Efficient Discontinuous Galerkin Methods For Miscible Displacement Problems

Discontinuous Galerkin methods are very attractive for numerical simulations in computational fluid dynamics because of their physical and numerical properties. Firstly,it is flexible that allows for general non-conforming meshes with variable degrees of approximation and of higher accuracy.Secondly,it is locally mass conservative.Thirdly,it has less numerical diffusion and can also handle rough and discontinuous coefficient problems.Fourthly,it is easier for hp-adaptivity than conforming approaches because the information over cell boundaries is almost decoupled. In addition,with appropriate meshing,varying p can yield exponential convergence rates.Incompressible and compressible miscible displacement problems in porous media are investigated.Several discontinuous Galerkin(DG) finite element methods are used for spatial discretization.For the incompressible case,many scholars such as S.Sun and M.F.Wheeler,have done a lot of work.They established the error estimates in a energy norm for four semi-discrete primal discontinuous Galerkin methods,i.e.,Oden-Babu(?)ka-Baumann DG(OBB-DG),non-symmetric interior penalty Galerkin(NIPG),symmetric interior penalty Galerkin(SIPG), and incomplete interior penalty Galerkin(IIPG) without duality assumption.The estimates in L~2(L~2) norm and in negative norm are also derived for SIPG by them. Based on the duality argument,we work out a unified a posteriori error estimate in L~2(L~2) norm for four semi-discrete and full-discrete primal DG methods above. H.Chen and M.R.Cui present hp error analysis for a combined mixed and discontinuous Galerkin method(MFE/DG) for compressible miscible displacement problems.But only the dispersion-free case is considered.In practical,the effects of molecular diffusion and dispersion are included,which makes our analysis more complicate and difficult.Based on the work of H.Chert and M.R.Cui,we consider the complete compressible case with no restrictions on the diffusion/dispersion tensor. Several DG approximations and combined MFE/DG methods are applied.A priori error estimates and a posteriori one are obtained using the induction hypothesis instead of the cut-off operator and duality technique,respectively.Finally, some superconvergence analysis of combined MFE/DG methods are investigated for incompressible and compressible miscible displacement problems.