Finite Element Approximation to a Finite-Size Modified Poisson-Boltzmann Equation

Authors:
Jehanzeb Hameed Chaudhry;Stephen D. Bond;Luke N. Olson
Affiliations:
Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, USA 61801;Applied Mathematics and Applications Group, Sandia National Laboratories, Albuquerque, USA 87185;Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, USA 61801
Venue:
Journal of Scientific Computing
Year:
2011

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Abstract

The inclusion of steric effects is important when determining the electrostatic potential near a solute surface. We consider a modified form of the Poisson-Boltzmann equation, often called the Poisson-Bikerman equation, in order to model these effects. The modifications lead to bounded ionic concentration profiles and are consistent with the Poisson-Boltzmann equation in the limit of zero-size ions. Moreover, the modified equation fits well into existing finite element frameworks for the Poisson-Boltzmann equation. In this paper, we advocate a wider use of the modified equation and establish well-posedness of the weak problem along with convergence of an associated finite element formulation. We also examine several practical considerations such as conditioning of the linearized form of the nonlinear modified Poisson-Boltzmann equation, implications in numerical evaluation of the modified form, and utility of the modified equation in the context of the classical Poisson-Boltzmann equation.