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2013 | 1 | 90-108
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Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations

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rithms for solving the 3D electro-diffusion equations such as the Poisson-Nernst-Planck equations and the size-modified Poisson-Nernst-Planck equations in simulations of biomolecular systems in ionic liquid. A set of transformation methods based on the generalized Slotboom variables is used to solve the coupled equations. Calculations of the diffusion-reaction rate coefficients, electrostatic potential and ion concentrations for various systems verify the method’s validity and stability. The iterations between the Poisson equation and the Nernst- Planck equations in the primitive method and in the transformation method are compared to illustrate how the new method accelerates the convergence of the solution. To speed up the convergence, we introduce the DIIS (direct inversion of the iterative subspace) method including Simple Mixing and Anderson Mixing as under-relaxation techniques, the effectiveness of which on acceleration is shown by numerical tests. It is worth noting that the primitive method fails to solve the size-modified Poisson-Nernst-Planck equations for real protein systems but the transformation method succeeds in the simulations of the ACh-AChE reaction system and the DNA fragment. To improve the accuracy of the solution, we introduce high order elements and mesh adaptation based on an a posteriori error estimator. Numerical results indicate that our mesh adaptation process leads to quasi-optimal convergence. We implement our algorithms using the parallel adaptive finite element package PHG [53] and high parallel efficiency is obtained.
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  • J. Ahrens, B. Geveci, and C. Law. Paraview: An end user tool for large data visualization. the VisualizationHandbook. Edited by CD Hansen and CR Johnson. Elsevier, 2005.
  • R. E. Bank, D. J. Rose, and W. Fichtner. Numerical methods for semiconductor device simulation. SIAM J. Sci.Statist. Comput., 4:416–435, 1983.[Crossref]
  • M. J. Berger and P. Colella. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82(1):64–84,1989.[Crossref]
  • A. H. Boschitsch and M. O. Fenley. Hybrid boundary element and finite difference method for solving the nonlinearPoisson-Boltzmann equation. J. Comput. Chem., 25(7):935–955, 2004.
  • A. E. Cardenas, R. D. Coalson, and M. G. Kurnikova. 3D Poisson-Nernst-Planck theory studies: Influence ofmembrane electrostatics on gramicidin A channel conductance. Biophys. J., 79(1):80–93, 2000.[Crossref][PubMed]
  • D. Chen and R. Eisenberg. Charges, currents, and potentials in ionic channels of one conformation. Biophys. J.,64(5):1405–1421, 1993.[Crossref][PubMed]
  • Duan P. Chen. Competitive permeation of calcium ions through the calcium release channel (ryanodine receptor)of cardiac muscle: An application of the coupled Poisson-Nernst-Planck system of equations. J. Phys. Chem. B,107(34):9139–9145, 2003.
  • M. X. Chen and B. Z. Lu. TMSmesh: A robust method for molecular surface mesh generation using a trace technique.J. Chem. Theory Comput., 7(1):203–212, 2011.[Crossref]
  • Zhiming Chen, Yuanming Xiao, and Linbo Zhang. The adaptive immersed interface finite element method for ellipticand Maxwell interface problems. J. Comput. Phys., 228(14):5000–5019, 2009.[Crossref]
  • I-Liang Chern, Jian-Guo Liu, and Wei-Cheng Wang. Accurate evaluation of electrostatics for macromolecules insolution. Methods Appl. Anal., 10(2):309–328, 2003.
  • Vincent B. Chu, Yu Bai, Jan Lipfert, Daniel Herschlag, and Sebastian Doniach. Evaluation of ion binding to DNAduplexes using a size-modified Poisson-Boltzmann theory. Biophys. J., 93(9):3202–3209, 2007.
  • H. Cohen and J. W. Cooley. The numerical solution of the time-dependent Nernst-Planck equations. Biophys. J.,5(2):145–162, 1965.[PubMed][Crossref]
  • Ben Corry, Serdar Kuyucak, and Shin-Ho Chung. Dielectric self-energy in Poisson-Boltzmann and Poisson-Nernst-Planck models of ion channels. Biophys. J., 84(6):3594–3606, 2003.[Crossref]
  • H. Daiguji, Y. Oka, and K. Shirono. Nanofluidic diode and bipolar transistor. Nano Lett., 5(11):2274–2280, 2005.[Crossref][PubMed]
  • H. Daiguji, P. Yang, and A. Majumdar. Ion transport in nanofluidic channels. Nano Lett., 4(1):137–142, 2004.[Crossref]
  • P. Debye and E. Hückel. De la theorie des electrolytes. I. abaissement du point de congelation et phenomenesassocies. Physikalische Zeitschrift, 24(9):185–206, 1923.
  • B. Eisenberg, Y. Hyon, and C. Liu. Energy variational analysis of ions in water and channels: Field theory forprimitive models of complex ionic fluids. J. Chem. Phys., 133:104104, 2010.
  • R. S. Eisenberg. Computing the field in proteins and channels. J. Membr. Biol., 150(1):1–25, 1996.
  • Q. Fang and D. A. Boas. Tetrahedral mesh generation from volumetric binary and grayscale images. In BiomedicalImaging: From Nano to Macro, 2009. ISBI’09. IEEE International Symposium on, pages 1142–1145, 2009.
  • W. Fichtner, D.J. Rose, and R.E. Bank. Semiconductor device simulation. IEEE T. Electron Dev., 30(9):1018–1030,1983.[Crossref]
  • U. Hollerbach, D. P. Chen, D. D. Busath, and B. Eisenberg. Predicting function from structure using the Poisson-Nernst-Planck equations: sodium current in the gramicidin A channel. Langmuir, 16(13):5509–5514, 2000.[Crossref]
  • M. Holst, N. Baker, and F. Wang. Adaptive multilevel finite element solution of the Poisson-Boltzmann equation i:Algorithms and examples. J. Comput. Phys., 21:1319–1342, 2000.
  • M. J. Holst. The Poisson Boltzmann equation: Analysis and multilevel numerical solution. Ph.D thesis, Universityof Illinois at Urbana-Champaign, 1994.
  • M. J. Holst. Finite element toolkit., 2010.
  • Michael Holst, Jeffrey S. Ovall, and Ryan Szypowski. An efficient, reliable and robust error estimator for ellipticproblems in. Appl. Numer. Math., 61(5):675–695, 2011.[Crossref]
  • W. Im and B. Roux. Ion permeation and selectivity of OmpF porin: a theoretical study based on molecular dynamics,Brownian dynamics, and continuum electrodiffusion theory. J. Mol. Biol., 322(4):851–869, 2002.
  • A. Jungel and C. Pohl. Numerical simulation of semiconductor devices: energy-transport and quantum hydrodynamicmodeling. In Computational Electronics, 1998. IWCE-6. Extended Abstracts of 1998 Sixth International Workshopon, pages 230–233, 1998.
  • C. E. Korman and I. D. Mayergoyz. A globally convergent algorithm for the solution of the steady-state semiconductordevice equations. J. App. Phys., 68(3):1324–1334, 1990.
  • K. KrabbenhÃÿft and J. KrabbenhÃÿft. Application of the Poisson-Nernst-Planck equations to the migration test.Cement Concrete Res., 38(1):77–88, 2008.[Crossref]
  • Witold Kucza, Marek Danielewski, and Andrzej Lewenstam. Eis simulations for ion-selective site-based membranesby a numerical solution of the coupled Nernst-Planck-Poisson equations. Electrochem. Commun., 8(3):416–420,2006.[Crossref]
  • Maria G. Kurnikova, Rob D. Coalson, Peter Graf, and Abraham Nitzan. A lattice relaxation algorithm for threedimensionalPoisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel.Biophys. J., 76(2):642–656, 1999.
  • B. Z. Lu, Y. C. Zhou, M. J. Holst, and J. A. McCammon. Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications. Commun. Comput. Phys., 3(5):973–1009, 2008.
  • Benzhuo Lu, Michael J. Holst, J. Andrew McCammon, and Y.C. Zhou. Poisson-Nernst-Planck equations for simulatingbiomolecular diffusion-reaction processes I: Finite element solutions. J. Comput. Phys., 229(19):6979–6994, 2010.
  • Benzhuo Lu and J. Andrew McCammon. Molecular surface-free continuum model for electrodiffusion processes.Chem. Phys. Lett., 451(4-6):282–286, 2008.
  • Benzhuo Lu and Y.C. Zhou. Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processesII: Size effects on ionic distributions and diffusion-reaction rates. Biophys. J., 100(10):2475–2485, 2011.
  • Benzhuo Lu, Y.C. Zhou, Gary A. Huber, Stephen D. Bond, Michael J. Holst, and J. Andrew McCammon. Electrodiffusion:A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution. J. Chem.Phys., 127(13), 2007.
  • Benzhuo Lu and Yongcheng Zhou. Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reactionprocesses II: Size effects on ionic distributions and diffusion-reaction rates. Biophys. J., 100(10):2475–2485, 2011.
  • P.A. Markowich. The stationary semiconductor device equations, volume 1. Springer Verlag, 1986.
  • A. Nicholls and B. Honig. A rapid finite difference algorithm, utilizing successive over-relaxation to solve thePoisson-Boltzmann equation. J. Comput. Chem., 12(4):435–445, 1991.
  • Juan Manuel Paz-García, BjÃurn Johannesson, Lisbeth M. Ottosen, Alexandra B. Ribeiro, and JosÃl MiguelRodríguez-Maroto. Modeling of electrokinetic processes by finite element integration of the Nernst-Planck-Poisson system of equations. Sep. Purif. Technol., 79(2):183–192, 2011.
  • Pulay Peter. Convergence acceleration of iterative sequences. Chem. Phys. Lett., 73(2):393–398, 1980.[Crossref]
  • C. S. Rafferty, M. R. Pinto, and R. W. Dutton. Iterative methods in semiconductor device simulation. IEEE T.Electron Dev., 32(10):2018–2027, 1985.[Crossref]
  • W. Rocchia, E. Alexov, and B. Honig. Extending the applicability of the nonlinear Poisson-Boltzmann equation:Multiple dielectric constants and multivalent ions. J. Phys. Chem. B, 105(28):6507–6514, 2001.[Crossref]
  • B. Roux. Theoretical and computational models of ion channels. Curr. Opin. Struc. Biol., 12(2):182–189, 2002.[Crossref]
  • Isaak Rubinstein. Electro-diffusion of ions, volume 11. SIAM, 1990.
  • Y. Saad and Y. Saad. Iterative methods for sparse linear systems, volume 20. PWS publishing company Boston,1996.
  • Z. Schuss, B. Nadler, and R.S. Eisenberg. Derivation of Poisson and Nernst-Planck equations in a bath and channelfrom a molecular model. Phys. Rev. E, 64(3):036116, 2001.[Crossref]
  • H. Si and A. TetGen. A quality tetrahedral mesh generator and three-dimensional delaunay triangulator. WEIERSTRASSINST FOR APPLIED ANALYSIS AND STOCHASTICS BERLIN (GERMAN), 2006.
  • Hang Si. Tetview: A tetrahedral mesh and piecewise linear complex viewer., 2004.
  • B. F. Smith, P. E. Bjørstad, and W. Gropp. Domain decomposition: parallel multilevel methods for elliptic partialdifferential equations. Cambridge Univ. Pr., 2004.
  • Tomasz Sokalski and Andrzej Lewenstam. Application of Nernst-Planck and Poisson equations for interpretationof liquid-junction and membrane potentials in real-time and space domains. Electrochem. Commun., 3(3):107–112,2001.
  • A. Zaharescu, E. Boyer, and R. Horaud. Transformesh: A topology-adaptive mesh-based approach to surfaceevolution. In Proceedings of the 8th Asian conference on Computer vision-Volume Part II, pages 166–175. Springer-Verlag, 2007.
  • Linbo Zhang. A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection. Numer.Math. Theor. Meth. Appl., 2(1):65–89, 2009.
  • Q. Zheng, D. Chen, and G. W. Wei. Second-order Poisson-Nernst-Planck solver for ion transport. J. Comput. Phys.,230(13):5239–5262, 2011.[PubMed][Crossref]
  • Konstantin Zhurov, Edmund J.F. Dickinson, and Richard G. Compton. Dynamic simulation of the moving boundarymethod for measuring transference numbers. Chem. Phys. Lett., 513(1-3):136–138, 2011.108
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