Symmetric Finite Volume Element Methods For Flow Problems In Porous Media

The finite volume element method (FVE) is a discretization technique for partial differential equations, especially for those that arise from physical conservation laws including mass, momentum, and energy. FVE ( also called generalized difference method [10] or box method [2]) uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discretize the equations. Meanwhile, the approximate solution is chosen out of a finite element space [3-5]. It possesses the important property of inheriting the physical conservation laws of the original problem locally. Thus it can be expected to capture shocks, to produce simple stencils, or to study other physical phenomena more effectively. FVE has been widely used in computational fluid mechanics and heat transfer problems [9].In 1978, R. H. Li [10] utilized finite element spaces and generalized characteristic functions on dual elements, i.e., the common terms of the local Taylor expansions, to rewrite integral interpolation methods in a form of generalized Galcrkin methods, and thus obtained a generalization of difference methods on irregular networks, that is, the so-called generalized difference methods (GDM for short) . Since then, extensive research has been carried out on the theory and application of GDM, such as constructing linear and high order difference schemes for elliptic, parabolic and hyperbolic equations. For numerical analysis, they have given the optimal order H~1(Ω) error estimates. Under the assumption that u∈H~3(Ω), Chou and Li [17] have given the optimal L~2 norm error estimates for a special dual grid obtained by connecting barycentcrs and medians. Both the theoretical observations and the computational experiments show that FVE has not only the simplicity of difference methods but also the accuracy of finite element methods. FVE also has some other virtues; see [2-5,11] for more details.However, in general case the coefficient matrix of the linear system obtained from the finite volume element method is not symmetric. This introduces some difficulties in real implementations; the method suitable to symmetric linear systems cannot be used in this case. Rui [12] has given symmetric modified finite volume element methods for general self-adjoint elliptic and parabolic problems with optimal energy norm error estimate. For symmetric finite volume method, we can get more properties in [13,14]In this thesis, we propose several kinds of symmetric finite volume element methods for several time-dependent equations, the optimal error estimate can be obtained. Finite volume element method is widely applied in scientific research and engineering technique, so the development of the finite volume element method has important significance. The innovative aspects of this thesis are as follows:(1) We put forward a kind of symmetric finite volume element method for nonlinear parabolic problems. We symmetrize the coefficient matrix of system, so we can use any method which is suitable for solving symmetric system. We give the optimal order energy norm error estimates for full discrete schemes. We also prove that the difference between the solutions of the finite volume element method and the solutions of symmetric finite volume element method is a high order term.(2) We propose the symmetric finite volume element method of characteristics for complicated coupled miscible displacement in porous media. The problems include the incompressible and compressible miscible displacement in porous media. For incompressible case, a sequential time-stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated by a mixed finite element method and the concentration is approximated by a combination of a symmetric finite volume element method and the method of characteristics. Optimal order convergence in H~1 and in L~2 are proved for full discrete schemes. For compressible case, we use symmetric finite volume element method for pressure equation, and the second concentration equation is approximated by a combination of a symmetric finite volume element method and the method of characteristics. Optimal order convergence in H~1 is proved for full discrete schemes. (3) We put forward a kind of symmetric finite volume element discretizations for an initial boundary value parabolic integro-differential equation. This kind of equation is widely used in various engineering models, such as nonlocal reactive flows in porous media, heat conduction, radioactive nuclear decay in fluid flows, or viscoclastic deformations of materials with memory and biotechnology. We introduce the finite volume element Ritz-Volterra projection. It is proved that symmetric finite volume element approximations arc convergent with optimal order in L~2-norm.This dissertation is divided into six chapters.In chapter 1, in§1.1, we mainly introduce the finite volume element method. In§1.2, the symmetric finite volume element method for elliptic problem is formulated, which is obtained by adding a high order term to the traditional finite element method. Its coefficient matrix is symmetric, so we can use any of the methods for solving symmetric linear system, and we need one prediction and one correction with the same symmetric system. In§1.3, we introduce symmetric finite volume element method for parabolic problem. At each time step, we only need to solve the symmetric system one time.In chapter 2, we put forward a kind of symmetric finite volume element method for nonlinear parabolic problems. In§2.2, on the basic of an important lemma of chapter 1, we extend symmetric finite volume element method to nonlinear parabolic equations. In§2.3, we give some auxiliary results. In§2.4, we analyze convergence of the method give the optimal order energy norm error estimates for full discrete schemes. We also prove that the difference between the solutions of the finite volume element method and the solutions of symmetric finite volume element method is a high order term.In chapter 3, we propose symmetric finite volume element method for incompressible miscible displacement in porous media. Numerical simulation for incompressible flow in porous media is extremely important for developing oil fields reasonably and understanding fluid dynamics of oil and water under the ground. Miscible displacement of one incompressible fluid by another in porous media is modeled by a nonlinear system of two coupled partial differential equations. Elliptic equation, hereafter called the pressure equation, embodies mass conservation for the system and uses the empir- ical Darcy law of fluid mechanics. It accounts for convection and externally imposed flows. Parabolic equation, the concentration equation, conserves mass for the injected fluid and allows for convection, diffusion, dispersion, and externally imposed flows. A modified method of characteristics was introduced and analyzed for a single parabolic equation by Douglas and Russell [23], using either finite differences or finite element to discretize in space. For incompressible miscible displacement problem, a sequential backward-difference time-stepping scheme is defined in [24]: it approximate the pressure by a standard Galerkin procedure and the concentration by a combination of a Galerkin method and the method of characteristics. For pressure equation, the Raviart-Thomas space and the mixed method was introduced and analyzed in combination with a standard Galerkin procedure for concentration equation in [25,26]. In [27], the author used a mixed finite element method for pressure equation and a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L~2 is proved. The organization of the rest of this chapter is as follows: In§3.2, a sequential time-stepping procedure is defined, in which the the pressure and Darcy velocity of the mixture are approximated by a mixed finite element method and the concentration is approximated by a combination of a symmetric finite volume element method and the method of characteristics. In§3.3, we give some auxiliary lemmas. In§3.4, optimal order convergence in H~1 and in L~2 are proved for full discrete schemes.In chapter 4, we propose symmetric finite volume element method for compressible miscible displacement in porous media. Douglas and Robert [37,38] have formulated and analyzed numerical schemes for approximating the solution of the system. Yuan [41,42] has present and analyzed time stepping along characteristics of the finite element approximation and finite difference approximation of compressible miscible displacement in porous media respectively. The organization of the rest of this chapter is as follows: In§4.2, we use symmetric finite volume element method for pressure equation, and the second concentration equation is approximated by a combination of a symmetric finite volume element method and the method of characteristics. In§4.3, we give some auxiliary lemmas. In§4.4, optimal order convergence in H~1 is proved for full discrete schemes. In chapter 5, we put forward a kind of symmetric finite volume element discretizations for an initial boundary value parabolic integro-differential equation. This model is very important in the transport of reactive and passive contaminates in aquifers, an area of active interdisciplinary research of mathematicians, engineers, and life scientists. We refer to [45,46] for the derivation of the mathematical models and for the precise hypotheses and analysis. Mathematical formulations of this kind also arise naturally in various engineering models, such as nonlocal reactive transport in underground water flows in porous media [47], heat conduction, radioactive nuclear decay in fluid flows [48], non-Newtonian fluid flows, or viscoelastic deformations of materials with memory (in particular polymers) [49], semi-conductor modeling [50], and biotechnology. One very important characteristic of all these models is that they all express a conservation of a certain quantity (mass, momentum, heat, etc.)in any moment for any subdomain. This in many applications is the most desirable feature of the approximation method when it comes to the numerical solution of the corresponding initial boundary value problem.This type of equations has been extensively treated by finite element, finite difference, and collocation methods in the last years [51-56]. The finite element method conserves the flux approximately: therefore, in the asymptotic limit (i.e., when the grid step-size tends to zero) it produces adequate results. However, this could be a disadvantage when relatively coarse grids are used. Perhaps the most important property of the finite volume method is that it exactly conserves the approximate flux(heat, mass, etc.) over each computational cell. This important property combined with adequate accuracy and ease of implementation has contributed to the recent renewed interest in the method. The theoretical [59] has given the finite volume element approximations of this kind of problem. But in general case the coefficient matrix of the linear system obtaining from the finite volume element method is not symmetric. In this article, we study symmetric finite volume element approximations for two-dimensional parabolic integro-differential equations, arising in modeling of nonlocal reactive flows in porous media. In§5.2, we set up symmetric finite volume element schemes. In§5.3, we give some auxiliary lemmas. In§5.4. we introduce the finite volume element Ritz-Volterra projection. It is proved that symmetric finite volume element approximations are con- vergent with optimal order in L~2-norm.In chapter 6, we give a numerical simulation of beam vibration on an elastic foundation. A vehicle running on the bridge can make bridge vibrate, which brings potential problem. The vibration problems of rail foundation and the building near the railway arouse our attention with the train speed higher and higher. Vibration becomes one of seven environmental social effects of pollution in international. People set about to study the rule and reason of vibration and find the method to control vibration. The basic theory of beam vibration on an elastic foundation has been shown in [60]. [61] uses Green's function to compute the displacement of the beam induced by moving load, but it needs large computational work and has some restrictions to solve high-speed problems. [62] gives numerical simulation and visualization of high-speed train on the railway structure from an engineering point of view. [63] uses the finite element method with B-spline basis functions for an Euler-Bernoulli beam with dynamic contact. In this chapter, we put forward a finite element method with cubic hermite element to solve the problem of beam vibration on an elastic foundation. This method has been found applicable to assess ground vibration effects caused by vehicle speed and load condition. Our numerical results give some evidence.