  Numerical solutions for flows in partially saturated porous media pose
challenges related to the non-linearity and elliptic-parabolic degeneracy of
the governing Richards' equation. Iterative methods are therefore required to
manage the complexity of the flow problem. Norms of successive corrections in
the iterative procedure form sequences of positive numbers. Definitions of
computational orders of convergence and theoretical results for abstract
convergent sequences can thus be used to evaluate and compare different
iterative methods. We analyze in this frame Newton's and $L$-scheme methods for
an implicit finite element method (FEM) and the $L$-scheme for an explicit
finite difference method (FDM). We also investigate the effect of the Anderson
Acceleration (AA) on both the implicit and the explicit $L$-schemes.
Considering a two-dimensional test problem, we found that the AA halves the
number of iterations and renders the convergence of the FEM scheme two times
faster. As for the FDM approach, AA does not reduce the number of iterations
and even increases the computational effort. Instead, being explicit, the FDM
$L$-scheme without AA is faster and as accurate as the FEM $L$-scheme with AA.
