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The Poisson-Boltzmann equation is a widely used model to study theelectrostatics in molecular solvation. Its numerical solution using a boundaryintegral formulation requires a mesh on the molecular surface only, yieldingaccurate representations of the solute, which is usually a complicatedgeometry. Here, we utilize adjoint-based analyses to form two goal-orientederror estimates that allows us to determine the contribution of eachdiscretization element (panel) to the numerical error in the solvation freeenergy. This information is useful to identify high-error panels to then refinethem adaptively to find optimal surface meshes. We present results for spheresand real molecular geometries, and see that elements with large error tend tobe in regions where there is a high electrostatic potential. We also find thateven though both estimates predict different total errors, they have similarperformance as part of an adaptive mesh refinement scheme. Our test casessuggest that the adaptive mesh refinement scheme is very effective, as we areable to reduce the error one order of magnitude by increasing the mesh sizeless than 20\%. This result sets the basis towards efficient automatic meshrefinement schemes that produce optimal meshes for solvation energycalculations.

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