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Numerical analysis of a method for high Peclet number transport in porous media. (English) Zbl 1153.76041

Summary: A variationally consistent eddy viscosity discretization is presented in [W. Layton, Appl. Math. Comput. 133, No. 1, 147–157 (2002; Zbl 1024.76026)] for the stationary convection-diffusion problem. This discretization is extended to the evolutionary problem in [N. Heitmann, J. Math. Anal. Appl. 331, No. 1, 38–50 (2007; Zbl 1147.76039)] with a near optimal error bound. In the following, we couple this discretization with the porous medium problem. We present a comprehensive analysis of stability and error for velocity field derived from the porous medium problem. Next, using a backward Euler approximation for the time derivative, we follow the inherited error in velocity through the coupling with the convection-diffusion problem. The method is shown to be stable and the error near optimal and independent of the diffusion coefficient.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76R99 Diffusion and convection
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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References:

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