The Poisson-Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well-known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem-size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second-kind equations are better for larger problems. We also present a fast Calderón preconditioner for first-kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson-Boltzmann equation for a given problem.
Reliable computer simulation methods for electrostatic biomolecular models based on the Poisson-Boltzmann equation
University of Duisburg-Essen（杜伊斯堡-埃森大学）；RICAM, Austrian Academy of Sciences（奥地利科学院计算与应用数学研究所）；University of Jyväskylä（于韦斯屈莱大学）；V.A. Steklov Institute of Mathematics at St. Petersburg（圣彼得堡瓦列里·阿诺尔多维奇·斯捷克洛夫数学研究所）
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
TOBB University of Economics and Technology（TOBB经济与技术大学）；Louisiana State University（路易斯安那州立大学）；Sandia National Laboratories（桑迪亚国家实验室）；University of California San Diego（加州大学圣地亚哥分校）
Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation
University of Duisburg-Essen（杜伊斯堡-埃森大学）；RICAM, Austrian Academy of Sciences（奥地利科学院RICAM研究所）；University of Jyväskylä（于韦斯屈莱大学）；V. A.˜Steklov Institute of Mathematics in St.˜Petersburg（圣彼得堡V.A.斯捷洛夫数学研究所）
