A Mortar Domain Decomposition Method for Quasilinear Problems

Abstract

The equation of interest in this work is the Richards equation which describes the flow of water in a porous medium. Since the equation contains two nonlinearities, stable and efficient methods have to be developed. In Berninger (ph.D. thesis, Freie Universität Berlin, 2008) a monotone multigrid method is considered in the case of a homogeneous soil. In the case of a heterogeneous soil, domain decomposition methods are discussed. In this work a different approach is considered. As already mentioned, the Richards equation contains two nonlinearities. In Berninger et al. (Comput Geosci 19(1):213-232, 2015) and Schreiber (Nichtüberlappende Gebietszerlegungsmethoden für lineare und quasilineare (monotone und nichtmonotone) Probleme, Universität Kassel, 2009) the so called Kirchhoff transformation is used to shift one nonlinearity from the domain to the boundary in the homogeneous setting. To carry this idea over to the heterogeneous case, a different formulation is needed to ensure compatibility with the Kirchhoff transformation. The Primal–Hybrid formulation is used to derive such a formulation. After applying local Kirchhoff transformations a coupled system of equations with one nonlinearity within the domain and nonlinear coupling conditions is derived. To approximate the solution, the mortar finite element method is applied.