We prove a substructuring result for a variational inequality concerning - but not restricted to - the Richards equation in homogeneous soil and including boundary conditions of Signorini's type. This generalizes existing results for the linear case and leads to interface conditions known from linear variational equalities: continuity of Dirichlet and flux values in a weak sense. In case of the Richards equation these are the continuity of the physical pressure and of the water flux, which is hydrologically reasonable. Therefore, we also apply these interface conditions in the heterogeneous case of piecewise constant soil parameters, which we address by the Robin method. We prove that, for a certain time discretization, the homogeneous problems in the subdomains including Robin and Signorini-type boundary conditions can be solved by convex minimization. As a consequence we are able to apply monotone multigrid in the discrete setting as an efficient and robust solver for the local problems. Numerical results demonstrate the applicability of our approach.

Connectedness, disconnectedness, and light factorization structures are studied in the realm of the topological constructs \textbf{FPUConv} and \textbf{FSUConv} of fuzzy preuniform convergence spaces and fuzzy semiuniform convergence spaces respectively which have been introduced by the author in \cite{23} using fuzzy filters in the sense of Eklund and Gähler \cite{7}. The presented theory profits from the fact that both constructs have hereditary quotients. Additionally, there are special features, e.g. a product theorem for the investigated connectedness concept and the existene of a proper class of light factorization structures on FPUConv as well as on FSUConv.

We present globally convergent nonsmooth Schur-Newton methods for the solution of discrete vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems as arising from discretization by implicit Euler methods in time and first order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments.

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.

Let C[0,1] be the Banach algebra of real valued continuous functions on [0,1], provided with the supremum norm. For f,g\in C[0,1] and balls B_{f}, B_{g} with center f and g, respectively, it is not necessarily true that f\cdot g is in the interior of B_{f}\cdot B_{g}. In the present paper we characterize those pairs f, g where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way.

We propose a variational formulation of rate- and state-dependent models for the dynamic sliding of a linearly elastic block on a rigid surface in terms of two coupled variational inequalities. Classical Dieterich-Ruina models are covered as special cases. We show existence and uniqueness of solutions for the two spatial subproblems arising from time discretisation. Existence of solutions to the coupled spatial problems is established for Dieterich's state equation through a fixed point argument.We conclude with some numerical experiments that suggest mesh independent convergence of the underlying fixed point iteration, and illustrate quasiperiodic occurrence of stick/slip events.

Eines der zur Zeit schnellsten Verfahren zur Faktorisierung ganzer Zahlen ist das ``Quadratische Sieb'' (engl. ``quadratic sieve factorization method''), das 1981 von Carl Pomerance entwickelt wurde. Wir beschreiben im Folgenden die Basisversion des Quadratischen Siebs sowie die Variante des Quadratischen Siebs mit mehrfachen Polynomen, das sogenannte ``Multiple Polynomial Quadratic Sieve'' MPQS, das unabhängig von J. Davis und D. Holdridge bzw. P. Montgomery gefunden wurde. Bei der Darstellung der Verfahren orientieren wir uns an [Buchmann 2010], [Crandall & Pomerance 2005], [Esslinger et al. 2011], [Pomerance 1996], 'Quadratisches Sieb' in [Wikipedia de] und 'quadratic sieve' in [Wikipedia en].

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.

We derive and analyse a solver-friendly finite element discretiza- tion of a time discrete Richards equation based on Kirchhoff transformation. It can be interpreted as a classical finite element discretization in physical variables with non-standard quadrature points. Our approach allows for non- linear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the non-degenerate case, of the discrete physical pressure. The associated discrete algebraic problems can be formu- lated as discrete convex minimization problems and, therefore, can be solved efficiently by monotone multigrid methods. In numerical examples for two and three space dimensions we observe L2-convergence rates of order O(h2) and H1-convergence rates of order O(h) as well as robust convergence behaviour of the multigrid method with respect to extreme choices of soil parameters.

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.

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.

In Analysis two modes of non-topological convergence are interesting: order convergence and convergence almost everywhere. It is proved here that oder convergence of sequences can be induced by a limit structure, even a finest one, whenever it is considered in sigma-distributive lattices. Since convergence almost everywhere can be regarded as order convergence in a certain sigma-distributive lattice, this result can be applied to convergence of sequences almost everywhere and thus generalizing a former result of U. Höhle obtained in a more indirect way by using fuzzy topologies.