The saturated-unsaturated flow of fluid (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.