# A Posteriori Analysis and Efficient Refinement Strategies for the Poisson-Boltzmann Equation

## TL;DR

This paper employs adjoint-based a posteriori analysis to quantify errors in solving the Poisson-Boltzmann equation and introduces efficient refinement strategies to improve the accuracy of solvation free energy calculations.

## Contribution

It presents a novel application of a posteriori error analysis to the PBE and develops new refinement strategies for finite element solutions.

## Key findings

- Accurate error quantification of solvation free energy
- Identification of key error sources in PBE solutions
- Development of effective adaptive refinement methods

## Abstract

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.