Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media

We present an analysis of expanded mixed finite element methods applied to Richards' equation, a nonlinear parabolic partial differential equation mod eling the flow of water into a variably saturated porous medium. We conside r the full range of saturated to completely unsaturated media. In the case of the lowest order Raviart-Thomas spaces and the range of all possible sat urations, we bound the H+1-norm of the error in capacity in terms of approx imation error. This estimate uses a time-integrated scheme and the Kirchhof f transformation to handle a degeneracy in the case of completely unsaturat ed flow. Optimal convergence is then shown for a nonlinear form related to the error in the capacity for the case of saturated to partially saturated flow. Convergence rates depending on the Holder continuity of the capacity term are derived. Last, optimal convergence of pressures and fluxes is stat ed for the case of strictly partially saturated flow.