Browsing by Author "Arbogast, Todd"

Now showing 1 - 11 of 11
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    A Characteristic-Mixed Method for Contaminant Transport and Miscible Displacement
    (1992-02) Arbogast, Todd; Chilakapati, Ashokkumar; Wheeler, Mary F.
    Recently, Arbogast and Wheeler have formulated and analyzed a modified method of characteristics-mixed method for approximating solutions to convection-diffusion equations. This scheme is theoretically mass conservative over each grid cell; it is approximately so in implementations. We consider application of this procedure to contaminant transport and to miscible displacement with unfavorable mobility ratio. Results in one, two, and three space dimensions are discussed.
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    A Characteristics-Mixed Finite Element Method for Advection Dominated Transport Problems
    (1992-11) Arbogast, Todd; Wheeler, Mary F.
    We define a new finite element method, called the characteristics-mixed method, for approximating the solution to an advection dominated transport problem. The method is based on a space-time variational form of the advection-diffusion equation. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the advective (i.e., hyperbolic) part of the equation. Thus the scheme uses a characteristic approximation to handle advection in time. This is combined with a low order mixed finite element spatial approximation of the equation. Boundary conditions are incorporated in a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. A post-processing step is included in the scheme in which the approximation to the scaler unknown is improved by utilizing the approximate vector flux. This has the effect of improving the rate of convergence of the method. We show that it is optimally convergent to order one in time and at least suboptimally convergent to order 3/2 in space.
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    A New Formulation of Mixed Finite Element Methods for Second Order Elliptic Problems
    (1991-05) Arbogast, Todd
    In this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation. We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method's finite element space Mh satisfies two conditions, then the two approximation methods are equivalent. These two conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. For any such mixed spaces defined on a geometrically regular partition of the domain, we can then easily construct appropriate conforming spaces Mh. We also present for several mixed methods alternative nonconforming spaces Mh that also satisfy the two conditions for equivalence.
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    Gravitational Forces in Dual-Porosity Models of Single Phase Flow
    (1991-03) Arbogast, Todd
    A dual porosity model is derived by the normal theory of homogenization. The model properly incorporates gravity in that it respects the equilibrium states of the medium.
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    Mixed Finite Element Methods as Finite Difference Methods for Solving Elliptic Equations on Triangular Elements
    (1993-11) Arbogast, Todd; Dawson, Clint; Keenan, Phil
    Several procedures of mixed finite element type for solving elliptic partial differential equations are presented. The efficient implementation of these approaches using the lowest-order Raviart-Thomas approximating spaces defined on triangular elements is discussed. A quadrature rule is given which reduces a mixed method to a finite difference method on triangles. This approach substantially reduces the complexity of the mixed finite element matrix, but may also lead to a loss of accuracy in the solution. An enhancement of this method is derived which combines numerical quadrature with Lagrange multipliers on certain element edges. The enhanced method regains the accuracy of the solution, with little additional cost if the geometry is sufficiently smooth. Numerical examples in two dimensions are given comparing the accuracy of the various methods.
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    Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Finite Differences
    (1994-01) Arbogast, Todd; Wheeler, Mary F.; Yotov, Ivan
    We develop the theory of an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, its gradient, and its flux (the tensor coefficient times the gradient). The expected optimal order approximations are obtained in the L² and H^{-s}-norms, and superconvergence is obtained between the L²-projection of the scalar variable and its approximation. The scheme is suitable for the case in which the coefficient is a tensor that may have zeros, since it does not need to be inverted.The resulting linear system is a saddle point problem. In the case of the lowest order Raviart-Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method, requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. We show that the optimal rates of convergence are retained; moreover, superconvergence is obtained for the scalar unknown as well as for its gradient and flux at certain discrete points. Computational results illustrate these theoretical results.
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    Numerical Methods for the Simulation of Flow in Root-Soil Systems
    (1991-04) Arbogast, Todd; Obeyesekere, Mandri; Wheeler, Mary F.
    We consider the numerical properties of approximation schemes for a model that simulates water transport in root-soil systems. The model given in this paper is a reformulation of a previously proposed model now defined completely in terms of the water potential. The system of equations consists of a parabolic partial differential equation which contains a nonlinear capacity term coupled to two linear ordinary differential equations. A closed form solution is obtained for one of the latter equations. Finite element and finite difference schemes are defined to approximate the solution of the coupled system, and optimal order error estimates are derived. A postprocessed water mass flux computation is also presented and shown to be superconvergent to the true flux. Computational results which verify the theoretical convergence rates are given.
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    Simplified Dual-Porosity Model for Two-Phase Flow
    (1992-02) Arbogast, Todd
    A model for two-phase, incompressible, immiscible fluid flow in a highly fractured porous medium is derived as a simplification of a much more detailed dual-porosity model. This simplified model has a nonlinear matrix-fracture interaction, and it is more general than similar existing "transfer function" models. It is computationally less complex than the detailed model, and simulation results are presented which assess any loss in accuracy. It is shown that the new model approximates capillary effects quite well, and better than similar existing models.
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    Simulation of Flow in Root-Soil Systems
    (1991-05) Arbogast, Todd; Obeyesekere, Mandri; Wheeler, Mary F.
    In this paper we develop a mathematical model of a root-soil system, and also accurate and efficient finite element and finite difference algorithms for approximating this model. The goal of our work is to develop an understanding of the properties of root systems, which can be modified by using genetic engineering techniques, in order to improve the performance of plants when water availability is limited. The results of some numerical simulations are presented, which demonstrate the effectiveness of genetic and physical changes to the root-soil system.
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    The Existence of Weak Solutions to Single Porosity and Simple Dual-Porosity Models of Two-Phase Incompressible Flow
    (1992-02) Arbogast, Todd
    It is shown that there exists a weak solution to a degenerate and singular elliptic-parabolic partial integro-differential system of equations. These equations model two-phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium. The full model is of dual-porosity type, though the single porosity case is covered by setting the matrix-to-fracture flow terms to zero. This matrix-to-fracture flow is modeled simply in terms of fracture quantities; that is, no distinct matrix equations arise. The equations are considered in global pressure formulation that is justified by appealing to a physical relation between the degeneracy of the wetting fluid's mobility and the singularity of the capillary pressure function. In this formulation, the elliptic and parabolic parts of the system are separated; hence, it is natural to consider various boundary conditions, including mixed nonhomogeneous, saturation dependent ones of the first three types. A weak solution is obtained as a limit of solutions to discrete time problems. The proof makes no use of the corresponding regularized system. The hypotheses required for various earlier results on single-porosity systems are weakened so that only physically relevant assumptions are made. In particular, the results cover the cases of a singular capillary pressure function, a pure Neumann boundary condition, and an arbitrary initial condition.