The saturated-unsaturated
flow of fl
uid (water) through a porous medium can be described by the
Richards equation which was introduced by the American physicist Lorenzo Adolph Richards in 1931.
Since the Richards equation is a highly nonlinear elliptic-parabolic partial differential equation, straight-
forward approximation methods have to be handled with care or are not applicable at all. In this work
we consider a new approach to compute the approximate solution.
In a first step, we use the primal hybrid formulation to derive a system of nonlinear equations with linear
coupling conditions. To simplify the resulting system, we apply the Kirchhoff transformation to shift
the nonlinearity of the principal part from the subdomains to the interface. After the transformation, a
coupled system with a linear principal part within the subdomains and nonlinear coupling conditions is
obtained. Solvability and uniqueness of the system are discussed.
The analogy to the discrete mortar finite element method was decisive for its application to compute
the approximate solution. We use the Newton method to solve the discrete nonlinear system. In view
efficiency, domain decomposition methods for the mortar finite element method are of special interest.
Finally we present numerical examples in two and three space dimensions.