Mortar Domain Decomposition Methods for Quasilinear Problems and Applications

The saturated-unsaturated flow of fluid (water) through a porous medium can be described by theRichards 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 workwe 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 linearcoupling conditions. To simplify the resulting system, we apply the Kirchhoff transformation to shiftthe nonlinearity of the principal part from the subdomains to the interface. After the transformation, acoupled system with a linear principal part within the subdomains and nonlinear coupling conditions isobtained. Solvability and uniqueness of the system are discussed.The analogy to the discrete mortar finite element method was decisive for its application to computethe approximate solution. We use the Newton method to solve the discrete nonlinear system. In viewefficiency, domain decomposition methods for the mortar finite element method are of special interest.Finally we present numerical examples in two and three space dimensions.