A Combined Mixed Finite Element And Discontinuous Galerkin Method For Compressible Miscible Displacement Problem

The flow and transport of fluids in porous media is of great importance socially and economically in the oil recovery and environmental pollution problem. The compressible miscible displacement problem is the most perplexing one among this kind of problems. It can be described as a nonlinear partial differential equations system of equations which contains pressure equation (or continuity equation) and concentration equation (or transport equation). Among the variety methods to treat the problem, mixed finite element method (MFE) can obtain the same accuracy for both pressure and Darcy velocity and the Discontinuous Galerkin (DG) method allows more general meshes construction. Thus charactics make the above two methods gained great popular in the last two decades. The aim of this paper is to investigate the compressible miscible displacement problem by the MFE and DG method.The whole paper consists of two parts. In the first part, the mathematical model of compressible miscible displacement is introduced ana the equations as well as the parameters in the equations are explained in the physical sence. Then we put forward the combined scheme, mixed finite element method for pressure equation and discontinuous Galerkin method with interior penalty term for concentration equation. We also analyse the existence and uniqueness of the solution for the scheme. In the second part, we make our efforts to prove the convergence of the scheme. We use the method of induction hypothesis and a cut-off operator ".M" to make the combined scheme converge. And the the technique of integrating by part is often used to keep the accuracy of the convergency. At last we obtain the optimal prior error estimate for both concentration and pressure.In the end of the paper, we draw a conclusion of this work and make some comments on the prospect of the convergence analysis.