Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel pro
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Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem Ying Yang1 · Ming Tang2 · Chun Liu3 · Benzhuo Lu4 · Liuqiang Zhong2 Received: 27 December 2019 / Accepted: 16 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. An error estimate in H 1 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. Keywords Nonlinear Poisson-Nernst-Planck equations · Steady state · Finite element method · Error estimate · Superconvergent gradient recovery · Ion channel Mathematics subject classification (2010) 65N30
1 Introduction Ion channels are a special integral protein on the cell membrane with characteristic of ion selectivity. They are involved in many physiological activities in bodies, such as the release of neurotransmitters, the contraction of muscles, and other more complex learning and memory [21]. Poisson-Nernst-Planck (PNP) equations are an important theoretical model for simulating the permeation mechanism of ion channels.
Communicated by: Long Chen Liuqiang Zhong
[email protected]
Extended author information available on the last page of the article.
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Adv Comput Math
(2020) 46:78
Although PNP model is widely applied in ion channel area and has some success in dealing with experimental data, limitations are also recognized in it. For example, it does not include correlations introduced by the finite diameter of ions, and these are of great importance in determining selectivity of channels and the properties of ionic solutions in general [22]. Some modified PNP models are then developed to deal with them. For example, Lu and Zhou [29] improved the PNP equations by the addition of the size effect, which simulates the biomolecular diffusion-reaction processes well. Hyon et al. [23] derived a modified PNP system for the ion channel taking into account the protein (ion channel) structure compared with the primitive PNP model. These modifications in PNP models always produce strong nonlinearity, which brings some difficulties in analysis and computation for these models. Generally speaking, it is difficult to find the analytic solutions for PNP equations. There appears many literatures on numerical methods for PNP equations, including finite difference method, finite volume method, and finite element method. Finite difference method (FDM) and finite volume method (FVM) have the advantages of
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