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We derive a numerical method for Darcy flow, hence also for Poisson’s equation in first order form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is its discretization on simplicial complexes such as triangle and tetrahedral meshes. We start by rewriting the governing… (More)

- Stephen D. Bond, Jehanzeb Hameed Chaudhry, Eric C. Cyr, Luke N. Olson
- Journal of Computational Chemistry
- 2010

The Poisson-Boltzmann equation is an important tool in modeling solvent in biomolecular systems. In this article, we focus on numerical approximations to the electrostatic potential expressed in the regularized linear Poisson-Boltzmann equation. We expose the flux directly through a first-order system form of the equation. Using this formulation, we propose… (More)

- Jehanzeb Hameed Chaudhry, Jeffrey Comer, Aleksei Aksimentiev, Luke N. Olson
- Communications in computational physics
- 2014

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and… (More)

- Jehanzeb Hameed Chaudhry, Stephen D. Bond, Luke N. Olson
- J. Sci. Comput.
- 2011

The inclusion of steric effects is important when determining the electrostatic potential near a solute surface. We consider a modified form of the Poisson-Boltzmann equation, often called the Poisson-Bikerman equation, in order to model these effects. The modifications lead to bounded ionic concentration profiles and are consistent with the… (More)

Implicit Explicit (IMEX) schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations. We develop accurate a posteriori error estimates for a user-defined quantity of interest for two classes of multistep IMEX schemes for advection-diffusion-reaction problems. The analysis… (More)

- Jehanzeb Hameed Chaudhry, Stephen D. Bond, Luke N. Olson
- Applied Mathematics and Computation
- 2012

The finite element methodology has become a standard framework for approximating the solution to the Poisson-Boltzmann equation in many biological applications. In this article, we examine the numerical efficacy of least-squares finite element methods for the linearized form of the equations. In particular, we highlight the utility of a first-order form,… (More)

- THROUGH A QUANTITY-OF-INTEREST, Jehanzeb Hameed Chaudhry, Eric C. Cyr, Thomas A. Manteuffel, Luke N. Olson, Lei Tang
- 2014

In this paper we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-ofinterest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest.… (More)

The finite element methodology has become a standard framework for approximating the solution to the Poisson–Boltzmann equation in many biological applications. In this article, we examine the numerical efficacy of least-squares finite element methods for the linearized form of the equations. In particular, we highlight the utility of a first-order form,… (More)

We derive, implement, and test a posteriori error estimates for numerical methods for a non-autonomous linear system that involve iterative solution of the discrete equations. We consider two iterations: the Picard iteration and the Jacobi iteration for solving the discrete matrix-vector equations. To carry out the analysis, we define an appropriate adjoint… (More)

- Jehanzeb Hameed Chaudhry, Eric C. Cyr, Kuo Liu, Thomas A. Manteuffel, Luke N. Olson, Lei Tang
- SIAM J. Numerical Analysis
- 2014

In this paper we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-ofinterest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest.… (More)