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Linearized Conservative Finite Element Methods for the Nernst–Planck–Poisson Equations

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Abstract

The aim of this paper is to present and study new linearized conservative schemes with finite element approximations for the Nernst–Planck–Poisson equations. For the linearized backward Euler FEM, an optimal \(L^2\) error estimate is provided almost unconditionally (i.e., when the mesh size h and time step \(\tau \) are less than a small constant). Global mass conservation and electric energy decay of the schemes are also proved. Extension to second-order time discretizations is given. Numerical results in both two- and three-dimensional spaces are provided to confirm our theoretical analysis and show the optimal convergence, unconditional stability, global mass conservation and electric energy decay properties of the proposed schemes.

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References

  1. Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012). doi:10.1007/978-3-642-23099-8

    MATH  Google Scholar 

  2. Bochev, P., Lehoucq, R.: On the finite element solution of the pure Neumann problem. SIAM Rev. 47, 50–66 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)

    Book  MATH  Google Scholar 

  4. Brezzi, F., Marini, L., Pietra, P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26, 1342–1355 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezzi, F., Marini, L., Pietra, P.: Numerical simulation of semiconductor devices. Comput. Methods Appl. Mech. Eng. 75, 493–514 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brunk, M., Kværnø, A.: Positivity preserving discretization of time dependent semiconductor drift-diffusion equations. Appl. Numer. Math. 62, 1289–1301 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Y., Wu, L.: Second-order Elliptic Equations and Elliptic Systems, Translations of Mathematical Monographs 174, AMS (1998)

  8. Evans, L.: Partial Differential Equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, RI (1998)

  9. 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)

    Google Scholar 

  10. Gajewski, H., Gröger, K.: On the basic equations for carrier transport in semiconductors. J. Math. Anal. Appl. 113, 12–35 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. He, D., Pan, K.: An energy preserving finite difference scheme for the Poisson–Nernst–Planck system. Appl. Math. Comput. 287–288, 214–223 (2016)

    MathSciNet  Google Scholar 

  12. Heywood, J., 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hou, Y., Li, B., Sun, W.: Error analysis of splitting Galerkin methods for heat and sweat transport in textile materials. SIAM J. Numer. Anal. 51, 88–111 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, B., Lu, B., Wang, Z., McCammon, J.A.: Solutions to a reduced Poisson–Nernst–Planck system and determination of reaction rates. Phys. A 389, 1329–1345 (2010)

    Article  Google Scholar 

  15. Liu, Y., Shu, C.W.: Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices. Sci. China Math. 59, 115–140 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, H., Wang, Z.: A free energy satisfying finite difference method for Poisson–Nernst–Planck equations. J. Comput. Phys. 268, 363–376 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lu, B., Holst, M., McCammon, J., Zhou, Y.: Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: finite element solutions. J. Comput. Phys. 229, 6979–6994 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Metti, M., Xu, J., Liu, C.: Energetically stable discretizations for charge transport and electrokinetic models. J. Comput. Phys. 306, 1–18 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mirzadeh, M., Gibou, F.: A conservative discretization of the Poisson–Nernst–Planck equations on adaptive Cartesian grids. J. Comput. Phys. 274, 633–653 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mock, M.: An initial value problem from semiconductor device theory. SIAM J. Math. Anal. 5, 597–612 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3) 20, 733–737 (1966)

    MathSciNet  MATH  Google Scholar 

  22. Prohl, A., Schmuck, M.: Convergent discretizations for the Nernst–Planck–Poisson system. Numer. Math. 111, 591–630 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Prohl, A., Schmuck, M.: Convergent finite element for discretizations of the Navier–Stokes–Nernst–Planck–Poisson system. M2AN Math. Model. Numer. Anal. 44, 531–571 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38, 437–445 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Scharfetter, D., Gummel, H.: Large signal analysis of a silicon read diode oscillator. IEEE Trans. Electron. Dev. 16, 64–77 (1969)

    Article  Google Scholar 

  26. Schmuck, M.: Analysis of the Navier–Stokes–Nernst–Planck–Poisson system. Math. Models Methods Appl. Sci. 19, 993–1015 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, Y., Sun, P., Zheng, B., Lin, G.: Error analysis of finite element method for Poisson–Nernst–Planck equations. J. Comput. Appl. Math. 301, 28–43 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sun, W., Sun, Z.: Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials. Numer. Math. 120, 153–187 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)

    MATH  Google Scholar 

  30. Wei, G., Zheng, Q., Chen, Z., Xia, K.: Variational multiscale models for charge transport. SIAM Rev. 54, 699–754 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

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Acknowledgements

Huadong Gao was supported in part by a grant from the National Natural Science Foundation of China (NSFC) under Grant No. 11501227. Dongdong He was supported in part by a Grant from the National Natural Science Foundation of China (NSFC) under Grant No. 11402174. The authors would like to thank Dr. Weifeng Qiu for useful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Huadong Gao

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, People’s Republic of China

    Dongdong He

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  1. Huadong Gao
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  2. Dongdong He
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Correspondence to Dongdong He.

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Gao, H., He, D. Linearized Conservative Finite Element Methods for the Nernst–Planck–Poisson Equations. J Sci Comput 72, 1269–1289 (2017). https://doi.org/10.1007/s10915-017-0400-4

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  • DOI: https://doi.org/10.1007/s10915-017-0400-4

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