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Degenerate two-phase incompressible flow. V: Characteristic finite element methods. (English) Zbl 1077.76527

Summary: This is the fifth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we study a finite element procedure for numerically solving this system. This procedure uses a mixed finite element method for the elliptic equation for pressure and velocity, and the modified method of characteristics for the degenerate parabolic equation for the saturation. Sharp error estimates in energy norms are obtained for this procedure. The error analysis does not use any regularization of the saturation equation and does not impose any restriction on the nature of degeneracy; these error estimates are derived directly from this degenerate saturation equation. Other characteristic finite element methods such as the modified method of characteristics with adjusted advection are also considered.
For Part IV, see Z. Chen and R. E. Ewing, J. Sci. Comput. 18, No. 3, 329–360 (2003; Zbl 1032.76590)

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76T30 Three or more component flows

Citations:

Zbl 1032.76590
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