https://refubium.fu-berlin.de/handle/fub188/17741 Thu, 30 Apr 2026 17:40:48 GMT 2026-04-30T17:40:48Z https://refubium.fu-berlin.de/handle/fub188/18578 A multidomain discretization of the Richards equation in layered soil Berninger, Heiko; Kornhuber, Ralf; Sander, Oliver We consider the Richards equation on a domain that is decomposed into nonoverlapping layers, i.e., the decomposition has no cross points. We assume that the saturation and permeability functions are space-independent on each subdomain. Kirchhoff transformation of each subdomain problem separately then leads to a set of semi-linear equations, which can each be solved efficiently using monotone multigrid. The transformed subdomain problems are coupled by nonlinear continuity and flux conditions. This nonlinear coupled problem can be solved using substructuring methods like the Dirichlet-Neumann or Robin iteration. We give several numerical examples showing the discretization error, the solver robustness under variations of the soil parameters and a hydrological example with four soil layers and surface water. Tue, 01 Jan 2013 00:00:00 GMT https://refubium.fu-berlin.de/handle/fub188/18578 2013-01-01T00:00:00Z https://refubium.fu-berlin.de/handle/fub188/19169 A polynomial chaos approach to stochastic variational inequalities Forster, Ralf; Kornhuber, Ralf We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show equivalence to a related collocation method. Numerical experiments illustrate the efficiency of our approach and suggest similar error estimates as for linear elliptic problems. Fri, 01 Jan 2010 00:00:00 GMT https://refubium.fu-berlin.de/handle/fub188/19169 2010-01-01T00:00:00Z https://refubium.fu-berlin.de/handle/fub188/18273 Adaptive modelling of coupled hydrological processes with application in water management Bastian, Peter; Berninger, Heiko; Dedner, Andreas; Engwer, Christian; Henning, Patrick; Kornhuber, Ralf; Kröner, Dietmar; Ohlberger, Mario; Sander, Oliver; Schiffler, Gerd; Shokina, Nina; Smetana, Kathrin This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated-unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet-Neumann method, using the dune-grid-glue module in the numerics software Dune. Fri, 01 Jan 2010 00:00:00 GMT https://refubium.fu-berlin.de/handle/fub188/18273 2010-01-01T00:00:00Z https://refubium.fu-berlin.de/handle/fub188/18076 An adaptive Newton multigrid method for a model of marine ice sheets Jouvet, Guillaume; Gräser, Carsten In this paper, we consider a model for the time evolution of marine ice sheets. This model combines the Shallow Ice Approximation (SIA) for the ice deformation, the Shallow Shelf Approximation (SSA) for the basal sliding and the mass conservation principle. At each time step, we solve a generalized p-Laplace minimization-type problem with obstacle (SIA), a vectorial p-Laplace minimization-type problem (SSA) and a transport equation (mass conservation). The two minimization problems are solved using a truncated nonsmooth Newton multigrid method while the transport equation is solved using a vertex-centred finite volume method. Our approach is combined to a mesh adaptive refinement procedure to face the large gradients of the solution that are expected close to the grounding line which separates the ice sheet and the ice shelf. As applications, we present some simulations of the marine ice sheet model inter- comparison project MISMIP in two and three space dimensions. In particular, we test the ability of our model to reproduce a reversible grounding line after being perturbed in model parameters. Sun, 01 Jan 2012 00:00:00 GMT https://refubium.fu-berlin.de/handle/fub188/18076 2012-01-01T00:00:00Z