Browsing by Author "Wheeler, Mary F."
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Item 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.Item 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.Item A numerical model of processes governing groundwater contaminant transport(1983) Springer, Nina K.; Bedient, Philip B.; Wheeler, Mary F.; Tomson, Mason B.The purpose of this research was to develop a numerical model to simulate transport of solutes in groundwater which could be used to 1) predict existing or potential groundwater contamination and 2) aid determination of governing transport mechanisms in the field. The numerical model developed successfully uses a Laplace Modified Alternating Direction Implicit solution scheme to solve, for the first time in this application, a finite-difference expression for the governing transport equations. The model incorporates transport processes of advection, dilution, physical dispersion, reaction, and adsorption. The major advantages of the model are its non-iterative solution scheme which allows quick solutions, its non-time-dependent stability, and its wide and flexible range of transport process simulation capabilities. The transport model was linked to an existing groundwater flow model, GWSIM-II. This model, a modification of the Prickett-Lonnquist model used by the Texas Department of Water Resources, calculates groundwater heads over time by solving a finite difference form of the unsteady state pressure equation. The resulting combined model was called MPACTS; a model of physical and chemical transport in the subsurface. The accuracy of the model was verified by comparing MPACTS solutions and analytical solutions to simple problems. The model solutions to these problems are quite accurate when reasonable model parameters are used. In order to demonstrate application of the model to field data, MPACTS was used to simulate groundwater contamination at a rapid soil infiltration sewage treatment facility at Fort Devens, Massachusetts. Contamination by chlorides and a trace level organic, tetrachloroethylene, was modeled. The site was fairly well documented with respect to groundwater heads and chloride concentration in the immediate vicinity of the basins, but was not defined elsewhere. The lack of good definition over the whole modeled area, typical of field studies, was reflected in inaccuracies in simulated contaminant concentrations. However, the simulations were sufficiently accurate to be used for estimation of contaminant contours at the site.Item A priori error estimates of finite element models of systems of shallow water equations(1998) Martinez, Monica Lucia; Wheeler, Mary F.; Dawson, Clint N.In recent years, there has been much interest in the numerical solution of shallow water equations. The numerical procedure used to solve the shallow water equations must resolve the physics of the problem without introducing spurious oscillations or excessive numerical diffusion. Staggered-grid finite difference methods have been used extensively in modeling surface flow without introducing spurious oscillations. Finite element methods, permitting a high degree of grid flexibility for complex geometries and facilitating grid refinement near land boundaries to resolve important processes, have become much more prevalent. However, early finite element simulations of shallow water systems were plagued by spurious oscillations and the various methods introduced to eliminate these oscillations through artificial diffusion were generally unsuccessful due to excessive damping of physical components of the solution. Here, we give a brief overview on some finite element models of the shallow water equations, with particular attention given to the wave and characteristic formulations. In the literature, standard analysis, based on Fourier decompositions of these methods, has always neglected contributions from the nonlinear terms. We derive ${\cal L}\sp{\infty} ((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for both the continuous-time and discrete-time Galerkin approximation to the nonlinear wave model, finding these to be optimal in ${\cal H}\sp1(\Omega).$ Finally, we derive ${\cal L}\sp{\infty}((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for our proposed Characteristic-Galerkin approximation to the nonlinear primitive model. We find these estimates to be optimal in ${\cal H}\sp1(\Omega)$ but with less restrictive time-step constraints when compared to the Galerkin estimates for the wave model.Item A Raster scan visibility algorithm for parametric surfaces(1984) Montgomery, Jerome; Akin, J. E.; Wierum, Frederic A.; Wheeler, Mary F.A raster scan visibility algorithm is presented for contouring on parametric surfaces. The algorithm entails three-dimensional aspects of coloring, shading, and visibility of points. The approach uses parametric interpolation functions as the primary basis in computations. This technique along with a predictor-corrector method makes contour line tracing on each element more accurate. Continuous color variations on each surface are produced through the use of interpolation functions. The color at every display point can be assigned in proportion to the value of the quantity of interest at that pixel. Shading is included involving several possibilities of the shading rule. They may include ambient light, diffuse reflection, specular reflection, shadows, or transparency. A combination of these may also be included. All of the graphics generated using this algorithm can be displayed on color graphics, as well as, non-color graphics display terminals, and line printers. The algorithm could be used as a segment of a complete computer graphics package.Item A study of reactive transport phenomena in porous media(1997) Saaf, Fredrik Edvard; Wheeler, Mary F.The numerical modeling of reactive transport in a porous medium has important applications in hydrology, the earth sciences and in numerous industrial processes. However, realistic simulations involving a large number of chemical species undergoing simultaneous transport and chemical transformation present a significant computational challenge, particularly in multiple spatial dimensions. A framework for analyzing the chemical batch problem is first introduced, which is sufficiently general to allow for reactions of both equilibrium and kinetic type. The governing equations for reactive transport of a single flowing phase through a porous medium are presented next, and a classification based on the nature of the reactive system is established. A computer module for the equilibrium problem is developed, based on a novel application of the interior-point algorithm for nonlinear programming. Among its advantages are good global convergence and automatic selection of mineral phases. To handle kinetic reactions, the equilibrium module is embedded in a time-integration framework using explicit ODE integrators. Reactive transport of species is achieved through operator-splitting, which enables a straightforward incorporation of the batch module into the existing parallel, three-dimensional, single-phase flow and transport simulator PARSim1. Numerical results are presented which demonstrate the correctness of the computer program for major classes of geochemistry problems, including ion-exchange, precipitation/dissolution, adsorption, aqueous complexation and redox reactions.Item Error estimates for Godunov mixed methods for nonlinear parabolic equations(1988) Dawson, Clint; Wheeler, Mary F.Many computational fluids problems are described by nonlinear parabolic partial differential equations. These equations generally involve advection (transport) and a small diffusion term, and in some cases, chemical reactions. In almost all cases they must be solved numerically, which means approximating steep fronts, and handling time-scale effects caused by the advective and reactive processes. We present a time-splitting algorithm for solving such parabolic problems in one space dimension. This algorithm, referred to as the Godunov-mixed method, involves splitting the differential equation into its advective, diffusive, and reactive components, and solving each piece sequentially. Advection is approximated by a Godunov-type procedure, and diffusion by a mixed finite element method. Reactions split into an ordinary differential equation, which is handled by integration in time. The particular scheme presented here combines the higher-order Godunov MUSCL algorithm with the lowest-order mixed method. This splitting approach is capable of resolving steep fronts and handling the time-scale effects caused by rapid advection and instantaneous reactions. The scheme as applied to various boundary value problems satisfies maximum principles. The boundary conditions considered include Dirichlet, Neumann and mixed boundary conditions. These maximum principles mimic discretely the classical maximum principles satisfied by the true solution. The major results of this thesis are discrete L$\sp\infty$(L$\sp2$) and L$\sp\infty$(L$\sp1$) error estimates for the method assuming various combinations of the boundary conditions mentioned above. These estimates show that the scheme is essentially first-order in space and time in both norms; however, in the L$\sp1$ estimates, one sees a much weaker dependence on the lower bound of the diffusion coefficient than is usually derived in standard energy estimates. All of these estimates hold for uniform and non-uniform grid. Error estimates for a lower-order Godunov-mixed method for a fully nonlinear advection-diffusion-reaction problem are also considered. First-order estimates in L$\sp1$ are derived for this problem.Item Finite Element Approximations to the System of Shallow Water Equations, Part I: Continuous Time a Priori Error Estimates(1995-12) Chippada, S.; Dawson, Clint N.; Martinez, M.L.; Wheeler, Mary F.Various sophisticated finite element models for surface water flow exist in the literature. Gray, Kolar, Luettich, Lynch and Westerink have developed a hydrodynamic model based on the generalized wave continuity equation (GWCE) formulation, and have formulated a Galerkin finite element procedure based on combining the GWCE with the nonconservative momentum equations. Numerical experiments suggest that this method is robust, accurate and suppresses spurious oscillations which plague other models. We analyze a slightly modified Galerkin model which uses the conservative momentum equations (CME). For this GWCE-CME system of equations, we present an a priori error estimate based on an L² projection.Item Krylov-secant methods for solving large-scale systems of coupled nonlinear parabolic equations(1997) Klie, Hector Manuel; Wheeler, Mary F.This dissertation centers on two major aspects dictating the computational time of applications based on the solution of systems of coupled nonlinear parabolic equations: nonlinear and linear iterations. The former aspect leads to the conception of a novel way of reusing the Krylov information generated by GMRES for solving linear systems arising within a Newton method. The approach stems from theory recently developed on a nonlinear version of the Eirola-Nevanlinna, algorithm (originally for solving non-symmetric linear systems) which is capable of converging twice as fast as Broyden's method. A secant update strategy of the Hessenberg matrix resulting from the Arnoldi process in GMRES amounts to reflecting a secant update of the current Jacobian with the rank-one term projected onto the generated Krylov subspace (Krylov-Broyden update). This allows the design of a new nonlinear Krylov-Eirola-Nevanlinna (KEN) algorithm and a higher-order version of Newton's method (HOKN) as well. The underlying development is also auspicious to replace the use of GMRES by cheaper Richardson iterations for the sake of fulfilling the inexact Newton condition. Hence, three algorithms derived from Newton's method, Broyden's method and the nonlinear Eirola-Nevanlinna algorithm are proposed as a part of a new family of hybrid Krylov-secant methods. All five algorithms are shown to be computationally more economical than their Newton and quasi-Newton counterparts. The aspect of linear iterations complements the present research with an analysis on nested or inner-outer iterations to efficiently precondition Krylov subspace iterative solvers for linear systems arising from systems of coupled nonlinear equations. These preconditioners are called two-stage preconditioners and are developed on the basis of a simple but effective decoupling strategy. Their analysis is restricted to the particular class of problems arising in multi-phase flow phenomena modeled by systems of coupled nonlinear parabolic equations. The resulting approach outperforms fairly robust and standard preconditioners that "blindly" precondition the entire coupled linear system. Theoretical discussion and computational experiments show the suitability that both linear and nonlinear aspects undertaken in this research have for large scale implementations.Item Krylov-Secant Methods for Solving Systems of Nonlinear Equations(1995-09) Klíe, Héctor; Ramé, Marcelo; Wheeler, Mary F.We present a novel way of reusing the Krylov information generated by GMRES for solving the linear system arising within a Newton method. Our approach departs from the theory of secant preconditioners developed by Martinez and then combines secant updates of the Hessenberg matrix generated by the Arnoldi process in GMRES, the Richardsn iteration and limited memory quasi-Newton compact representations to generate descent directions for each Newton step. The proposed method allows to reflect - without explicitly computing them - secant updates of the Jacobian matrix that lead us to skip GMRES for the benefit of satisfying the Dembo-Eisenstat-Steihaug condition. Hence, the resulting method turns out to be computationally more economical than traditional inexact Newton implementations. Computational experiments reveal the suitability of this approach for large scale problems in several application contexts.Item Mixed finite element methods for flow in porous media(1996) Yotov, Ivan Petrov; Wheeler, Mary F.Mixed finite element discritizations for problems arising in flow in porous medium applications are considered. We first study second order elliptic equations which model single phase flow. We consider the recently introduced expanded mixed method. Combined with global mapping techniques, the method is suitable for full conductivity tensors and general geometry domains. In the case of the lowest order Raviart-Thomas space, quadrature rules reduce the method to cell-centered finite differences, making it very efficient computationally. We consider problems with discontinuous coefficients on multiblock domains. To obtain accurate approximations, we enhance the scheme by introducing Lagrange multiplier pressures along subdomain boundaries and coefficient discontinuities. This modification comes at no extra computational cost, if the method is implemented in parallel, using non-overlapping domain decomposition algorithms. Moreover, for regular solutions, it provides optimal convergence and discrete superconvergence for both pressure and velocity. We next consider the standard mixed finite element method on non-matching grids. We introduce mortar pressures along the non-matching interfaces. The mortar space is chosen to have higher approximability than the normal trace of the velocity spaces. The method is shown to be optimally convergent for all variables. Superconvergence for the subdomain pressures and, if the tensor coefficient is diagonal, for the velocities and the mortar pressures is also proven. We also consider the expanded mixed method on general geometry multiblock domains with non-matching grids. We analyze the resulting finite difference scheme and show superconvergence for all variables. Efficiency is not sacrificed by adding the mortar pressures. The computational complexity is shown to be comparable to the one on matching grids. Numerical results are presented, that verify the theory. We finally consider the mixed finite element discretizations for the nonlinear multi-phase flow system. The system is reformulated as a pressure and a saturation equation. The methods described above are directly applied to the elliptic or parabolic pressure equation. We present an analysis of a mixed method on non-matching grids for the saturation equation of degenerate parabolic type.Item Mixed finite element methods for variably saturated subsurface flow(1996) San Soucie, Carol Ann; Dawson, Clint N.; Wheeler, Mary F.The flow of water through variably saturated subsurface media is commonly modeled by Richards' equation, a nonlinear and possibly degenerate partial differential equation. Due to the nonlinearities, this equation is difficult to solve analytically and the literature reveals dozens of papers devoted to finding numerical solutions. However, the literature also reveals a lack of two important research topics. First, no a priori error analysis exists for one of the discretization schemes most often used in discretizing Richards' equation, cell-centered finite differences. The expanded mixed finite element method reduces to cell-centered finite differences for the case of the lowest-order discrete space and certain quadrature rules. Expanded mixed methods are useful because this simplification occurs even for the case of a full coefficient tensor. There has been no analysis of expanded mixed methods applied to Richards' equation. Second, no results from parallel computer codes have been published. With parallel computer technology, larger and more computationally intensive problems can be solved. However, in order to get good performance from these machines, programs must be designed specifically to take advantage of the parallelism. We present an analysis of the mixed finite element applied to Richards' equation accounting for the two types of degeneracies that can arise. We also consider and analyze a two-level method for handling some of the nonlinearities in the equation. Lastly, we present results from a parallel Richards' equation solve code that uses the expanded mixed method for discretization.Item Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Finite Differences(1994-01) Arbogast, Todd; Wheeler, Mary F.; Yotov, IvanWe 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.Item 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.Item Parallel Domain Decomposition Method for Mixed Finite Elements for Elliptic Partial Differential Equations(1990-11) Cowsar, Lawrence C.; Wheeler, Mary F.In this paper we develop a parallel domain decomposition method for mixed finite element methods. This algorithm is based on a procedure first formulated by Glowinski and Wheeler for a two subdomain problem. This present work involves extensions of the above method to an arbitrary number of subdomains with an inner product modification and multilevel acceleration. Both Neumann and Dirichlet boundary conditions are treated. Numerical experiments performed on the Intel iPSC/860 Hypercube are presented and indicate that this approach is scalable and fairly insensitive to variation in coefficients.Item Preconditioner schemes for elliptic saddle-point matrices based upon Jacobi multi-band polynomial matrices(1995) Parr, Victor J.; Wheeler, Mary F.Simulation of flow in porous media requires the numerical approximation of elliptic partial differential equations. Mixed finite element methods are frequently employed, because of local mass conservation and accurate approximation of both pressure and velocity. Mixed methods give rise to "elliptic" saddle-point (ESP) matrices, which are difficult to solve numerically. In addition, the problems to be modelled in ground water flow require that the hydraulic conductivity or absolute permeability be a tensor, which adds additional complexity to the resulting saddle-point matrices. This research develops several preconditioners for restarted GMRES solution of the ESP linear systems. These preconditioners are based on a new class of polynomial matrices, which we refer to as Multi-band Jacobi Polynomial (JMP) matrices. Applications of these preconditioners to the numerical solution of two and three spatial dimensional flow equations with tensor coefficients using rectangular lowest order Raviart-Thomas spaces are presented.Item 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.Item Some domain decomposition and multigrid preconditioners for hybrid mixed finite elements(1994) Cowsar, Lawrence Charles; Wheeler, Mary F.Discretizations of self-adjoint, linear, second-order, uniformly elliptic partial differential equations by hybrid mixed finite elements lead to large, ill-conditioned saddle-point problems. By eliminating the flux variable, a reduced problem is formed that is symmetric and positive definite but still large and ill-conditioned. Several domain decomposition and multigrid preconditioners are applied to the reduced problem, and bounds on their asymptotic rates of convergence are derived. Two Schwarz domain decomposition methods are shown to converge at least as fast asymptotically as the same methods applied to conforming linear finite element discretizations. In particular, for both the standard additive overlapping Schwarz method of Dryja and Widlund and one of the interfacial Schwarz methods of Smith, it is proven that the rates of convergence of the methods are uniformly bounded with respect to the mesh size in both two and three dimensions under standard assumptions. Several multigrid preconditioners are constructed for the reduced problem including a generalization of a method due to Bramble, Pasciak and Xu and an adaptation of methods of Wohlmuth and Hoppe. A common feature of these multigrid methods is the use of conforming finite element spaces on the coarser grids. Uniform convergence rates are proven for most of the methods and numerical results that verify the bounds are reported. A mixed finite element discretization of a simplified model of sediment transport in a two dimensional periodic channel is also described. The results of two simulations that employ one of the multigrid preconditioners are reported.Item Superconvergence of Recovered Gradients of Discrete Time/Piecewise Linear Galerkin Approximations for Linear and Nonlinear Parabolic Problems(1992-03) Wheeler, Mary F.; Whiteman, John R.Superconvergent error estimates in l2(H¹) and linfinity(H¹) norms are derived for recovered gradients of finite difference in time/piecewise linear Galerkin approximations in space for linear and quasi-nonlinear parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context, and covers problems in regions with non-smooth boundaries under certain assumptions on the regularity of the solutions.Item Time-Splitting Methods for Advection-Diffusion-Reaction Equations Arising in Contaminant Transport(1992-03) Dawson, Clint N.; Wheeler, Mary F.Two time-splitting methods for advection-diffusion-reaction problems are discussed and analyzed. Numerical results for the first approach applied to bioremediation of contaminants in groundwater are presented.