Two-grid Method For Incompressible Miscible Displacement Problem By Mixed Finite Element Methods And Standard Finite Element Methods

Miscible displacement problem in porous media is a typical and complex phe-nomenon of fluid, which is closely related to environmental pollution and oil reser-voir exploitation, and it is a research field interests many scientists. We simulate this kind of problem through the mass conservation equation and variational form of Darcy's law, in which the mathematical model is mainly composed of a set of nonlinear partial differential equations. The problem we discussed contains two subproblems . They are fluid transport and flow respectively , in which one is the pressure equation, the other is the concentration equation. In Ewing's paper [62],they had given comprehensive overview about mathematical model and numerical simulation of miscible displacement problem in porous media. Because of the large amount of calculations about this problem, we hope to investigate more effective numerical methods to solve this problem quickly and improve the computational efficiency. We put our emphasis on efficient numerical methods of this problem. In this paper, we firstly construct two-grid methods for incompressible miscible dis-placement problem by mixed finite element methods and standard finite element methods, and the convergence properties of our two-grid algorithm are discussed and analyzed then.In this paper, we consider the following three types of incompressible mis-cible displacement problems . First, we discuss a simple displacement problem only involving the molecular diffusion phenomenon, then, an incompressible dis-placement problem with coefficient of gravity is study for general term of r(c);finally, we consider the complex problem with molecular diffusion and mechanical dispersion in detail. In this paper, based on the incompressible displacement prob-lems we propose, two-grid method which is composed of RT mixed finite element method and standard finite element method is constructed, and we also obtain a series of approximation properties and error estimates for the above three types of problems. It is found that two-grid methods could achieve same accuracy as the variational problem of the source problem but with much more less time cost. The whole paper consists of three parts as follows.In the first part of this paper, we construct the discrete system by using the mixed finite element approximation for pressure and velocity equation, and the standard finite element approximation for the concentration equation. Based on the properties of elliptic projection and Xu's duality argument techniques of[112], the optimal Lq-norm error estimates of mixed finite element solutions are obtained. Then, we design fast algorithms by using the idea of Newton iteration on the fine grid, coupled nonlinear problem on fine grid is transformed into a nonlinear problem in rough space and a (or two) linear problem in fine space by the two-grid. At the same time, we also obtain the error estimation of Lq-norm for the concentration and the convergence for the pressure and velocity about the algorithms. Finally, numerical experiments are provided to verify the theoretical results.In the second part, we study incompressible displacement problem with coef-ficient of gravity r(c) and design fast algorithms by combining mixed finite element method with standard finite element method. First, we obtain the Lq-norm es-timates of the mixed finite element solutions by using the properties of elliptic projection and duality argument. Then, a two-grid method for mixed finite ele-ment is established, and corresponding convergence estimates are given in detail.It is found that two-grid method can achieve same accuracy but with much less time cost as mixed finite method through the convergence analysis and numerical examples .In the last part, we mainly discuss the complex incompressible miscible dis-placement problem with molecular diffusion and mechanical dispersion. We try to consider an effective two-grid algorithm of nonlinear diffusion coefficient. We present the weak formulas of our model, then we give an analysis of the mixed finite element solutions by using the mixed finite element approximation for pres-sure and velocity equation and the standard finite element approximation for the concentration equation. In the following, our main algorithms are advocated, er-ror estimates and convergence properties of pressure and velocity can be obtained through the analysis and discussion of the algorithms.