Journal of Computational and Applied Mathematics
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods
Journal of Computational Physics
Modeling electrokinetic flows by the smoothed profile method
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Finite Element Approximation to a Finite-Size Modified Poisson-Boltzmann Equation
Journal of Scientific Computing
SIAM Journal on Scientific Computing
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
Journal of Scientific Computing
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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. A priori error estimates for the finite element approximation are 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 an 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.