##
**A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media.**
*(English)*
Zbl 0856.76033

Summary: We study a model nonlinear, degenerate, advection-diffusion equation having application in petroleum reservoir and groundwater aquifer simulation. The main difficulty is that the true solution is typically lacking in regularity; therefore, we consider the problem from the point of view of optimal approximation. Through time integration, we develop a mixed variational form that respects the known minimal regularity, and then we develop and analyze two versions of a mixed finite element approximation, a simpler semidiscrete (time-continuous) version and a fully discrete version. Our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error. The exceptional term is a nonstandard approximation error term. We also consider our new formulation for the nondegenerate problem, showing the usual optimal \(L_2\)-error bounds; moreover, superconvergence is obtained under special circumstances.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76S05 | Flows in porous media; filtration; seepage |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35K65 | Degenerate parabolic equations |

86A05 | Hydrology, hydrography, oceanography |