Slides of selected talks
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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.
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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.
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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.
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Homogenization of hysteresis problems: Hysteresis2010. This presentation regards homogenization methods for problems with hysteresis which could be applied to Hydromechanics and to Plasticity.
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Plasticity: Homogenization of plasticity equations is presented e.g. in Plasticity-Rome-2016
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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 transmission. MetaMat2013
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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.
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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. A more recent talk from a broader perspective is Long-time-waves-2025.
- Maxwell equations: Existence and uniqueness for Maxwell's equations in a periodic waveguide. Maxwell-Waveguide-2025.
- Neural Networks: Slides for a one-day course on neural networks and deep learning held in Regensburg by Ben Schweizer and Klaas Poelstra. NeuralNetworks2024
- Long time homogenization in stochastic media: On what time-scales does homogenization work in stochastic media? On what time-scales does homogenization fail? This is the topic of this talk on the one-dimensional wave equation: Time-horizon-stoch-2025
- Homogenization of Maxwell's equations: In this talk, some older results concerning bulk homogenization of Maxwell's equations with small rings are presented. They are combined with new results on the homogenization of thin structures that give rise to effective interface conditions: Phenomena-Maxwell-2025.pdf