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Title:Finite Element Methods for Implicit Solvent Models
Author(s):Hameed Chaudhry, Jehanzeb
Contributor(s):Olson, Luke
Subject(s):Finite Element
Poisson Nernst Planck
Least Squares
Quantity of interest
Abstract:In this thesis, we develop finite element methods (FEMs) for implicit solvent models. Implicit models treat the solvent in biomolecular systems as bulk continuum and are thus computationally efficient. The Poisson-Boltzmann equation (PBE) is one important example. We design a FEM for the linear PBE by directly exposing the flux through a first-order system. We propose a system that yields a tractable least-squares finite element formulation and establish theory to support this approach. Moreover, we explore the impact of weighting and the choice of elements on conditioning and adaptive refinement. In a series of numerical experiments, we compare the finite element methods when applied to the problem of computing the solvation free energy for realistic molecules of varying size. The inclusion of steric effects is important in regions of high potential. Hence, we consider a modified PBE, in order to model these effects. We establish well-posedness of the weak problem along with convergence of an associated finite element formulation. We also examine several practical considerations such as conditioning of the linearized form of the nonlinear modified Poisson-Boltzmann equation, implications in numerical evaluation of the modified form, and utility of the modified equation in the context of the classical Poisson-Boltzmann equation. The modified Poisson-Nernst-Planck equations model the dynamics of ions in solvated systems, while accounting for steric effects. We develop an efficient SUPG FEM method to prevent spurious values in regions there are high repulsive potentials. Our method takes into account conservation of ions and periodic boundary conditions common in molecular dynamics simulations. We apply our method to compute current due to the flow of ions through a nanopore. Finally, we develop the theoretical and algorithmic framework for efficiently including quantities-of-interest (QOIs) into the least-squares framework. We show existence, uniqueness, and implications from a linear algebra point-of-view. Furthermore, we develop bounds on the error of the solution and the QOI. We outline an adaptive refinement algorithm based on our approach and support our approach with numerical result for several application areas, including solvation free energy for the PBE.
Issue Date:2011-07-18
Genre:Dissertation / Thesis
Publication Status:unpublished
Peer Reviewed:not peer reviewed
Date Available in IDEALS:2011-07-18

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