## Finite element method for incompressible miscible displacement in porous media.(English)Zbl 0787.76043

Summary: Under the assumptions of nonlinear finite element and $$\Delta t=o(h)$$, R. E. Ewing and M. F. Wheeler [SIAM J. Numer. Anal. 17, 351- 365 (1980; Zbl 0458.76092)] discussed a Galerkin method for the single phase incompressible miscible displacement of one fluid by another in porous media. In this paper we give a finite element scheme which weakens the $$\Delta t=o(h)$$-restriction to $$\Delta t=o(h^ \varepsilon)$$, $$0<\varepsilon \leq {1\over 2}$$. Furthermore, this scheme is suitable for both linear and nonlinear element. We also derive the optimal approximation estimates for concentration $$c$$, its gradient $$\nabla c$$ and the gradient $$\nabla p$$ of the pressure $$p$$.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage

### Keywords:

optimal approximation estimates; concentration; pressure

Zbl 0458.76092
Full Text:

### References:

 [1] J. Douglas, Jr, R.E. Ewing, M.F. Wheeler, The Approximations of the Pressure by a Mixed Method in the Simulation of Miscible Displacement,RIARO Anal. Numer.,17 (1983), 17–33. · Zbl 0516.76094 [2] R.E. Ewing, M.F. Wheeler, Galerkin Method for Miscible Displacement Problems in a Porous Media,SIAM J. Numer. Anal.,17 (1980), 351–365. · Zbl 0458.76092 [3] Hong Minchun, Galerkin Method for Compressible Miscible Displacement in a Porous Media,J. Compu. Math.,2 (1988), 119–128 (in Chinese). · Zbl 0661.76097 [4] J. Douglas, Jr, T. Dupont, Galerkin Methods for Parabolic Equations with Nonlinear Boundary Conditions,Numer. Math.,20 (1973), 213–237. · Zbl 0234.65096 [5] M.F. Wheeler, A PrioriL 2-error Estimates for Galerkin Approximations to Parabolic Partial Differential Equation,SIAM J. Numer. Anal.,10 (1973), 723–759. · Zbl 0258.35041
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