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Fakultät für Mathematik
Angewandte Analysis (LSI)

Slides of selected talks

  • On the effect of oil-trapping and the derivation of a homogenized equation: Oiltrapping2008. This regards the two-phase flow equations in a one-dimensional domain with an interface condition at (many) internal interfaces.

  • A system with forward- and backward diffusion: Backward2008. This talk is about a  scalar equation in one space dimension, including diffusion with both signs. Together with D. Horstmann I compared two different solution concepts and characterized one of them with a free boundary problem.

  • Outflow boundary conditions for various porous media equations: Outflow2009. For various bulk equations (degenerate and non-degenerate Richards, two-phase flow) a regularization scheme for outflow boundary conditions is analyzed. Results are in parts obtained together with M. Lenzinger and S. Pop.

  • Homogenization of hysteresis problems: Hysteresis2010. This presentation regards homogenization methods for problems with hysteresis which could be applied to Hydromechanics and to Plasticity.

  • Plasticity: Homogenization of plasticity equations is presented e.g. in Plasticity-Rome-2016

  • Meta-Materials are studied in the context of Maxwell equations. Together with G. Bouchitte and later with A. Lamacz I studied the question whether materials with a negative optical index can be constructed with a complex micro-structure. Indeed, such a construction was proposed by Pendry and others. In our contribution we give a detailed analysis of the microscopic behavior of electric and magnetic field and derive an effective Maxwell equation with negative index. A related effect is perfect transmissionMetaMat2013

  • Fingering effect for Richards equation with hysteresis: Fingering2013. When we introduce static hysteresis in the Richards equation, planar front solutions become unstable. Together with the dynamic term, true fingering occurs for both Richards and two-phase flow in porous media under the influence of gravity.

  • Dispersion for  waves in heterogeneous materials: Waves2015. We describe waves with a linear wave equation, the elliptic operator is in divergence form and with highly oscillatory coefficients. For finite times, the effective equation is again a linear second order wave equation -- and cannot describe the dispersion effects that are observed numerically and experimentally. Instead, for large times, the effective equation is a dispersive wave equation

  • Wave-guides and interfaces are studied in terms of uniqueness and with the aim of practical schemes with Bloch-waves. An important subject is the definition of an outgoing wave condition in such media. For experts in Blochwaves-Korsika-2016 and in a more general language in Blochwaves-Heidelberg2017. Existence of solutions to Helmholtz equations with energy methods was presented in Existence-WavePhen2022.

  • Homogenization of a layer perforation is studied using measures, see Layerhomogen-Valencia-2019 and Perforation-WavePhen-2022.

  • Representation with profile functions: At large times, the solution of a wave equation can be represented as a ring (or shell) with a profile function that solves an easy equation Representation-SIAM-2022.