Mathematical And Numerical Analysis For Two Types Of Flows

The Crouzeix-Raviart (C-R) finite element appeared in the 1973 pa-per [23] entitled conforming and nonconforming finite element methods for solving the stationary Stokes equations. It is a triangular/tetrahedral P1 finite element where the degrees of freedom consist of the values of the shape functions at the centers of the edges/faces. The functions in the finite ele-ment space defined by this element over a triangulation of the computational domain Ω are therefore only continuous at the middle points/centers of the edges/faces, which means that the finite element space is not a subspace of the Sobolev space H1(Ω) where second order elliptic boundary value prob-lems are posed. For this reason the C-R element is often referred to as the nonconforming P1 finite element, compared with the conforming PI or Courant element [22] where the degrees of freedom are associated with the vertices. It is also known as a Loof element [52] in the engineering com-munity. But the mathematical community came to know this finite element largely through the paper by Crouzeix and Raviart. For far more detailed information on C-R element, we could refer to [16,17]. In this thesis, we in-vestigate two types of fluid flows from both the mathematical and numerical viewpoints in which the velocity and pressure are approximated by noncon-forming C-R element and piecewise constants, respectively. We begin with the the Darcy-Forchheimer equation combined with incompressible/slightly compressible mass conservation equation while another case is the coupling of free fluid and porous media flows with mass transfer.The mathematical model for fluid flow in porous media is widely used in many scientific fields such as groundwater, environment science, the ex-ploration and production of oil and gas in petroleum reservoir and so on [5,7,63]. The model is based on the conservation laws for mass, momen-tum and energy of fluids. Usually the engineers focus on physical quantities such as pressure, fluid velocity, temperature, strain, stress and concentration of one component in miscible flow. However, many geological and fluid pa-rameters such as gravity, viscosity, compressibility, density, porosity, absolute permeability, relative permeability play an important role in this process and thus are introduced following the conservation laws, the physical quantities above considered. Due to heterogeneity and anisotropy, the introduced pa-rameters vary sharply, thus many physical quantities are in average sense and the model with the corresponding approximate form coincides with the practical physical process in a certain range. Under some reasonable assump-tions, the mathematical model for the fluid flow and transport processes can be simplified, but it is still shown to be a highly coupled time-dependent nonlinear partial differential equations. Because of the complex structure of the partial differential equations, analytical solutions only exist in some specific cases, thus it is significant and necessary to propose high-accurate and effective numerical schemes that keep physical properties of the system.Darcy’s law, provides the simplest linear relationship between the veloc-ity of creep flow and the gradient of pressure, which is supported by years of experimental data. The relationship is valid under the physically reasonable assumption that fluid flows are usually very slow and all the inertial (nonlin-ear) terms can be neglected [5]. A theoretical derivation of Darcy’s law can be found in [58,78]. Darcy flow is of great importance in many fields such as oil recovery and prevention of groundwater pollution.Forchheimer [37] conducted flow experiments in sandpacks and recog-nized that for moderate Reynolds numbers (Re> 0.1 approximately), Dar-cy’s law is not adequate and a nonlinear relationship is developed. In fact, he observed for a wide range of experimental data, the nonlinear term ap- peared to be quadratic, suggested by Forchheimer in 1901 [5]. While there continues to be some debate over the functionality of the Darcy-Forchheimer equation [6], nonlinearity has been verified experimentally, numerically [46] and theoretically [55,69]. The Darcy-Forchheimer equation continues to be used for predicting high velocity flow in porous media, especially in the vicinity of gas wells [3]. The derivation of the Forchheimer model can be found in [2,18,39,48,69,78]. The theoretical researches on the Forch-heimer model can be referred to [4,47,33,74]. This problem is nonlinear and non-degenerate of monotone type. This study is similar to the treatment of p-Laplacian model, quasi-Newtonian flows and Ladyzhenskaya flows. The mathematical tools and techniques dealing with such above models can be found in [30,31,34,35,44,72].In recent years, several numerical discretizations have been proposed for the Darcy-Forchheimer model. In [43,53] a primal nonconforming mixed finite element method is demonstrated while in [71] conforming mixed finite element method is given, which makes the resulting finite element matrix sparser and reduces the computer memory and CPU time required for solv-ing the problem iteratively. Park [62] adopts the dual mixed element for the slightly compressible case. Girault and Wheeler [43] prove the existence and uniqueness of a weak solution for the Darcy-Forchheimer problem. They pro-pose a discretization in mixed elements in which the velocity and the pressure are approximated by discontinuous constant elements and non-continuous linear functions, respectively (see [23]). They also present an alternating di-rections iterative method to solve the system. The convergences of both the iterative algorithm and the mixed element scheme are proved. Lopez et al. [53] carry out numerical tests of the methods studied in [43]. The system of nonlinear equations is also solved via the Newton method to compare the results with the method proposed in [43]. Furthermore, another mixed finite element space is proposed in which the approximation to the pressure is s-moother than the one demonstrated in [53]. Specifically, the finite element space approximates the velocity by constant functions in each triangle and the pressure by continuous functions whose restriction to each triangle is a degree-one polynomial. Furthermore, in [71], the authors present a theoret-ical study on the method and two new mixed finite element spaces are also proposed in which the approximation to the pressure with standard finite elements P1 are still kept and the velocity is approximated with C-R finite elements in one case and standard finite elements P1 in the other. However, the numerical and theoretical studies on the two newly proposed methods have not been done.This thesis aims to propose a new dual mixed finite element approx-imation for the Darcy-Forchheimer equation combined with incompressible mass conservation equation. The velocity and pressure are respectively ap-proximated by nonconforming C-R element and piecewise constants, which is usually regarded as a promising candidate for dealing with the Darcy-Stokes problem. In this case, we can obtain a second order elliptic partial differential equation of the pressure by eliminating the velocity. Then we proceed to give a research on the Darcy-Forchheimer equation coupled with slightly compressible mass conservation equation utilizing the same discrete numerical method in space while a second order parabolic partial differential equation of the pressure will be derived by eliminating the velocity.The coupling of free flow and porous media flow has been extensively s-tudied. The model of this problem is based on imposing the Stokes equations in the fluid region and Darcy’s law in the porous media region, coupled with the appropriate interface conditions, which consist of the Beavers-Joseph-Saffman condition [8,70], the continuity of flux and the balance of forces. Also it is challenging in both the mathematical and the numerical aspects: the Stokes and Darcy solutions have very different regularity properties, the tangential velocity may be discontinuous on the interface between the two regions, and because the dimension of the interface is one less than that of the domain, it is difficult to guarantee no loss in the regularity and the conver-gence rates when handling the weak formulation and the numerical scheme. There are a number of stable and convergent numerical methods developed for the coupled Stokes-Darcy flow system; see, e.g., [24,38,51,54,66]. One serious problem today is surface water and groundwater contamination re-sulting from leaky underground storage tanks, chemical spills, and various human activities. A model coupling the Stokes-Darcy equations with a trans-port equation can be used to study the spread of pollution released in the water and assess the danger. However, the model was only analyzed in one work [76], while the viscosity of the fluid was assumed to be independently of the concentration. The assumption decouples the flow equations from the concentration equation.The rest of this paper studies the coupling of the Stokes-Darcy flow sys-tem with an advection-diffusion equation that models transport of a chemical in the case of two space dimensions, where the viscosity is concentration-dependent. The coupled problem actually has two coupling meanings:one is the coupling of two domains, different domains have different flow equations, different diffusion coefficients, and different sources or sinks, and two domain interact with each other only through the interface; the other is the coupling of flow and mass transport. Therefore, the fully coupled system becomes very complicated. In general, there are two kinds of numerical methods based on the mixed weak formulation:one employs different finite element spaces in different domain; the other uses the same spaces. The use of the same finite element allows to write an efficient code with little regard to the nature of the underlying equations, i. e., whether an element lies in the Stokes region or the Darcy region, and makes the treatment of the interface conditions more straightforward. It would be advantageous if the same element could be used in both Stokes and Darcy regions. A seemingly promising candidate for such an element is the nonconforming C-R element, which has several nice properties:in combination with piecewise constant pressure it satisfies the inf-sup condition and is elementwise mass conserving; it is also easy to implement. However, in an article by Mardal et al. [54] it is shown that the C-R element does not converge when applied to Darcy problem. It is also well known that the C-R element does not fulfill a discrete Korn’s in- equality, which precludes the use of the physically more realistic form of the Stokes operator. Stabilization similar to [16] penalizing the jumps over the element edges of the piecewise continuous velocities sufficiently for fulfilling the Korn’s inequality was introduced by Hansbo and Larson [45]. For the numerical approximation of the transport equation, we employ the classical Lagrange Galerkin finite element. The coupling of flow and transport in a single domain, i.e. a model for miscible displacement in porous media, has been extensively studied(see, e.g., ([9,73,49,36,19,27,28,68,20,47,32,29] and the references therein).The outline of the thesis is as follows.In Chapter 1, the mathematical model obeying Darcy-Forchheimer law in a porous medium is introduced and the mass conservation equation is derived based on some physical properties of the fluid. Furthermore, we give the advection-diffusion equation describing the transport of the solute for the fluid in a porous medium. Then, some useful notations are recommended including definitions of function spaces and corresponding norms and we display a number of lemmas which are frequently used in the next chapters.In Chapter 2, we propose a stabilized dual mixed finite element approx-imation for the Darcy-Forchheimer equation combined with incompressible mass conservation equation. The velocity and pressure are respectively ap-proximated by nonconforming Crouzeix-Raviart element and piecewise con-stants. We present a discrete inf-sup condition and existence and uniqueness of its discrete solution are demonstrated due to the discrete inf-sup condi-tion and properties of the nonlinear operator which is monotone, coercive and hemi-continuous. The a priori error estimates in L2 and L3 norms for the ve-locity and L2 norm for the pressure are presented via the monotonicity of the nonlinear term. Numerical tests in order to verify our theoretical predictions are carried out and a comparison between the discretization proposed in our thesis and the one in [59] is given to demonstrate that our discretization has better properties.In Chapter 3, we apply the stabilized mixed finite element method pro- posed in Chapter 2 to the Darcy-Forchheimer equation combined with slight-ly compressible mass conservation equation. The a priori error estimates for both semi-discrete finite element scheme and a fully discrete scheme are pre-sented by introducing the projections of the Darcy-Forchheimer velocity and pressure.In Chapter 4, we consider the numerical approximation for the coupling of the Stokes-Darcy flow system with a mass transfer equation that models transport of a chemical in the coupled fluid flow region with porous media region, where the viscosity is concentration-dependent. We present a fully discrete finite element scheme, which decouples the flow equations and the transport equations. We use the C-R element that has the advantages men-tioned above for the velocities and piecewise constant for the pressures in both regions and apply a stabilization term similar to [16] penalizing the jumps over the element edges of the piecewise continuous velocities. For the numerical approximation of the transport equation, we employ the conform-ing Lagrange finite element. We derive a priori error estimates under some suitable regularity assumptions on the weak solutions.