Energetically stable discretizations for charge transport and electrokinetic models

A finite element discretization using a method of lines approached is proposed for approximately solving the Poisson-Nernst-Planck (PNP) equations. This discretization scheme enforces positivity of the computed solutions, corresponding to particle density functions, and a discrete energy estimate is established that takes the same form as the energy law for the continuous PNP system. This energy estimate is extended to finite element solutions to an electrokinetic model, which couples the PNP system with the incompressible Navier-Stokes equations. Numerical experiments are conducted to validate convergence of the computed solution and verify the discrete energy estimate.

[1]  C. Schmeiser,et al.  Semiconductor equations , 1990 .

[2]  YunKyong Hyon,et al.  A mathematical model for the hard sphere repulsion in ionic solutions , 2011 .

[3]  Joseph W. Jerome,et al.  CONSISTENCY OF SEMICONDUCTOR MODELING: AN EXISTENCE/STABILITY ANALYSIS FOR THE STATIONARY VAN ROOSBROECK SYSTEM* , 1985 .

[4]  Rolf J. Ryham,et al.  Existence, Uniqueness, Regularity and Long-term Behavior for Dissipative Systems Modeling Electrohydrodynamics , 2009, 0910.4973.

[5]  Martin Stynes,et al.  Finite Element Methods for Convection-Diffusion Problems using Exponential Splines on Triangles , 1998 .

[6]  Chun Liu,et al.  PNP equations with steric effects: a model of ion flow through channels. , 2012, The journal of physical chemistry. B.

[7]  Peter A. Markowich,et al.  The Stationary Semiconductor Device Equations. , 1987 .

[8]  J. Slotboom,et al.  Computer-aided two-dimensional analysis of bipolar transistors , 1973 .

[9]  W. Nernst,et al.  Die elektromotorische Wirksamkeit der Jonen , 1889 .

[10]  H. Gummel,et al.  Large-signal analysis of a silicon Read diode oscillator , 1969 .

[11]  In Seok Kang,et al.  Mixing enhancement by using electrokinetic instability under time-periodic electric field , 2005 .

[12]  P N Scharbach,et al.  A Dynamical Theory of the Electromagnetic Field , 1983 .

[13]  Randolph E. Bank,et al.  Numerical Methods for Semiconductor Device Simulation , 1983 .

[14]  Diana Adler,et al.  Electronic Transport In Mesoscopic Systems , 2016 .

[15]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[16]  Joseph W. Jerome,et al.  L∞ stability of finite element approximations to elliptic gradient equations , 1990 .

[17]  Guido Kanschat,et al.  A locally conservative LDG method for the incompressible Navier-Stokes equations , 2004, Math. Comput..

[18]  M. S. Mock,et al.  Analysis of mathematical models of semiconductors devices , 1983 .

[19]  Andreas Prohl,et al.  CONVERGENT FINITE ELEMENT DISCRETIZATIONS OF THE NAVIER-STOKES-NERNST-PLANCK-POISSON SYSTEM , 2010 .

[20]  Blanca Ayuso de Dios,et al.  A Simple Preconditioner for a Discontinuous Galerkin Method for the Stokes Problem , 2012, Journal of Scientific Computing.

[21]  M. Planck,et al.  Ueber die Erregung von Electricität und Wärme in Electrolyten , 1890 .

[22]  Minxin Chen,et al.  A parallel finite element simulator for ion transport through three‐dimensional ion channel systems , 2013, J. Comput. Chem..

[23]  Piotr Biler,et al.  Long Time Behavior of Solutions to Nernst – Planck and Debye – Hückel Drift – Diffusion Systems , 1999 .

[24]  Chun Liu,et al.  Transport of Charged Particles: Entropy Production and Maximum Dissipation Principle , 2014, 1407.8245.

[25]  Michael J. Holst,et al.  Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions , 2010, J. Comput. Phys..

[26]  Piotr Biler,et al.  The Debye system: existence and large time behavior of solutions , 1994 .

[27]  A. H. Schatz,et al.  A weak discrete maximum principle and stability of the finite element method in _{∞} on plane polygonal domains. I , 1980 .

[28]  Randolph E. Bank,et al.  A new discretization scheme for the semiconductor current continuity equations , 1989, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[29]  Joseph W. Jerome.  Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices , 1995 .

[30]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[31]  Jinchao Xu,et al.  Newton-Krylov-Multigrid Algorithms for Battery Simulation , 2002 .

[32]  Ludmil T. Zikatanov,et al.  A monotone finite element scheme for convection-diffusion equations , 1999, Math. Comput..

[33]  Andreas Prohl,et al.  Convergent discretizations for the Nernst–Planck–Poisson system , 2009, Numerische Mathematik.

[34]  Dongqing Li.  Electrokinetics in Microfluidics , 2004 .

[35]  Minxin Chen,et al.  Modeling and Simulating Asymmetrical Conductance Changes in Gramicidin Pores , 2014 .

[36]  YunKyong Hyon,et al.  Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids. , 2010, The Journal of chemical physics.

[37]  D. Kwak,et al.  Energetic variational approach in complex fluids: Maximum dissipation principle , 2009 .

[38]  A. Majumdar,et al.  Rectification of ionic current in a nanofluidic diode. , 2007, Nano letters.

[39]  D. Erickson,et al.  Influence of Surface Heterogeneity on Electrokinetically Driven Microfluidic Mixing , 2002 .

[40]  Randolph E. Bank,et al.  The Finite Volume Scharfetter-Gummel method for steady convection diffusion equations , 1998 .

[41]  M. Sushko,et al.  Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment , 2014, 1407.0674.

[42]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[43]  Ludmil T. Zikatanov,et al.  An exponential fitting scheme for general convection-diffusion equations on tetrahedral meshes , 2012, 1211.0869.