Substructuring of a Signorini-type problem and Robin's method for the Richards equation in heterogeneous soil

We prove a substructuring result for variational inequalities. It concerns but is not restricted to the Richards equation in heterogeneous soil, and it includes 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. We use these interface conditions in a heterogeneous problem with 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

Heterogeneous domain decomposition of surface and porous media flow

We present a heterogeneous domain decomposition approach to the Richards equation coupled with surface water flow. Assuming piecewise constant soil parameters in the constitutive equations for saturation and relative permeability, we present a novel domain decomposition approch to the Richards equation involving on fast and robust subdomain solver based on optimization techniques. The coupling of ground and surface water is resolved by a Dirichlet - Neumann-type iteration

Fast and robust numerical solution of the Richards equation in homogeneous soil

We derive and analyze a solver-friendly finite element discretization of a time discrete Richards equation based on Kirchhoff transformation. It can be interpreted as a classical finite element discretization in physical variables with nonstandard quadrature points. Our approach allows for nonlinear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the nondegenerate case, of the discrete physical pressure. The associated discrete algebraic problems can be formulated 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 $L^2$-convergence rates of order $\mathcal{O}(h^2)$ and $H^1$-convergence rates of order $\mathcal{O}(h)$ as well as robust convergence behavior of the multigrid method with respect to extreme choices of soil parameters

On nonlinear Dirichlet-Neumann Algorithms for jumping nonlinearities. In: Domain Decomposition Methods in Science and Engineering XVI

We consider a quasilinear elliptic transmission problem where the nonlinearity changes discontinuously across two subdomains. By a reformulation of the problem via Kirchhoff transformation we first obtain linear problems on the subdomains together with nonlinear transmission conditions and then a nonlinear Steklov–Poincar´e interface equation. We introduce a Dirichlet–Neumann iteration for this problem and prove convergence to a unique solution in one space dimension. Finally we present numerical results in two space dimensions suggesting that the algorithm can be applied successfully in more general cases

Heterogeneous substructuring methods for coupled surface and subsurface flow

The exchange of ground- and surface water plays a crucial role in a variety of practically relevant processes ranging from flood protection measures to preservation of ecosystem health in natural and human-impacted water resources systems

BlogForever D3.2: Interoperability Prospects

This report evaluates the interoperability prospects of the BlogForever platform. Therefore, existing interoperability models are reviewed, a Delphi study to identify crucial aspects for the interoperability of web archives and digital libraries is conducted, technical interoperability standards and protocols are reviewed regarding their relevance for BlogForever, a simple approach to consider interoperability in specific usage scenarios is proposed, and a tangible approach to develop a succession plan that would allow a reliable transfer of content from the current digital archive to other digital repositories is presented

Non-overlapping domain decomposition for the Richards equation via superposition operators

Simulations of saturated-unsaturated groundwater flow in heterogeneous soil can be carried out by considering non-overlapping domain decomposition problems for the Richards equation in subdomains with homogeneous soil. By the application of different Kirchhoff transformations in the different subdomains local convex minimization problems can be obtained which are coupled via superposition operators on the interface between the subdomains. The purpose of this article is to provide a rigorous mathematical foundation for this reformulation in a weak sense. In particular, this involves an analysis of the Kirchhoff transformation as a superposition operator on Sobolev and trace spaces

Magnetically Controlled Exchange Process in an Ultracold Atom-Dimer Mixture

We report on the observation of an elementary exchange process in an optically trapped ultracold sample of atoms and Feshbach molecules. We can magnetically control the energetic nature of the process and tune it from endoergic to exoergic, enabling the observation of a pronounced threshold behavior. In contrast to relaxation to more deeply bound molecular states, the exchange process does not lead to trap loss. We find excellent agreement between our experimental observations and calculations based on the solutions of three-body Schr\"odinger equation in the adiabatic hyperspherical representation. The high efficiency of the exchange process is explained by the halo character of both the initial and final molecular states.Comment: 4 pages, 4 figure