Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations

Yan Xie; Jie Cheng; Benzhuo Lu; Linbo Zhang

Molecular Based Mathematical Biology (2013)

  • Volume: 1, page 90-108
  • ISSN: 2299-3266

Abstract

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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.

How to cite

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Yan Xie, et al. "Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations." Molecular Based Mathematical Biology 1 (2013): 90-108. <http://eudml.org/doc/267189>.

@article{YanXie2013,
abstract = {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.},
author = {Yan Xie, Jie Cheng, Benzhuo Lu, Linbo Zhang},
journal = {Molecular Based Mathematical Biology},
keywords = {Poisson-Nernst-Planck equations; Adaptive Finite Element Method; DIIS; Parallel Processing; Slotboom variables; PHG; adaptive finite element method; parallel processing; slotboom variables; software package PHG},
language = {eng},
pages = {90-108},
title = {Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations},
url = {http://eudml.org/doc/267189},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Yan Xie
AU - Jie Cheng
AU - Benzhuo Lu
AU - Linbo Zhang
TI - Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations
JO - Molecular Based Mathematical Biology
PY - 2013
VL - 1
SP - 90
EP - 108
AB - 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.
LA - eng
KW - Poisson-Nernst-Planck equations; Adaptive Finite Element Method; DIIS; Parallel Processing; Slotboom variables; PHG; adaptive finite element method; parallel processing; slotboom variables; software package PHG
UR - http://eudml.org/doc/267189
ER -

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