Mixed Finite Elements and Newton-type Linearization for the Solution of Richard's Equation

We present the development of a two-dimensional Mixed-Hybrid Finite Element (MHFE) model for the solution of the non-linear equation of variably saturated flow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest-order Raviart-Thomas (RT0) elements and is 'exactly' mass conserving. Hybridization is used to overcome the ill-conditioning of the mixed system. The scheme is globally first-order in space. Nevertheless, numerical results employing non-uniform meshes show second-order accuracy of the pressure head and normal fluxes on specific grid points. The non-linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE-Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation techniques