Order of Convergence Estimates for an Euler Implicit, Mixed Finite Element Discretization of Richards' Equation

Abstract

We analyze a discretization method for a class of degenerate parabolic problems that includes the Richards' equation. This analysis applies to the pressure-based formulation and considers both variably and fully saturated regimes. To overcome the difficulties posed by the lack in regularity, we first apply the Kirchhoff transformation and then integrate the resulting equation in time. We state a conformal and a mixed variational formulation and prove their equivalence. This will be the underlying idea of our technique to get error estimates. A regularization approach is combined with the Euler implicit scheme to achieve the time discretization. Again, equivalence between the two formulations is demonstrated for the semidiscrete case. The lowest order Raviart--Thomas mixed finite elements are employed for the discretization in space. Error estimates are obtained, showing that the scheme is convergent.