| In this thesis, we investigate the coupling of free fluid and porous media flows from both the mathematical and numerical viewpoints. We begin with a basic situation:the coupling of Stokes and Darcy flows. 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 consists of the Beavers-Joseph-Saffman condition [7,54], the continuity of flux and the balance of forces. The research on the coupled Stokes and Darcy flows problem gets more and more attention due to its significance in hydrology, environment science and biofluid dynamics. 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 error orders when handling the weak formulation and the numerical scheme.The early studies of the coupled Stokes and Darcy flows problem have developed some numerical techniques [32,55], and both articles use Beavers-Joseph condition on the interface. Beavers-Joseph condition is the law on the tangential fluid velocity on the interface between fluid domain and porous me-dia, which is in best agreement with experiment evidence worked by Beavers and Joseph [7]. Before that, no-slippage along the interface is a commonly used condition for the free fluid. Jones [39] extended this to multidimensional flows. Saffman [54] justified this law theoretically and pointed out that the Darcy velocity on the interface was much smaller than other quantities ap-pearing in the law of Beavers-Joseph and it could be dropped. Therefore, he proposed the Beavers-Joseph-Saffman condition.Using Beavers-Joseph-Saffman condition, two independent papers in 2002 and 2003, respectively, proved the well-posedness of weak formulation of the coupled problem from different points. In [22], Discacciati, Miglio. and Quarteroni propose an iterative subdomain method which is based on a standard finite element method for two order elliptic problem in the Darcy domain and a standard mixed element method in the Stokes domain. In [43], Layton, Schieweck, and Yotov use a Lagrange multiplier on the interface to prove the existence and uniqueness of a weak solution and to allow the use of existing Stokes and Darcy flow simulators in a domain decomposition pro-cedure. After that, many effective numerical methods were developed and many numerical experiments were also done. There are two kinds of numer-ical methods based on the mixed weak formulation:one uses 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 inter-face conditions more straightforward. We present a stabilized mixed finite element method for the coupled Stokes and Darcy flows problem. We use the Crouzeix-Raviart element for the velocities and piecewise constant for the pressures in both the Stokes and Darcy regions, and apply a stabilization term penalizing the jumps over the element edges of the piecewise continuous velocities. The nice properties of C-R element are presented:in combina-tion with piecewise constant pressures it satisfies the inf-sup condition; it is elementwise mass conserving; it is easy to implement both in two and three dimensions. For details see Chapter 1 and Chapter 2.However, in practice mass and heat transfer gets more attention than the velocity and pressure in fluid field. We wish to analyze a model for motion of a salt or pollutant dissolved in a fluid in a coupled free fluid and porous medium region. This model has interesting applications in practice. It can be used to predict how contaminants transport through rivers into the aquifers, simulate the miscible displacement of one fluid by another in a vuggy porous medium, and describe various industrial processes involving filtration. 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 each other only on the interface; the other is the coupling of flow and transport. Therefore, the fully coupled system becomes very complicate. The coupling of flow and transport in a single domain has been extensively studied(see, e.g., [8,56,41,27,16,23,24,53] and the references therein). The model describing solute transport in the coupled free/porous media flow was only analyzed in one work [61], where the viscosity of the fluid was assumed to be independent of the concentration. The assumption decouples the flow equations from the concentration equation.The rest of this thesis studies the fully coupled Navier-Stokes/Darcy system with a transport equation in the case of two space dimensions, where the viscosity is concentration-dependent. We first propose a mathematical model and its equivalent weak formulation. To prove existence of weak so-lutions, we introduce an iterative scheme to decouple the system. Then, for the decoupled problems, weak solutions are obtained as a limit of solutions to discrete time problems. It follows Temam [58] for treating Navier-Stokes equations. Then we obtain a sequence of solutions of the iterative scheme and pass to the limit in the iterative scheme to find a weak solution of the coupled system. We establish uniqueness for the semiclassical solution of the system. To design a numerical scheme for the system, we propose a dif-ferent weak formulation by adding some limits for the diffusion coefficient. For the numerical approximation of the fully coupled system, thanks to the nature of time-dependent problems, we decouple the flow equations and the transport equations and employ a domain decomposition method to allow a simulation of the flow equations to be decoupled into steps involving porous media and free fluid flow subproblems. There are two motivations for the use of the domain decomposition strategy. First, it allows to apply the existing single-model solvers for the two subproblems. Second, it is useful to be able to use different time steps in different subdomains because in most practical cases the velocity field in the porous medium varies slower in time than other physical quantities in the free fluid domain. When analyzing the approximation error, we find the error terms related to the interface reduce the error orders due to the reduction in the dimension of integrated domain. Thus we cannot obtain an optimal a priori error esti-mate in the sense of L2-norm in general. However, if some regularity theory holds for the corresponding partial differential equations, then an optimal estimate can be derived by a projection technique and a negative norm es-timate technique. Then we discuss a sufficient condition for the regularity theory. Finally, we present some numerical examples to verify the theoretical predictions.The outline of the thesis is as follows.In Chapter 1, a stabilized mixed finite element method for solving the stationary coupled Stokes and Darcy flows problem is formulated and an-alyzed. The approach utilizes the same nonconforming Crouzeix-Raviart element discretization on the entire domain. A discrete inf-sup condition and an optimal a priori error estimate are derived. Finally, some numerical examples verifying the theoretical predictions are presented.In Chapter 2, we apply the unified stabilized mixed finite element method proposed in Chapter 1 to non-stationary coupled Stokes and Darcy flows problem. We present a semidiscrete finite element scheme and a fully dis-crete scheme and show that the method approximates both the true velocity and pressure to the optimal error order in the energy norm.In Chapter 3, we consider a coupled system made of the Navier-Stokes and Darcy equations with a mass transfer equation which models the cou-pling of fluid and porous media flows with mass transfer. Under physically reasonable hypotheses on the data, we prove existence of a weak solution and establish uniqueness for the semiclassical solution of the system.In Chapter 4, we consider the numerical approximation for the coupled system proposed in Chapter 3 and present a fully discrete finite element scheme, which decouples the flow equations and the transport equations and allows a simulation of the flow equations to be decoupled into steps involving porous media and free fluid flow subproblems. We derive an optimal a pri- ori error estimate under some suitable conditions. Finally, some numerical examples verifying the theoretical predictions are presented. |
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