A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media

Authors:
Todd Arbogast;Mary F. Wheeler;Nai-Ying Zhang
Affiliations:
-;-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
1996

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Abstract

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.