, 373–409 (2001)
Adams, R., Fournier, J.F.: Sobolev spaces, Academic press (2003)
Brenner, S., Scott, L.: The mathematical theory of finite element methods. Springer, New York (2002)
Lin, Q., Lin, J.F.: Finite element methods: accuracy and improvement. Science Press, Beijing (2006)
Shi, D.Y., Wang, P.L., Zhao, Y.M.: Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation. Appl. Math. Lett. 38, 129–134 (2014)
Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)
Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38, 437–445 (1982)
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This work is supported by the National Natural Science Foundation of China (Nos. 12101568, 11801527) and the Doctoral Starting Foundation of Zhengzhou University of Aeronautics(No. 63020390).
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Communicated by: Long Chen
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Yang, H., Li, M. Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations. Adv Comput Math 50, 43 (2024). https://doi.org/10.1007/s10444-024-10145-4
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DOI: https://doi.org/10.1007/s10444-024-10145-4