Browsing by Author "Rivière, Béatrice M."

Now showing 1 - 4 of 4
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    Analysis of Two Mathematical Models for the Coupled Navier-Stokes/Darcy Problem
    (2009-04) Chidyagwai, Prince; Rivière, Béatrice M.
    This paper introduces and analyzes two models coupling of incompressible Navier-Stokes equations with the porous media flow equations. A numerical method that uses continuous finite elements in the incompressible flow region and discontinuous finite elements in the porous medium, is proposed. Existence and uniqueness results under small data condition of the numerical solution are proved. Optimal a priori error estimates are derived. Numerical examples comparing the two models are provided.
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    Analysis of Weak Solutions For the Fully Coupled Stokes-Darcy-Transport Problem
    (2009-11) Cesmelioglu, Aycil; Rivière, Béatrice M.
    This paper analyzes the surface/subsurface flow coupled with transport. The flow is modeled by the coupling of Stokes and Darcy equations. The transport is modeled by a convection-dominated parabolic equation. The two-way coupling between flow and transport is nonlinear and it is done via the velocity field and the viscosity. This problem arises from a variety of natural phenomena such as the contamination of the groundwater through rivers. The main result is existence and stability bounds of a weak solution.
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    Convergence of a Discontinuous Galerkin Method For the Miscible Displacement Under Minimal Regularity
    (2009-05) Rivière, Béatrice M.; Walkington, Noel
    Discontinuous Galerkin time discretizations are combined with the mixed finite element and continuous finite element methods to solve the miscible displacement problem. Stable schemes of arbitrary order in space and time are obtained. Under minimal regularity assumptions on the data, convergence of the scheme is proved by using compactness results for functions that may be discontinuous in time.
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    Numerical Analysis of Nonlinear Boundary Integral Equations Arising in Molecular Biology
    (2019-04-18) Klotz, Thomas S; Rivière, Béatrice M.; Knepley, Matthew G.
    The molecular electrostatics problem, which asks for the potential generated by a charged solute suspended in a dielectric solvent, is of great importance in computational biology. Poisson models, which treat the solvent as a dielectric continuum, have inherent inaccuracies which can ruin energy predictions. These inaccuracies are primarily due to the inability of continuum models to capture the structure of solvent molecules in close proximity to the solute. A common approach to overcome these inaccuracies is to adjust the dielectric boundary by changing atomic radii. This adjustment procedure can accurately reproduce the expected solvation free energy, but fails to predict thermodynamic behavior. The Solvation Layer Interface Method (SLIC) replaces the standard dielectric boundary condition in Poisson models with a nonlinear boundary condition which accounts for the small-scale arrangement of solvent molecules close to the dielectric interface. Remarkably, SLIC retains the accuracy of Poisson models and furthermore predicts solvation entropies and heat capacities, while removing the need to adjust atomic radii. In this thesis, we perform foundational numerical analysis for the SLIC model. The first major result is a proof that a solution exists for the nonlinear boundary integral equation arising in the SLIC model. We are able to do this by proving existence for an auxiliary equation whose solutions correspond to the SLIC model's equation. Next, we prove that solutions to the SLIC model are unique for spherical geometries, which are common in biological solutes. Finally, we have experimented with nonlinear solvers for the nonlinear BIE, such as Anderson Acceleration, as well as two discretization techniques, in order to provide scalable numerical methods which can be applied to a variety of problems in drug design and delivery.