[1] Marcicki J, Conlisk A T and Rizzoni G. Comparison of limiting descriptions of the electrical double layer using a simplified lithium-ion battery model[J]. ECS Transactions, 2012, 41(14):9-21. [2] Ciucci F. Derivation of Micro/Macro Lithium Battery Models from Homogenization[J]. Transport in Porous Media, 2011, 88(2):249-270. [3] Richardson G and King J R. Time-dependent modelling and asymptotic analysis of electrochemical cells[J]. Journal of Engineering Mathematics, 2007, 59(3):239-275. [4] Jerome J. Analysis of Charge Transport:A mathematical theory and approximation of semiconductor models[M]. Springer-Verlag, New York, 1996. [5] 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[J]. Plos Computational Biology, 2009, 5(1):e1000277. [6] Lu B, Zhou Y, 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]. Journal of Chemical Physics, 2007, 127(13):135102. [7] Singer A and Norbury J. Singer. A poisson-nernst-planck model for biological ion channels an asymptotic analysis in a three-dimensional narrow funnel[J]. Siam Journal on Applied Mathematics, 2009, 70(3):949-968. [8] Cárdenas 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[J]. Biophysical Journal, 2000, 79(1):80-93. [9] Lu B, Holst M, McCammon J and Zhou Y C. Poisson-Nernst-Planck eauations for simulating biomolecular diffusion-reaction process I:Finite element solutions[J]. Journal of Computational Physics, 2010, 229(19):6979-6994. [10] 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]. Journal of Physical Chemistry B, 2008, 112(2):270-275. [11] Lu B, Zhou Y. Poisson-Nernst-Planck equations for simulating biomolecular diffffusion-reaction processes II:size effffects on ionic distributions and diffffusion-reaction rates[J]. Biophysical Journal, 2011, 100(10):2475-2485. [12] Lu B, Zhou Y C, Huber G A and Bond S D. Electrodiffusion:a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution[J]. Journal of Chemical Physics, 2007, 127(13):10B604-78. [13] Tu B, Chen M, Xie Y, Zhang L, Eisenberg B and Lu B. A parallel finite element simulator for ion transport through three-dimensional ion channel systems[J]. Journal of Computational Chemistry, 2013, 34(24):2065-2078. [14] Xie Y, Cheng J and Lu B, Zhang L B. Parallel adaptive finite element algorithms for solving the coupled electro-diffusion equation[J]. Molecular Based Mathematical Biology, 2013, 1:90-108. [15] Sun Y Z, Sun P T, Zheng B and Lin G. Error a-nalysis of finite element method for PoissonNernst-Planck equations[J]. Journal of Computational and Applied Mathematics, 2016, 301:28-43. [16] Yang Y and Zhou A H. Local averaging based a posteriori finite element error control for quasilinear elliptic problems with application to electrical potential computation[J]. Computer Methods in Applied Mechanics Engineering, 2006, 196(1):452-465. [17] Brandts J and KŘÍŽEK M. Gradient superconvergence on uniform simplicial partitions of polytopes[J]. IMA Journal of Numerical Analysis, 2003, 23(3):489-505. [18] Brenner S C and Scott L R. The Mathematical Theory of Finite Element Methods[M]. SpringerVerlag, New York, 1994. [19] Shen R, Shu S, Yang Y and Lu B. A Decoupling Two-grid Method for the Time-dependent Poisson-Nernst-Planck Equations[J]. Numerical Algorithms, https://doi.org/10.1007/s11075-019-00744-4, 2019. [20] Yang Y and Lu B Z. An error analysis for the finite element approximation to the steadystate Poisson-Nernst-Planck equations[J]. Advances in Applied Mathematics and Mechanics, 2013, 5(01):113-130. [21] Verfurth R. A posteriori error estimators for convection-diffusion equations[J]. Numeriche Mathematik, 1998, 80:641-663. [22] Verfurth R. A Review of a Posteriori Error Estimates and Adaptive Mesh-refinement Techniques[M]. Wiley-Teubner, New York, 1996. |