一类Poisson-Nernst-Planck方程的两网格有限元离散方法
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摘要
应用两网格有限元方法离散求解一类Poisson-Nernst-Planck(PNP)方程.通过两网格离散,将耦合PNP系统解耦成较小规模的线性对称系统,可有效降低计算复杂度.理论结果表明,线性对称化的两网格算法具有与传统有限元方法相同的误差阶;数值结果表明,相比于传统有限元方法,该方法计算效率更高.
A two-grid finite element method was used to solve a class of Poisson-Nernst-Planck(PNP)equations discretely.By the two-grid discretization,the coupled PNP system was decoupled into a small scale linear symmetric system,which could effectively reduce the computational complexity.The theoretical results show that the linear symmetry two-grid algorithm has the same error order as the traditional finite element method.The numerical results show that the method has higher computational efficiency than the traditional finite element method.
A two-grid finite element method was used to solve a class of Poisson-Nernst-Planck(PNP)equations discretely.By the two-grid discretization,the coupled PNP system was decoupled into a small scale linear symmetric system,which could effectively reduce the computational complexity.The theoretical results show that the linear symmetry two-grid algorithm has the same error order as the traditional finite element method.The numerical results show that the method has higher computational efficiency than the traditional finite element method.
引文
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[3]EISENBERG B,HYON Y,LIU Chun.Energy Variational Analysis of Ions in Water and Channels:Field Theory for Primitive Models of Complex Ionic Fluids[J].Journal of Chemical Physics,2010,133(10):104104-1-104104-23.
[4]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[J].Biophysical Journal,2000,79(1):80-93.
[5]MATHUR S R,MURTHY J Y.A Multigrid Method for the Poisson-Nernst-Planck Equations[J].International Journal of Heat&Mass Transfer,2009,52(17/18):4031-4039.
[6]LU Benzhuo,HOLST M J,McCAMMON J A,et al.Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction ProcessesⅠ:Finite Element Solutions[J].Journal of Computational Physics,2010,229(19):6979-6994.
[7]XU Jinchao.A Novel Two-Grid Method for Semilinear Elliptic Equations[J].SIAM Journal on Scientific Computing,2006,15(1):231-237.
[8]XU Jinchao.Two-Grid Discretization Techniques for Linear and Nonlinear PDEs[J].SIAM Journal on Numerical Analysis,1996,33(5):1759-1777.
[9]JIN Jicheng,SHU Shi,XU Jinchao.A Two-Grid Discretization Method for Decoupling Systems of Partial Differential Equations[J].Mathematics of Computation,2006,75:1617-1626.
[10]MU Mo,XU Jinchao.A Two-Grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow[J].SIAM Journal on Numerical Analysis,2007,45(5):1801-1813.
[11]李雪芳,阳莺.一种求解稳态Poisson-Nernst-Planck方程的线性化的两网格方法[J].桂林电子科技大学学报,2017,37(2):169-172.(LI Xuefang,YANG Ying.A Linearized Two-Grid Method for Steady-State PoissonNernst-Planck Equations[J].Journal of Guilin University of Electronic Technology,2017,37(2):169-172.)
[12]YANG Ying,LU Benzhuo.An Error Analysis for the Finite Element Approximation to the Steady-State PoissonNernst-Planck Equations[J].Advances in Applied Mathematics&Mechanics,2013,5(1):113-130.
[13]ADAMS R A.Sobolev Spaces[M].New York:Academic Press,1975.
[14]SUN Yuzhou,SUN Pengtao,ZHENG Bin,et al.Error Analysis of Finite Element Method for Poisson-NernstPlanck Equations[J].Journal of Computational&Applied Mathematics,2016,301:28-43.
[15]PROHL A,SCHMUCK M.Convergent Discretizations for the Nernst-Planck-Poisson System[J].Numerische Mathematik,2009,111(4):591-630.
[16]HORNG T L,LIN T C,LIU C,et al.PNP Equations with Steric Effects:A Model of Ion Flow Through Channels[J].The Journal of Physical Chemistry B,2012,116(37):11422-11441.
[17]HYON Y K,EISENBERG B,LIU C.A Mathematical Model for the Hard Sphere Repulsion in Ionic Solutions[J].Communications in Mathematical Sciences,2010,9(2):459-475.
[18]HYON Y K,EISENBERG B,LIU C.An Energetic Variational Approach to Ion Channel Dynamics[J].Mathematical Methods in the Applied Sciences,2014,37(7):952-961.