A Discontinuous Finite Volume Element Method For Miscible Displacement Problems In Porous Media

In this paper, we firstly consider the discontinuous finite volume element methodon triangular grids for compressible miscible displacement problems: p(x, t)=p0(x), x∈,with boundary conditionsThe method does not require continuity of the approximation function acrossthe interelement boundary conditions, which makes it easy to construct the space.Besides, it has the advantages of a high order of accuracy, high parallelizability,easy to handle the complex boundary and so on, the study of discontinuous finitevolume method has been an efective method of dealing with the nonlinear partialdiferential equations. In this chapter, the concentration equation and the pres-sure equation are approximated with discontinuous finite volume element method,and for the above two equations, the fully discrete schemes are formulated by us-ing discontinuous finite volume element method. Based on interpolation projection properties and some induction hypotheses, by detailed theoretical analyses, we ob-tain the optimal error estimates of pressure|||p ph|||!concentration|||c ch|||andthe error estimate of velocity u uh.Then we consider the mixed-upwind discontinuous finite volume element methodfor the following incompressible miscible displacement problems in porous mediawith boundary conditionsIn this article, the concentration equation is approximated with the aid of a up-wind discontinuous finite volume element method, while for the pressure equation,we present a mixed finite volume element method. For the above incompressiblemiscible displacement problems in porous media, the fully discrete schemes are for-mulated by using mixed-upwind discontinuous finite volume element method. Thisschemes not only can approximate velocity and pressure at the same time but alsocan eliminate some combination of excessive numerical dispersion and nonphysicaloscillation because of the upwind technique. By detailed theoretical analysis, theerror estimates for the concentration c chin L~2-normal, the optimal order errorfor pressure p phin L~2-normal and the optimal error estimate for Darcy velocityu uh H(div)in H(div)-normal are obtained.