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Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem

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Abstract

Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. An error estimate in H1 norm is obtained for a piecewise finite element approximation to the solution of the nonlinear steady-state Poisson-Nernst-Planck equations. Some superconvergence results are also derived by using the gradient recovery technique for the equations. Numerical results are given to validate the theoretical results. It is also numerically illustrated that the gradient recovery technique can be successfully applied to the computation of the practical ion channel problem to improve the efficiency of the external iteration and save CPU time.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Andersen, O.S.: Ion movement through gramicidin a channels interfacial polarization effects on single-channel current measurements. Biophys. J. 41(2), 135–146 (1983)

    Google Scholar 

  3. Brandts, J., KŘÍžEK, M.: Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23(3), 489–505 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K., Copps, K.: Validation of a posteriori error estimators by numerical approach. Int. J. Numer. Meth. Eng. 37(7), 1073–1123 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part i: Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)

    MATH  Google Scholar 

  7. Cao, W.M.: Superconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes. Math. Comput. 84(291), 89–117 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Cardenas, A.E., Coalson, R.D., Kurnikova, M.G.: Three-dimensional Poisson-Nernst-Planck theory studies: influence of membrane electrostatics on gramicidin a channel conductance. Biophys. J. 79(1), 80–93 (2000)

    Google Scholar 

  9. Carstensen, C., Bartels, S.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. i: low order conforming, nonconforming, and mixed FEM. Math. Comput. 71(239), 945–969 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Chen, C.M.: Superconvergence and extrapolation of the finite element approximations to quasilinear elliptic problems. Northeastern Math. J. 2, 228–236 (1986)

    MathSciNet  MATH  Google Scholar 

  11. Chen, J., Wang, D., Du, Q.: Linear finite element superconvergence on simplicial meshes. Math. Comp. 83, 2161–2185 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Chen, L.: Superconvergence of tetrahedral linear finite elements. Int. J. Numer. Anal. Model. 3(3), 273–282 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Chen, M., Lu, B.Z.: TMSMesh: a robust method for molecular surface mesh generation using a trace technique. J. Chem. Theory Comput. 7(1), 203–212 (2011)

    Google Scholar 

  14. Chen, Y., Wu, L.: Second-Order Elliptic Equations and Elliptic Systems. Science Press, Beijing (1991). (in Chinese)

    Google Scholar 

  15. Cohen, H., Cooley, J.W.: The numerical solution of the time-dependent Nernst-Planck equations. Biophys. J. 5(2), 145–162 (1965)

    Google Scholar 

  16. Du, L., Yan, N.N.: Gradient recovery type a posteriori error estimate for finite element approximation on non-uniform meshes. Adv. Comput. Math. 14(2), 175–193 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Gao, H.D., He, D.D.: Linearized conservative finite element methods for the Nernst-Planck-Poisson equations. J Sci. Comput. 72(3), 1269–1289 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Gummel, H.K.: A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE T. Electron Dev. 11(10), 455–465 (1964)

    Google Scholar 

  19. Guo, H.L., Yang, X.: Gradient recovery for elliptic interface problem: i. body-fitted mesh. Commun. Comput. Phys. 23(5), 1488–1511 (2018)

    MathSciNet  Google Scholar 

  20. Guo, H.L., Xie, C., Zhao, R.: Superconvergent gradient recovery for virtual element methods. Math. Models Methods Appl. Sci. 29(11), 2007–2031 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Hille, B.: Ion Channels of Excitable Membranes, 3rd edn. Sinauer Associates, Sunderland (2001)

    Google Scholar 

  22. Horng, T.L., Lin, T.C., Liu, C., Eisenberg, B.: PNP Equations with steric effects: a model of ion flow through channels. J. Phys. Chem. B. 116(37), 11422–11441 (2012)

    Google Scholar 

  23. Hyon, Y.K., Eisenberg, B., Liu, C.: An energetic variational approach to ion channel dynamics. Math. Method. Appl. Sci. 37(7), 952–961 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Jerome, J.W., Brosowski, B.: Evolution systems in semiconductor device modeling: a cyclic uncoupled line analysis for the gummel map. Math. Method. Appl. Sci. 9(1), 455–492 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Li, B., Zhang, Z.: Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numer. Meth. Part. D. E. 15(2), 151–167 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Li, J., Ying, J.Y., Lu, B.Z.: A flux-jump preserved gradient recovery technique for accurately predicting the electrostatic field of an immersed biomolecule. J. Comput. Phys. 396, 193–208 (2019)

    MathSciNet  Google Scholar 

  27. Liu, J.H., Jia, Y.S.: Pointwise superconvergence patch recovery for the gradient of the linear tetrahedral element. J. Comput. Anal. Appl. 16(1), 455–460 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Lu, B.Z., Holst, M.J., McCammon, J.A., Zhou, Y.C.: Poisson-nernst-planck equations for simulating biomolecular diffusion-reaction processes i: finite element solutions. J. Comput. Phys. 229(19), 6979–6994 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Lu, B.Z., Zhou, Y.C.: Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes ii: size effects on ionic distributions and diffusion-reaction rates. Biophys. J. 100(10), 2475–2485 (2011)

    Google Scholar 

  30. Lu, B.Z., Zhou, Y.C., Huber, G.A., Bond, S.D., Holst, M.J., McCammon, J.A.: Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution. J. Chem. Phys. 127(13), 135102 (2007)

    Google Scholar 

  31. Mathur, S.R., Murthy, J.Y.: A multigrid method for the Poisson-Nernst-Planck equations. Int. J. Heat Mass Tran. 52(17-18), 4031–4039 (2009)

    MATH  Google Scholar 

  32. Naga, A., Zhang, Z.: A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42(4), 1780–1800 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Prohl, A., Schmuck, M.: Convergent discretizations for the Nernst-Planck-Poisson system. Numer. Math. 111(4), 591–630 (2009)

    MathSciNet  MATH  Google Scholar 

  34. Shen, R.G., Shu, S., Yang, Y., Lu, B.Z: A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations. Numer. Algo. https://doi.org/10.1007/s11075-019-00744-4 (2019)

  35. Shi, D.Y., Yang, H.J.: Superconvergence analysis of finite element method for Poisson-Nernst-Planck equations. Numer. Meth. Part. D. E. 35, 1206–1223 (2019)

    MathSciNet  MATH  Google Scholar 

  36. 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)

    MathSciNet  MATH  Google Scholar 

  37. Tu, B., Chen, M.X., Xie, Y., Zhang, L.B., Eisenber, B., Lu, B.Z.: A parallel finite element simulator for ion transport through three-dimensional ion channel systems. J. Comput. Chem. 34(24), 2065–2078 (2013)

    Google Scholar 

  38. Wu, J., Srinivasan, V., Xu, J., Wang, C.: Newton-krylov-multigrid algorithms for battery simulation. J. Electrochem. Soc. 149(10), 1342–1348 (2002)

    Google Scholar 

  39. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996)

    MathSciNet  MATH  Google Scholar 

  40. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comput. Math. 14(4), 293–327 (2001)

    MathSciNet  MATH  Google Scholar 

  41. Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Comput. Method. Appl. M. 190(32-33), 4289–4299 (2001)

    MathSciNet  MATH  Google Scholar 

  42. Yang, Y., Lu, B.Z.: 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)

    MathSciNet  MATH  Google Scholar 

  43. Yang, Y., Zhou, A.: Local averaging based a posteriori finite element error control for quasilinear elliptic problems with application to electrical potential computation. Comput. Method. Appl. M. 196(1-3), 452–465 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergence patch recovery and a posteriori error estimates. Int. J. Numer. Meth. Eng. 33(7), 1331–1364 (1992)

    MATH  Google Scholar 

  45. Zhang, Z.M., Naga: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26, 1192–1213 (2005)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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Acknowledgments

The authors would like to thank Dr. Minxin Chen and Dr. Shixin Xu for their valuable discussions on numerical experiments.

Funding

Y. Yang was supported by the National Natural Science Foundation of China (Nos. 11561016, 11701119, 11771105), the Guangxi Natural Science Foundation (2020GXNSFAA159098, 2017GXNSFFA198012), the Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation open project fund, and the Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University. C. Liu was partially supported by NSF grant # 1759535 and the United States - Israel Binational Science Foundation (BSF) # 2024246. B. Z. Lu was supported by the National Key Research and Development Program of China (2016YFB0201304), the Science Challenge Program (No. TZ2016003), and the National Natural Science Foundation of China (No. 11771435). L. Q. Zhong was supported by the National Natural Science Foundation of China (Nos. 11671159, 12071160), the Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515010724), the Characteristic Innovation Projects of Guangdong Colleges and Universities, China (No. 2018KTSCX044), and the General Project topic of Science and Technology in Guangzhou, China (No. 201904010117).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, 541004, Guangxi, People’s Republic of China

    Ying Yang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Ming Tang & Liuqiang Zhong

  3. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA

    Chun Liu

  4. LSEC, the National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Benzhuo Lu

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  1. Ying Yang
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  2. Ming Tang
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  3. Chun Liu
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  4. Benzhuo Lu
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Correspondence to Liuqiang Zhong.

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Communicated by: Long Chen

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Yang, Y., Tang, M., Liu, C. et al. Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem. Adv Comput Math 46, 78 (2020). https://doi.org/10.1007/s10444-020-09819-6

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  • DOI: https://doi.org/10.1007/s10444-020-09819-6

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