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# Poisson Boltzmann Equation

The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation. SIAM Journal on Numerical Analysis, 45(6):2298--2320, 2007.

ABSTRACT: A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson--Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson--Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson--Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson--Boltzmann equation does not appear to have been previously studied in detail theoret- ically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson--Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson--Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.

The Poisson-Boltzmann Equation: Analysis and Multilevel Numerical Solution. , ():, 1994.

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Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications. Commun. Comput. Phys., 3():973-1009, 2008.

Adaptive multilevel finite element solution of the {Poisson-Boltzmann} Equations {II}: Refinement at Solvent-Accessible Surfaces in Biomolecular Systems. J. Comput. Chem., 21():1343-1352, 2000.

The adaptive multilevel finite element solution of the {Poisson-Boltzmann} equations on massively parallel computers. IBM Journal of Research and Development, ():427-438, 2001.

Steric effects in electrolytes: A modified {Poisson-Boltzmann} equation. Physical Review Letters, 79(3):435--438, 1997.

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation.. Journal of Computational Chemistry, 25(7):935-955, 2004.

On removal of charge singularity in Poisson--Boltzmann equation. The Journal of Chemical Physics, 130(14):145101, 2009.

A mortar finite element approximation for the linear {Poisson-Boltzmann} equation. Applied Mathematics and Computation, 164(1):11--23, 2005.

Adaptive multilevel finite element solution of the Poisson-Boltzmann Equations I: Algorithms and Examples. J. Comput. Chem., 21():1319-1342, 2000.

Boundary-value problems of the {Poisson-Boltzmann} equation. preprint, ():, 2006.

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Numerical solution of the nonlinear Poisson-Boltzmann equation: Developing more robust and efficient methods. Journal computational chemistry, 16(3):337-364, 1995.

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The {Poisson-Boltzmann} equation: approximation theory, regularization by singular functions, and adaptive techniques. preprint, ():, 2006.

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Classical electrostatics in biology and chemistry. Science, 268(5214):1144 -- 1149, 1995.

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Continuum Solvation Model: computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation. Computer Physics Communications, 111():59--75, 1998.

{Sobolev preconditioning for the Poisson--Boltzmann equation}. Computer Methods in Applied Mechanics and Engineering, 181(4):425--436, 2000.

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Electrostatic interactions in macromolecules: theory and applications. Annu. Kev. Biophys. Biophys. Chem., 19():301--332, 1990.

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Solution of the Nonlinear Poisson-Boltzmann Equation Using Pseudo-transient Continuation and the Finite Element Method. Journal of Colloid and Interface Science, 247():62--79, 2002.

ABSTRACT: The nonlinear Poisson-Boltzmann (PB) equation is solved using Newton-Krylov iterations coupled with pseudo-transient continuation. The PB potential is used to compute the electrostatic energy and evaluate the force on a user-specified contour. The PB solver is embedded in a existing, 3D, massively parallel, unstructured-grid, finite element code. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions \“regulating\” the surface charge. Stability and robustness are proved using results for backward Euler differencing of diffusion equations. Potentials and energies of charged spheres and plates are computed and results compared to analysis. An approximation to the potential of the nonlinear, spherical charge is derived by combining two analytic formulae. The potential and force due to a conical probe interacting with a flat plate are computed for two types of boundary conditions: constant potential and constant charge. The second case is compared with direct force measurements by chemical force microscopy. The problem is highly nonlinear--surface potentials of the linear and nonlinear PB equations differ by over an order of magnitude. Comparison of the simulated and experimentally measured forces shows that approximately half of the surface carboxylic acid groups, of density 1/(0.2 nm2), ionize in the electrolyte implying surface charges of 0.4 C/m2, surface potentials of 0.27 V, and a force of 0.6 nN when the probe and plate are 8.7 nm apart.

Poisson-Boltzmann solvents in molecular dynamics simulations. Commun. Comput. Phys., 3():1010-1031, 2008.

A Jump Condition Capturing Finite Difference Scheme for Elliptic Interface Problems. SIAM Journal of Scientific Computing, 25(5):1479--1496, 2004.

ABSTRACT: We propose a simple finite difference scheme for the elliptic interface problem with a discontinuous diffusion coefficient using a body-fitted curvilinear coordinate system. The resulting matrix is symmetric and positive definite. Standard techniques of acceleration such as PCG and multigrid can be used to invert the matrix. The main advantage of the scheme is its simplicity: the entries of the matrix are simply the centered difference second order approximation of the metric tensor $g^{\alpha\beta}$. In addition, the interface jump conditions are naturally built into the finite difference discretization. No interpolation/extrapolation process is involved in the derivation of the scheme. Both the solution and the flux are observed to have second order accuracy.

A new minimization protocol for solving nonlinear {Poisson-Boltzmann} mortar finite element equation. accepted, BIT Numerical Mathematics, ():, 2007.

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Some Studies on Finite Element Computing for the Poisson-Boltzmann Equation. , ():, 2005.