Fast and robust numerical solution of the Richards equation in homogeneous soil

Abstract

We derive and analyse a solver-friendly finite element discretiza- tion of a time discrete Richards equation based on Kirchhoff transformation. It can be interpreted as a classical finite element discretization in physical variables with non-standard quadrature points. Our approach allows for non- linear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the non-degenerate case, of the discrete physical pressure. The associated discrete algebraic problems can be formu- lated as discrete convex minimization problems and, therefore, can be solved efficiently by monotone multigrid methods. In numerical examples for two and three space dimensions we observe L2-convergence rates of order O(h2) and H1-convergence rates of order O(h) as well as robust convergence behaviour of the multigrid method with respect to extreme choices of soil parameters.