Commun. Comput. Phys., 11 (2012), pp. 179-214.


Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

M. Holst 1*, J. A. McCammon 2, Z. Yu 3, Y. C. Zhou 4, Y. Zhu 5

1 Department of Mathematics, Department of Physics, Center for Theoretical Biological Physics (CTBP), and National Biomedical Computation Resource (NBCR), University of California at San Diego, La Jolla, CA 92093, USA.
2 Department of Chemistry & Biochemistry, Center for Theoretical Biological Physics (CTBP), National Biomedical Computation Resource (NBCR), and Howard Hughes Medical Institute, University of California at San Diego, La Jolla, CA 92093, USA.
3 Department of Mathematics and National Biomedical Computation Resource (NBCR), University of California at San Diego, La Jolla, CA 92093, USA.
4 Department of Mathematics and Center for Theoretical Biological Physics (CTBP), University of California at San Diego, La Jolla, CA 92093, USA.
5 Department of Mathematics and Howard Hughes Medical Institute, University of California at San Diego, La Jolla, CA 92093, USA.

Received 8 October 2009; Accepted (in revised version) 13 June 2011
Available online 8 September 2011
doi:10.4208/cicp.081009.130611a

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 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 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^\infty 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.

AMS subject classifications: 92-08, 65N06, 65N30, 65N38, 65N50, 92C05

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Key words: Poisson-Boltzmann equation, semi-linear partial differential equations, supercritical nonlinearity, singularity, a priori L^\infty estimates, existence, uniqueness, well-posedness, Galerkin methods, discrete a priori L^\infty estimates, quasi-optimal a priori error estimates, adaptive finite methods, contraction, convergence, optimality, surface and volume mesh generation, mesh improvement and decimation.

*Corresponding author.
Email: mholst@math.ucsd.edu (M. Holst)
 

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