S.D. Bond, J.H. Chaudhry, E.C. Cyr and L.N. Olson, A First-Order Systems Least-Squares
Finite Element Method for the Poisson-Boltzmann Equation, Journal of Computational
Chemistry, 31:1625-1635, 2010.
The Poisson-Boltzmann equation is an important tool in modeling solvent
in biomolecular systems. In this article, we focus on numerical approximations
to the electrostatic potential expressed in the regularized linear Poisson-Boltzmann
equation. We expose the flux directly through a first-order system form of the
equation. Using this formulation, we propose a system that yields a tractable
least-squares finite element formulation and establish theory to support this
approach. The least-squares finite element approximation naturally provides an a
posteriori error estimator and we present numerical evidence in support of the
method. The computational results highlight optimality in the case of adaptive
mesh refinement for a variety of molecular configurations. In particular, we
show promising performance for the Born ion, Fasciculin 1, methanol, and a dipole,
which highlights robustness of our approach.