A Stabilized Discontinuous Galerkin Method For The Incompressible Miscible Displacement Problem

.In the 1970',Galerkin methods using interior penalties for elliptic and parabolic equations were first introduced by Douglas, Dupont and Wheeler[10,31].B .Reviere and M .F. Wheeler[32,33] use the NIPG method for the nonlinear parabolic equation . Sun combined the method with mixed finite element method for the miscible displacement problem in porous media[34].Sun and Wheeler[35] also analysed the concentration equation by using the NIPG and SIPG method ,and both methods obtain the optimal error estimate .The discontinuous Galerkin (DG) method allows more general meshes construction and degree of non uniformity than permitted by the more convectional finite element method, it is easy to construct higher order element to obtain higher order accuracy and to derive highly parallel algorithms .Because of these advantages, the discontinuous Galerkin (DG) method has become a very active area of researchThe compressible miscible displacement problem can be described as a nonlinear partial differential equations system of equations which contains transport equation(or concentration equation) and continuity equation(or pressure equation).In the paper ,A new stabilized discontinuous Galerkin method is proposed to solve the incompressible miscible displacement problem .The first order and the second order of fully discrete discontinuous finite element schemes are proposed for the concentration equation.For the pressure equation ,we develop mixed ,stabilized ,discontinuous Galerkin formulation .We can obtain the optimal priori estimates for the both concentration and pressure.