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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

Citations:

Zbl 0458.76092
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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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