Multipoint Flux Mixed Finite Element Method And Anaiysis For Several Flow Models

In porous media, mathematical model of fluid flow is widely used in ar-eas such as groundwater and reservoir simulation[10,12,52]. Model is mainly based on the conservation law for mass, momentum and energy. Based on some reasonable assumptions, these models can be simplified, the simplified model mathematically expressed as a time-dependent strongly coupled non-linear partial differential equations. The structure of some simplified partial differential is quite complex, only special equations exist analytical solution. Therefore, fast and efficient numerical simulation of these mathematical mod-els has become the urgent needs of engineering.Mixed finite element (MFE) method is widely used in porous media, which has many advantages, such as it can preserve local mass conservation, simultaneously solving velocity and pressure thereby accurately approximate velocity, and proper treatment of discontinuous coefficients. There are many documents about the introduction and application of mixed finite element method in fluid, such as [17,54,19,33,42,9,53,34,66,11,16,45,57,23, 6,7,18,26]. However, since the MFE simultaneously approaching velocity and pressure, so it need to solve a saddle point type algebraic system, which is a computational drawback of these methods, it takes a lot of computer memory and time-consuming. To overcome this problem, in [58] introduced that in the case of diagonal tensor coefficients and rectangular grids by use a quadrature rule for the velocity mass matrix, MFE methods can be reduced to cell-centered finite differences (CCFD) scheme for the pressure. In [68] the authors explored this relationship and obtain the convergence of CCFD on rectangular grids. In [5,8] the authors introduced the expanded mixed finite element (EMFE) and extended the above result to full tensor coefficients and logically rectangular grids. Although the EMFE method is very accurate for smooth grids and coefficients, but due to it adopt the arithmetic average for the discontinuous coefficients, thus it loses accuracy near discontinuities. In-troduced the pressure Lagrange multipliers along the discontinuous interfaces can recovered higher order accuracy, but that lost the cell-centered structure.Although, there are some other methods to process rough grid and coef-ficients, such as [21,38] introduced the control volume mixed finite element (CVMFE) method and mimetic finite difference (MFD) method. Approx-imation space of these two methods are closely related to the lowest order Raviart-Thomas mixed finite element space RT0. [24,59] and [13,14] are given the convergence of the above two methods. However, the two methods will lead to a saddle point type algebraic system. [1,2.27,28] introduced multipoint flux approximation (MFA) method which has the advantages of the methods mentioned above, that is, for the rough grid and coefficient showed accurate convergence, and reduces to a cell-centered stencil for the pressures. However, since the MFA method nonvariational formulation, so the well posedness and convergence analysis of the method [43] only have limited theoretical results. Therefore, find a method which not only have well performance for rough grid and coefficient but also can decoupling the algebra system become very important.To solve these problems, Mary F. Wheeler and Ivan Yotov in [72] for second-order linear elliptic problem presents one type of MFE method, which can reduces to CCFD for pressure and performs well for full tensors, irreg-ular grids and discontinuous coefficients. This method is inspired by MFA method, so Mary F. Wheeler and Ivan Yotov [72] called it MFMFE method. The approximation space selected as the lowest order Brezzi-Douglas-Marini (BDM) Mixed Finite Element [19,17] space, that is, BDM1 space. For ex- ample, for two dimensional domain, when the partition is unit triangle or the unit square element, degrees of freedom in velocity space can be chosen to be the values of the velocity normal component at any two points on each edge, for three dimensional domain, when the partition is unit tetrahedron, the degree of freedom in velocity space can be chosen to be the values of the velocity normal component at any three points on each face. By use a special quadrature rule for velocity mass matrix that allows for local velocity elimi-nation and leads to a symmetric and positive definite cell-centered scheme for the pressures. There are many literature about the MFMFE method, such as [72] introduced MFMFE method for linear elliptic problem with the sim-plex or quadrilateral partition. [39], [70] and [69] present MFMFE method for linear elliptic problems with the hexahedral, curved quadrilateral, hex-ahedral and general grid partition, respectively. However, we learned most of the documents are processed linear elliptic problems, simulate for other issues is rarely, literature about the method solving nonlinear problems is more rare. In this paper we are committed to expanding this method to sev-eral types of flow model under general quadrilateral grid partition, including linear parabolic problem, parabolic integro differential equations, nonlinear incompressible and compressible Darcy-Forchheimer model.In this paper we first introduce the motion equations of fluid mechan-ics, presented the Darcy’s law and Darcy-Forchheimer law which describe the relationship between fluid velocity and pressure gradient, and presented the simplified incompressible and compressible form of the mass conserva-tion equation. As we all know, Darcy’s law is a empirical formula which is experimentally measured by Darcy in 1856, it describes the linear rela-tionship between velocity and pressure gradient with low flow velocity and small permeability [10], in this case the pressure gradient is mainly used to overcome the viscous resistance [50,51], Darcy’s law applies, a theoreti-cal derivation of Darcy’s law can be found in [46,73]. However, when the flow rate of the fluid is relatively high, the inertial force of the flow en-hance, in addition to overcome the viscous resistance, the pressure gradi- ent is also used to overcome inertial force, in this case the inertia force is proportional to the square of velocity, Darcy’s law is no longer applicable. Therefore, the greater the velocity, the more obvious the force of inertia, then the relationship between velocity and pressure gradients can be de-scribed by Darcy-Forchheimer [32] Law. The theoretical derivation of the Forchheimer equation can be found in [60,46,35,3,22,40,73]. Research on the Darcy-Forchheimer equation solving method has a large number of documents [47,56,48,36,44,62,67,4,4,55].This thesis is organized as follows.In Chapter 1, we introduce the mathematical model of porous media flow problems that will be studied in this paper and give Darcy’s law and Darcy-Forchheimer law, show a simplified form incompressible and compressible fluid of the mass conservation equation. We presents the basic definitions, commonly used lemma and some inequalities that will needed in the process of the theoretical derivation. At the end of the chapter, we give a brief introduction of MFMFE method, presents the bilinear mapping, BDM mixed finite element space, the projection on the space of velocity and pressure and the quadrature formula those will be used in the following discussion. Finally, we present the error estimate of the quadrature formula.In Chapter 2, for compressible Darcy flow problem, we give the discrete format and error analysis of the MFMFE method for solving this problem. MFMFE method is based on the lowest order BDM mixed finite element space, the mass matrix be replaced by numerical integration, thereby simpli-fying mass conservation equation to a cell-centered finite difference scheme. We give two discrete forms of the model, Backward Euler and Crank-Nicolson scheme. Also presents the convergence order analysis. Finally, we also give a numerical example to verify the theoretical analysis.In Chapter 3, we discuss the MFMFE method for parabolic integro-differential equations. Parabolic Integro-differential equations are common in application such as heat conduction in materials with memory and com-pression viscoelastic media, it can be seen as a non-local reaction diffusion models. There exists a lot of literature about Integra-differential equations, [29,31,30,65,41,64,49,37,63]. In this thesis we present MFMFE ap-proximate method for the model, including the semi-discrete method and full discrete Backward Euler method, we give the error analysis, we also give a numerical example to verify the theoretical analysis.In Chapter 4, the MFMFE method is applied to a class of nonlinear problem that is incompressible Darcy-Forchheimer equation. We give dis-crete format and well posedness analysis. It is worth mentioning that we proved Forchheimer equation nonlinear term quadrature formula portion is equivalent to L3-norm, which is very important for well posedness analysis and error estimates. We give the first order convergence for pressure in L2-norm, and for velocity in H(div)-norm. In the end, we present numerical examples, which verified the the theoretical results, and we also present the second order convergence of the discrete norm for pressure and velocity.In Chapter 5, we give the numerical schemes of the MFMFE method for compressible Darcy-Forchheimer equation. We develop two approxima-tion methods for the problems, the semi-discrete method and fully discrete method, where the fully discrete method includes the Backward Euler scheme and the Crank-Nicolson scheme. We present the error estimates, theoretical results indicate first-order convergence in spacial meshsize both for pressures and velocity, first order convergence for Backward Euler scheme and second order convergence for Crank-Nicolson scheme in time. Finally, numerical ex-amples verify the theoretical analysis.