Arbogast, Todd; Wheeler, Mary F.; Zhang, Nai-Ying A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media. (English) Zbl 0856.76033 SIAM J. Numer. Anal. 33, No. 4, 1669-1687 (1996). 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. Cited in 1 ReviewCited in 103 Documents 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 Keywords:advection-diffusion equation; petroleum reservoir; groundwater aquifer; optimal approximation; mixed variational form; error bounds; superconvergence × Cite Format Result Cite Review PDF Full Text: DOI