  The Poisson-Boltzmann equation (PBE) models the electrostatic interactions of
charged bodies such as molecules and proteins in an electrolyte solvent. The
PBE is a challenging equation to solve numerically due to the presence of
singularities, discontinuous coefficients and boundary conditions. Hence, there
is often large error in the numerical solution of the PBE that needs to be
quantified. In this work, we use adjoint based a posteriori analysis to
accurately quantify the error in an important quantity of interest, the
solvation free energy, for the finite element solution of the PBE. We identify
various sources of error and propose novel refinement strategies based on a
posteriori error estimates.
