Efficient mesh refinement for the Poisson-Boltzmann equation with boundary elements

TL;DR

This paper introduces an adaptive mesh refinement method for solving the Poisson-Boltzmann equation using boundary elements, significantly improving accuracy in electrostatics calculations for molecular solvation.

Contribution

It develops goal-oriented error estimates for adaptive mesh refinement, enabling efficient and accurate surface mesh generation for complex molecular geometries.

Findings

Error reduced by an order of magnitude with less than 20% increase in mesh size

High-error regions correlate with high electrostatic potential areas

Adaptive refinement outperforms uniform mesh refinement in efficiency

Abstract

The Poisson-Boltzmann equation is a widely used model to study the electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allows us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20\%. This result sets the basis towards efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.