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Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

Communications in Computational Physics 2012年第1期11卷 179-214页

作者： M.Holst J.A.McCammon Z.Yu Y.C.Zhou Y.Zhu Department of Mathematics University of California at San DiegoLa JollaCA 92093USA Department of Physics University of California at San DiegoLa JollaCA 92093USA Department of Chemistry&Biochemistry University of California at San DiegoLa JollaCA 92093USA Center for Theoretical Biological Physics(CTBP) University of California at San DiegoLa JollaCA 92093USA National Biomedical Computation Resource(NBCR) University of California at San DiegoLa JollaCA 92093USA Howard Hughes Medical Institute University of California at San DiegoLa JollaCA 92093USA

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 ***,in practical implementation,this two-term regularization exhibits numerical ***,we examine a variation of this regularization technique which can be shown to be less susceptible to such *** establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element *** show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original *** 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 *** 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¥*** 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 *** of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at *** stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regula

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

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