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

The Poisson‐Boltzmann equation is a widely used model to study 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 allow 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% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations. The Poisson‐Boltzmann equation is widely used to compute the solvation energies of molecules. It considers the solute as a cavity region in an infinite dielectric, interfaced by the molecular surface. The boundary element method offers an efficient numerical solution, as it discretizes the interface only. Here, we present an a posteriori error estimation method that detects high error elements, with which we generate a highly effective adaptive mesh refinement technique.