Cs. Woodward et Cn. Dawson, Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media, SIAM J NUM, 37(3), 2000, pp. 701-724
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.