@Article{CiCP-11-1,
author = {},
title = {Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation},
journal = {Communications in Computational Physics},
year = {2012},
volume = {11},
number = {1},
pages = {179--214},
abstract = {<p style="text-align: justify;">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 Poisson-Boltzmann equation, the
first provably convergent discretization and also allowed for the development of a
provably convergent AFEM. However, in practical implementation, this two-term 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 <em>a priori</em> 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&nbsp;continuous and discrete <em>a priori</em> <em>L</em><sup>∞</sup> estimates. 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.081009.130611a},
url = {https://global-sci.com/article/80689/adaptive-finite-element-modeling-techniques-for-the-poisson-boltzmann-equation}
}