Characteristics Finite Element Methods For Convection-Diffusion Problems

Convection-diffusion equations are an important class of partial differ-ential equations that arise in many scientific fields including fluid mechanics, gas dynamics and so on. Since these equations normally have no closed form analytical solutions, it is very important to have accurate numerical approx-imations. When diffusion dominates the physical process, standard finite difference methods and finite element methods work well in solving these equations. However, when convection is the dominant process, these equa-tions present many numerical difficulties including non-physical oscillations, excessive numerical diffusion which smears out sharp fronts, or a combination of both. In the framework of numerical solutions of convection dominated problems. a possible strategy is provided by the method of characteristics for time discretization. This approach is based on the discretization of the material derivative, and it takes advantage of the physics characteristics of the convoction-diffusion problems, has incomparable superiority for dealing with hyperbolic nature of convection-dominated diffusion problems. It, can not only substantially reduces the non-physical oscillations, excessive numer-ical diffusion, but also has no stability constraints on the time step, and the derivative value of the solution along the characteristic direction is smaller than the value along the time direction.Many authors have mathematically analyzed and applied this method to different problems. In the sixties of the last century, people construct a for-ward tracking method of characteristic (MOC)[32], until recent years, people continue to improve this method, and apply it to many practical problems. This type of method is relatively easy to implemented to simple problems, but the tracking along the characteristic line destroys the original space grid. and brought a lot of inconvenience to the calculation. In1982, Douglas and Russell [24]proposed a back tracking method of characteristic (modified method of characteristics MMOC), which overcomes the shortcomings of the original method of characteristics. This method has also been widely used, for example [46],[4] which combined this kind of method with mixed element method to handle the miscible displacement in porous media. But the back tracking method can not satisfy the mass conservation. Soon after that, Douglas and others constructed the modified method of characteristics with adjusted advection (MMOC A A)[22],[23] which can satisfy the overall mass conservation. In [51], professor Rui Hongxing and Tabata also constructed a new method of characteristics which also maintains the mass conservation of the convection diffusion process. In1990, Celia and others proposed the Euler-Lagrange localized adjoint method (ELLAM)[14], which not only can maintain the mass conservation, but also facilitates the treatment of boundary conditions, but the drawback is that the calculations of the inte-gral has certain difficulty. In [50], professor Rui Hongxing and Tabata first introduced a second order characteristic finite element method. This numer-ical scheme is of second order accuracy in time increment, symmetric and unconditionally stable.Mathematical and physical model for the fluid flow and transport pro-cesses in porous media leads to a highly coupled time-dependent nonlinear partial differential equations. Because of the complex structure of the partial differential equations, analytical solution only exists in some special cases. Therefore, large-scale, high-accuracy simulation for the model is a urgent necessity in science and engineering. In sight of the complexity of the model for fluid flow in porous media, mass conservation results in the mass balance of the injection and production, which can be described by the convection-diffusion equations, and the momentum conservation is often approximated by some simplified velocity and pressure relation, experimental formulas for example, the Darcy’s law. Assuming that the fluid is incompressible, the convection dominated diffusion equation combined with Darcy’s law is the widely used classical model. A lot of research and analysis have been done for the model, and also get a wide range of applications. Russell [53] has ana-lyzed the concentration equation by a combination of a Galerkin method and the method of characteristics and the press equation by a standard Galerkin procedure. Using the method of characteristics for solving the concentra-tion equation can make the equation symmetric, enhance stability, reduce the time truncation error, and the larger time step can be used. Pressure equation using the standard Galerkin method can only determine the pres-sure, but the speed can not be obtained directly. Ewing and Russell [28][25] have analyzed the coupled process by mixed finite elements and a modi-fied method of characteristics. This method can be obtained simultaneously by the pressure and velocity, without going through the differentiating the pressure to obtain an approximation of the velocity, thus reducing the speed error.Another research focus of this article is the moving grid method. Us-ing the finite element method for solving the time-dependent problems (such as parabolic), the usually effective practice in in the region of space using the finite element method, and in the time direction using a finite difference scheme. A fixed set of finite element mesh have been proposed in the past algorithms which is limited to the region of space Grid. However, in many practical computing problems, we often need in different layers of time using different finite element spaces. For example, the flame pass Broadcast. and oil-water frontier. Therefore, many mathematicians and engineers focus on the dynamic finite element spare method, and many dynamic finite element methods have been proposed. Liang Guoping [39]proposed a moving grid finite element method for general parabolic problems. The main idea of the method is that a different number of finite element spaces is adopted at differ-ent time level, and the approximate solution at the current time is projected in the L2-norm onto the next time finite clement space and make it as an initial value. By developing the idea of coordinate transformation, he then presents a full-discretization moving finite element scheme[41], which can ap- ply to parabolic equations in any dimension with variable domain and under certain continuity assumptions about moving grid the optimal convergence rate is preserved. Yang Daoqi [61]proposed a mixed finite element method with moving grid for the parabolic problems, and extended to the porous medium miscible displacement [62]. Professor Yuan Yirang [65] discussed the moving grid method for nonlinear convection-diffusion problems.This article is based on the finite element method of characteristics for the convection-diffusion equation. We mainly discussed the second or-der characteristics finite element method with moving grid, the second-order characteristics finite element method combined with the Galerkin finite ele-ment method in incompressible miscible displacement in porous media, and finally we gave an optimal-order error estimate for the mass-conservative characteristic finite clement scheme.In Chapter1, we give the mathematical model which is discussed in this paper. According to the physical background of the miscible displace-ment in porous media, from the nature of mass conservation, we derive the convection-diffusion equation describing the concentration of fluids in porous media, and combining with Darcy equations describing fluid velocity and pressure relationship, we derive the coupled nonlinear partial differential equations. Then we introduce the sobolev spaces and its norms and give several lemmas frequently used in the later chapters.In Chapter2, we disease the second order characteristics finite element method with moving grid. The approximation algorithm is of second order accuracy in time increment, and when the number of the moved grids M satisfies certain conditions, though it is not optimal accuracy in space incre-ment, but still have good convergence rate. When Mh is bounded, it has a convergence rate of hk in L2-norm.In Chapter3, we develop the numerical analysis of incompressible mis-cible displacement in porous media, we use the second-order characteristics finite element method to treat the concentration equation and the Galerkin finite element method in the press equation. We list our assumptions on the problem, define the second order characteristic-Galerkin method and give the error estimates in L2-norm. Finally, we prove that the approximation algorithm is of second order accuracy in time increment, and though it is not optimal accuracy in space increment, but still have good convergence rate.In Chapter3, an optimal-order error estimate is obtained for the model problem in one space dimension. The convergence rate is O(h2+Δt) in the l∞(L2) norm for the linear finite element. Some numerical examples are given to confirm the conclusion.