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 modeling the flow of water into a variably saturated porous medium. We consider 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 saturations, we bound the H-1-norm of the error in capacity in terms of approximation error. This estimate uses a time-integrated scheme and the Kirchhoff transformation to handle a degeneracy in the case of completely unsaturated 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 stated for the case of strictly partially saturated flow.