On developing stable finite element methods for pseudo-time simulation of biomolecular electrostatics

The Poisson-Boltzmann Equation (PBE) is a widely used implicit solvent model for the electrostatic analysis of solvated biomolecules. To address the exponential nonlinearity of the PBE, a pseudo-time approach has been developed in the literature, which completely suppresses the nonlinear instability through an analytic integration in a time splitting framework. This work aims to develop novel Finite Element Methods (FEMs) in this pseudo-time framework for solving the PBE. Two treatments to the singular charge sources are investigated, one directly applies the definition of the delta function in the variational formulation and the other avoids numerical approximation of the delta function by using a regularization formulation. To apply the proposed FEMs for both PBE and regularized PBE in real protein systems, a new tetrahedral mesh generator based on the minimal molecular surface definition is developed. With a body-fitted mesh, the proposed pseudo-time FEM solvers are more accurate than the existing pseudo-time finite difference solvers. Moreover, based on the implicit Euler time integration, the proposed FEMs are unconditionally stable for solvated proteins with source singularities and non-smooth potentials, so that they could be more efficient than the existing pseudo-time discontinuous Galerkin method based on the explicit Euler time stepping. Due to the unconditional stability, the proposed pseudo-time algorithms are free of blow-up or overflow issues, without resorting to any thresholding technique. Numerical experiments of several benchmark examples and free energy calculations of protein systems are conducted to validate the stability, accuracy, and robustness of the proposed PBE solvers.