A first-order system least-squares finite element method for the poisson-boltzmann equation

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 PoissonBoltzmann 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.