Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

Abstract

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the PoissonBoltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this twoterm regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L ∞ estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

A domain decomposition method for the polarizable continuum model based on the solvent excluded surface

“…First, one can obtain a generalized Poisson (GP) equation if ρ is independent on ψ as for the non-ionic solvent and second, a Poisson-Boltzmann equation [16] is obtained if the solvent contains ions whose movement is accounted for by Boltzmann statistics. Since the end of 1980s, many codes have been proposed for solving the Poisson-Boltzmann (or Poisson) equation for example using the boundary element method [7,35], the finite element method [2,24] or the finite difference method including UHBD [15], DelPhi [30], APBS [3,17,27] and the other work [18]. In particular, the APBS software is popular and widely used which can calculate the biomolecular electrostatics for large molecules.…”

Error Estimator Using Higher Order FEM for an Interface Problem

“…We will denote the piecewise constant function defined on Ω : The sum of (24) and (25) for every (27) Taking into account the flux jump (5) in the transmission equation, we have…”

“…That is, a treatment exclusively on the surface Γ without recourse to a solver in Ω is so far not sufficient. Holst [27] [28] [29] [30] is one of the most prominent specialists of PBE using FEM. His work seems to be extensively based on piecewise linear variational formulation.…”

Multilevel correction adaptive finite element method for semilinear elliptic equation

“…Besides for the linear boundary value problem, AFEM is also an efficient method for nonlinear elliptic equations (see, e.g., [10], [11]) and eigenvalue problems (see, e.g., [12], [19]). In this paper, a new adaptive scheme is designed for the semilinear elliptic problem based on the multilevel correction method (see [13]).…”