A NONLINEAR MIXED FINITE-ELEMENT METHOD FOR A DEGENERATE PARABOLIC EQUATION ARISING IN FLOW IN POROUS-MEDIA

We study a model nonlinear, degenerate, advection-diffusion equation h aving application in petroleum reservoir and groundwater aquifer simul ation. The main difficulty is that the true solution is typically lack ing in regularity; therefore, we consider the problem from the point o f view of optimal approximation. Through time integration, we develop a mixed variational form that respects the known minimal regularity, a nd 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 er ror. The exceptional term is a nonstandard approximation error term. W e also consider our new formulation for the nondegenerate problem, sho wing the usual optimal L(2)-error bounds: moreover, superconvergence i s obtained under special circumstances.