**Eldorado**

Resources for and from Research, Teaching and Studying

### Recent Submissions

We investigate the long time behavior of waves in crystals. Starting from a linear wave equation on a discrete lattice with periodicity ε > 0, we derive the continuum limit equation for time scales of order ε^(-2). The effective equation is a weakly dispersive wave equation of fourth order. Initial values with bounded support result in ring-like solutions and we characterize the dispersive long-time behavior of the radial profiles with a linearized KdV equation of third order.

We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the x2-direction, unbounded in the x1-direction and ε-periodic for large |x1|, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the ...

We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness.

We study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period η > 0; we study the behaviour of solutions ( E^η ,H^η ) in the limit η → 0 and derive an effective system. In geometries with a non-trivial topology, the limit system implies that certain components of the effective fields vanish. We identify the corresponding effective system and can predict, from topological propertie...

We analyze the time harmonic Maxwell's equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeability of the medium can be large and/or negative.

Meta-materials are assemblies of small components. Even though the single component consists of ordinary materials, the meta-material may behave effectively in a way that is not known from ordinary materials. In this text, we discuss some meta-materials that exhibit unusual properties in the propagation of sound or light. The phenomena are based on resonance effects in the small components. The small (sub-wavelength) components can be resonant to the wave-length of an external field if ...

We study connections between four different types of results that are concerned with vector-valued functions u : Ω→ℝ³ of class L²(Ω) on a domain Ω ⊂ ℝ³: Coercivity results in H^1(Ω) relying on div and curl, the Helmholtz decomposition, the construction of vector potentials, and the global div-curl lemma.

We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with x-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for x-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new r...

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length η > 0. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like η⁻² and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order η²; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system....

We investigate the deformation of heterogeneous plastic materials. The model uses internal variables and kinematic hardening, elastic and plastic strain are used in an infinitesimal strain theory. For periodic material properties with periodicity length scale n > 0, we obtain the limiting system as n -> 0. The limiting two-scale plasticity model coincides with well-known effective models. Our direct approach relies on abstract tools from two-scale convergence (regarding convex functiona...

We study a system of small strain visco-plasticity. We use an additive decomposition of the strain into elastic and plastic part, and allow for non-linear relations in the Hooke's law and in the flow rule. We show the existence of solutions, using a time-discrete approximation scheme. The limit procedure is based on a strong convergence result for the time-discrete solution sequence.

This work is devoted to an adaptive multiscale finite element method (MsFEM) for solving elliptic problems with rapidly oscillating coeefficients. Starting from a general version of the MsFEM with oversampling, we de- rive an a posteriori estimate for the H1-error between the exact solution of the problem and a corresponding MsFEM approximation. Our esti- mate holds without any assumptions on scale separation or on the type of the heterogeneity. The estimator splits into different contr...

We analyze the time harmonic Maxwell’s equations in a complex geometry. The homogenization process is performed in the case that many small, thin conductors are distributed in a subdomain of R^3. Each single conductor is, topologically, a split ring resonator, but we allow arbitrary flat shapes. In the limit of large conductivities in the rings and small ring diameters we obtain an effective Maxwell system. Depending on the frequency, the effective system can exhibit a negative effectiv...