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

Todd Arbogast, Mary F. Wheeler, and Nai-Ying Zhang

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.

KEYWORDS:
Mixed finite element, degenerate parabolic equation, nonlinear,
error estimates, porous media