Published

The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation

Long Chen, Michael Holst, and Jinchao Xu

SIAM Journal on Numerical Analysis, 45(6): 2298-2320, 2007

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ABSTRACT:

  A widely used electrostatics model in the biomolecular modeling
  community, the nonlinear Poisson-Boltzmann equation, along with its
  finite element approximation, are analyzed in this paper. A
  regularized Poisson-Boltzmann equation is introduced as an auxiliary
  problem, making it possible to study the original nonlinear equation
  with delta distribution sources. {\it A priori} error estimates for
  the finite element approximation is obtained for the regularized
  Poisson-Boltzmann equation based on certain quasi-uniform grids in
  two and three dimensions. Adaptive finite element approximation
  through local refinement driven by {\it a posteriori} error estimate
  is shown to converge. The Poisson-Boltzmann equation does not appear
  to have been previously studied in detail theoretically, and it is
  hoped that this paper will help provide molecular modelers with a
  better foundation for their analytical and computational work with
  the Poisson-Boltzmann equation. Note that this article apparently
  gives the first rigorous convergence result for a numerical
  discretization technique for the nonlinear Poisson-Boltzmann
  equation with delta distribution sources, and it also introduces the
  first provably convergent adaptive method for the equation.  This
  last result is currently one of only a handful of existing
  convergence results of this type for nonlinear problems.