Browsing by Author "Chidyagwai, Prince"

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    A Two-grid Method for Coupled Free Flow with Porous Media Flow
    (2010-06) Chidyagwai, Prince; Riviere, Beatrice
    This paper presents a two-grid method for solving systems of partial differential equations modelling free flow coupled with porous media flow. This work considers both the coupled Stokes and Darcy as well as the coupled Navier-Stokes and Darcy problems. The numerical schemes proposed are based on combinations of the continuous finite element method and the discontinuous Galerkin method. Numerical errors and convergence rates for solutions obtained from the two-grid method are presented. CPU times for the two-grid algorithm are shown to be significantly less than those obtained by solving the fully coupled problem.
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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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    Coupling surface flow with porous media flow
    (2010) Chidyagwai, Prince; Riviere, Beatrice M.
    This thesis proposes a model for the interaction between ground flow and surface flow using a coupled system of the Navier-Stokes and Darcy equations. The coupling of surface flow with porous media flow has important applications in science and engineering. This work is motivated by applications to geo-sciences. This work couples the two flows using interface conditions that incorporate the continuity of the normal component, the balance of forces and the Beaver-Joseph-Saffman Law. The balance of forces condition can be written with or without inertial forces from the free fluid region. This thesis provides both theoretical and numerical analysis of the effect of the inertial forces on the model. Flow in porous media is often simulated over large domains in which the actual permeability is heterogeneous with discontinuities across the domain. The discontinuous Galerkin method is well suited to handle this problem. On the other hand, the continuous finite element is adequate for the free flow problems considered in this work. As a result this thesis proposes coupling the continuous finite element method in the free flow region with the discontinuous Galerkin method in the porous medium. Existence and uniqueness results of a weak solution and numerical scheme are proved. This work also provides derivations of optimal a priori error estimates for the numerical scheme. A two-grid approach to solving the coupled problem is analyzed. This method will decouple the problem naturally into two problems, one in the free flow domain and other in the porous medium. In applications for this model, it is often the case that the areas of interest (faults, kinks) in the porous medium are small compared to the rest of the domain. In view of this fact, the rest of the thesis is dedicated to a coupling of the Discontinuous Galerkin method in the problem areas with a cheaper method on the rest of the domain. The finite volume method will be coupled with the Discontinuous Galerkin method on parts of the domain on which the permeability field varies gradually to decrease the problem sizes and thus make the scheme more efficient.
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    On the Coupling of Finite Volume and Discontinuous Galerkin Method for Elliptic Problems
    (2010-03) Chidyagwai, Prince; Mishev, Ilya; Riviere, Beatrice
    The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.