### Inhalt des Dokuments

Koordination: | Prof. Dr.
Rupert Klein (FU Berlin), Homepage [1] Prof. Dr. Barbara Wagner (TU Berlin), Homepage [2] Dr. Kersten Schmidt (TU Berlin), Homepage [3] Dr. Sergiy Nesenenko (TU Berlin), Homepage [4] |

Termine: | 1-2 Termine
pro Semester mit jeweils 2 Vorträgen |

Inhalt: | Vorträge zu
aktuellen Themen der Mehrskalenmodellierung mit partiellen
Differentialgleichungen |

Datum und Uhrzeit | Ort und
Raum | Vortragende(r) und
Vortragstitel | Poster |
---|---|---|---|

17.02.2016 14.00 Uhr | TU Berlin, MA 313 | Prof. Dr.
Nicola Popovic [5] (U Edinburgh)A geometric analysis of
fast-slow models for stochastic gene expression(Abstract) Dr. Martin Heida [6] (WIAS Berlin) On
Homogenization of Rate-independent Systems(Abstract) | Poster [7] |

11.12.2015, 9.15 Uhr | TU Berlin, MA 313 | Prof. Dr. Ralf
Kornhuber [8] (FU Berlin)Direct and iterative methods for
numerical homogenization (Abstract)Prof. Dr. Dirk Pauly [9] (Universität Duisburg-Essen) Low-frequency
asymptotics for time-harmonic Maxwell equations in exterior
domains (Abstract) | Poster
[10] |

30.10.2015,
9.15 Uhr | TU Berlin, MA 313 | Dr. Agnes Lamacz [11] (TU Dortmund)Effective
Maxwell's equations in a geometry with flat split-rings
(Abstract)Dr. Adrien Semin [12] (TU Berlin) (Abstract)When a thin periodic layer meets corners: asymptotic
analysis of a singular Helmholtz problem
| Poster [13] |

15.07.2015, 9.15 Uhr | TU Berlin, MA 313 | Prof.
Dr. Andreas Münch [14] (University of Oxford, UK)Asymptotic
analysis of phase-field models involving surface diffusion
(Abstract)Dr. Patrik Marschalik [15] (Universität Mainz) Fundamental concepts of the methods of matched
asymptotic approximations and multiple scales expansions
(Abstract) | Poster [16] |

03.12.2014, 9.15 Uhr | TU Berlin, MA 415 | Sina Reichelt
[17] (WIAS Berlin)Two-scale homogenization of nonlinear
reaction-diffusion systems involving different diffusion length
scales (Abstract)Dr. Anastasia Thöns-Zueva [18] (TU Berlin) Asymptotic expansion for nonlinear viscous acoustic
equations close to rigid wall (Abstract) | Poster [19] |

07.07.2014, 9.15 Uhr | TU Berlin,
MA 415 | Dr. Sergiy Nesenenko [20]
(Universität Duisburg-Essen) (Abstract)Homogenization in elasto-plasticity via a phase-shift technique Prof. Dr. Barbara Wagner [21] (TU Berlin) Unsteady
non-uniform base states and their stability (Abstract) | Poster [22] |

23.04.2014 16.00 Uhr | TU Berlin, MA 313 | Dr.
Alfonso Caiazzo [23] (WIAS Berlin)Multiscale modeling
of weakly compressible elastic materials in harmonic
regime(Abstract) Prof. Dr. Antonin Novotny [24] (University of Toulon, France) Discrete relative entropy and
error estimates for some finite volume/finite element schemes to
compressible Navier-Stokes equations(Abstract) | Poster [25] |

27.06.2013, 9.00 Uhr | TU
Berlin, MA 415 | Dr. Stefan Neukamm [26] (WIAS
Berlin)Quantitative results in stochastic
homogenization (Abstract)Dr. Maria Bruna [27] (University of Oxford, UK) Diffusion of finite-size
particles: multiple species and confined geometries (Abstract)Dr. Adrien Semin [28] (TU Berlin) Construction and
analysis of improved Kirchhoff conditions for acoustic wave
propagation in a junction of thin slots (Abstract) | Poster [29] |

08.04.2013, 14.15 Uhr | TU
Berlin, MA 313 | Prof. Dr. Alexander
Mielke [30] (WIAS / HU Berlin)Evolutionary Gamma convergence
and amplitude equations (Abstract)Prof. Dr. Carsten Hartmann [31] (FU Berlin) Optimal control of multiscale
diffusion (Abstract) | Poster [32] |

24.01.2013, 9.30 Uhr | TU
Berlin, MA 415 | Dr. Maciek Korzec [33]
(TU Berlin)Multiple scales in silicon type microstructure
growth (Abstract)Thomas Petzold [34] (WIAS Berlin) Modelling and simulation of multi-frequency induction hardening
of steel parts (Abstract) | Poster [35] |

30.11.2012, 9.30 Uhr | TU
Berlin, MA 415 | Dr. Daniel Peterseim [36]
(HU Berlin)A new multiscale method for (semi-) linear
elliptic problems (Abstract)Dr. Ludwig Gauckler (TU Berlin) Modulated Fourier expansion: Multiscale expansions
for analysing oscillatory Hamiltonian systems (Abstract) | Poster [37] |

21.06.2012, 9.30 Uhr | TU Berlin, MA 313 | Prof. Dr.
Rupert Klein [38] (FU Berlin)A three-scale asymptotic
problem in atmospheric flows (Abstract)Dr. Kersten Schmidt [39] (TU Berlin) High order asymptotic expansion for
viscous acoustic equations close to rigid walls (Abstract) | Poster
[40] |

### Nicola Popovic

**A geometric analysis of fast-slow models
for stochastic gene expression**

Mittwoch, 17.Februar
2016, 14.00 Uhr

Stochastic models for gene expression frequently exhibit dynamics on different time-scales. One potential scale separation is due to significant differences in the lifetimes of mRNA and the protein it synthesises, which allows for the application of perturbation techniques [1, 2]. Here [3], we develop a dynamical systems framework for the analysis of a family of "fast-slow" models for gene expression that is based on geometric singular perturbation theory [4]. We illustrate our approach by giving a complete characterisation of a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we develop a systematic expansion procedure for the resulting propagator probabilities that can in principle be taken to any order in the perturbation parameter. We verify our asymptotics by numerical simulation, and we explore its practical applicability, as well as the effects of a variation in the system parameters and the scale separation. Finally, we discuss the generalisation of our geometric framework to models for regulated gene expression that involve additional stages, which is a subject of ongoing research.

[1] V. Shahrezaei and P.S. Swain, Analytical distributions for
stochastic gene expression, *Proc. Natl. Acad. Sci USA* 105,
pp. 17256--17261, 2008.

[2] P. Bokes, J.R. King, A.T.A. Wood, and
M. Loose, Multiscale stochastic modelling of gene expression, *J.
Math. Biol.* 65, pp. 493--520, 2012.

[3] N. Popovic, C. Marr,
and P.S. Swain, A geometric analysis of fast-slow models for
stochastic gene expression, *J. Math. Biol.* 72, pp. 87--122,
2016.

[4] C. Jones, Geometric singular perturbation theory, in
Dynamical Systems, Lecture Notes in Math. 1609, pp. 44--118,
Springer-Verlag, Berlin, 1995.

### Martin Heida

**On Homogenization of Rate-independent
Systems**

Mittwoch, 17.Februar 2016, 15.20 Uhr

We study stochastic homogenization problems of the form

$0inpartialPsi_{epsilon}(partial_{t}u^epsilon)+D{cal
E}_{epsilon}(t,u^epsilon)$,

where ${cal E}_{epsilon}: [0,T]times
B_{epsilon}tooverline{mathbb R}$

is a proper, quadratic
functional and $Psi_{epsilon}: B_{epsilon}tooverline{mathbb R}$ is
proper and 1-homogeneous and $B_{epsilon}$ is an $epsilon$-dependent
Banach space. As usual in homogenization, the index $epsilon>0$ is
a smallnes parameter and (in general) relates to the scale of the
underlying geometry of the physical system, such as crystaline
structure, microscopic cracks etc..

We focus on Prandtl-Reuss
plasticity and on elasticity problems coupled with Coulomb-friction.

### Ralf Kornhuber

**Direct and iterative methods for numerical
homogenization**

Freitag, 11. Dezember 2015, 9.15 Uhr

We present a novel approach to the finite element discretization of
elliptic problems with oscillating coefficients based on basic
concepts of frequency spitting and subspace decomposition. In this
framework, we derive and analyze a class of new discretization schemes
and contribute to the analysis of existing methods as described,
*e.g.*, Efendiev & Hou [1] or Malquist & Peterseim [2].

[1] Y. Efendiev & T. Hou: Multiscale Finite Elements, Springer,
2009.

[2] A. Malquist & D. Peterseim: Localization of
Elliptic Multiscale Problems. *Math. Comp.* 83, pp. 2583-2603,
2014.

### Dirk Pauly

**Low-Frequency asymptotics for
time-harmonic Maxwell equations in exterior domains**

Freitag, 11. Dezember 2015, 10.35 Uhr

We will prove the complete low-frequency asymptotics for time-harmonic Maxwell equations in exterior domains. We start with introducing the solution theory for time-harmonic electro-magnetic scattering problems via a generalized Fredholm alternative using the limiting absorption principle and continue with proving an adequate corresponding electro-magneto static solution theory providing also special so-called towers of static solutions. In both cases we will work in polynomially weighted Sobolev spaces. Then a comparison with the whole space solution shows that a generalized asymptotic Neumann series gives the desired asymptotics for low frequencies up to a finite sum of degenerate operators, which can be described explicitly by strongly growing towers. Finally we compare these time-harmonic Maxwell radiation solutions with the corresponding solutions provided by the eddy-current model for low frequencies.

### Agnes Lamacz

**Effective Maxwell's equations in a
geometry with flat split-rings**

Freitag, 30. Oktober
2015, 9.15 Uhr

Propagation of light in heterogeneous media is a complex subject of
research. It has received renewed interest in recent years, since
technical progress demands smaller devices and offers new
possibilities. At the same time, theoretical ideas inspired further
research. Key research areas are photonic crystals, negative index
metamaterials, perfect imaging, and cloaking. The mathematical
analysis of negative index materials, which we want to focus on in
this talk, is connected to a study of singular limits in Maxwell's
equations. We present a result on homogenization of the time harmonic
Maxwell's equations in a complex geometry. The homogenization process
is performed in the case that many (order η^{-3}), small
(order η^{1}), thin (order η^{2}) and highly
conductive (order η^{-3}) metallic split-rings are
distributed in a domain Ω⊂R^{3}. We determine the effective
behavior of this metamaterial in the limit η→0. For η>0, each
single conductor occupies a simply connected domain, but the conductor
closes to a ring in the limit η→0. This change of topology allows
for an extra dimension in the solution space of the corresponding
cell-problem. Even though both original materials (metal and void)
have the same positive magnetic permeability μ_{0}>0, we
show that the effective Maxwell system exhibits, depending on the
frequency, a negative magnetic response. Furthermore, we demonstrate
that combining the split-ring array with thin, highly conducting wires
can effectively provide a negative index
metamaterial.

### Adrien Semin

**When a thin periodic layer meets corners:
asymptotic analysis of a singular Helmholtz problem.**

Freitag, 30. Oktober 2015, 10.35 Uhr

We study here the acoustic wave propagation in a bounded domain
made of a thin and periodic layer of finite length placed into an
medium that admits periodic inhomogeneities in a vicinity of the
layer. The difficulty of this problem is the presence of the
re-entrant corners at both ends of the layer. We are interested in the
construction and the analysis of the asymptotic expansion in the whole
domain, using different ansatz in the far field region (away from the
periodic layer), close to the periodic layer and away from its
corners, and in the near field region (close to the corners). The
interaction of the corner singularities and the periodic layer the
asymptotic expansion shows up in powers of δ and δ^{α},
where δ is the layer thickness and periodicity of the
inhomogeneities and α depends on the opening angle.

In the talk we will pay particular attention to analyze the
near-field problems close to the corner and the periodic layer. For
this, we will introduce a solution extending the Kondratiev theory for
corner singularities (which is based on the Mellin transform) in the
spirit of Nazarov. This will allow us give show rigorous estimates of
the error of the asymptotic expansion. The talk will be finished by a
series of numerical experiments illustrating the theoretical results
on the example of perforated liners.

### Andreas Münch

**Asymptotic analysis of phase-field models
involving surface diffusion**

Mittwoch, 15. Juli 2015,
9.15 Uhr

Phase field models frequently provide insights to phase transitions, and are robust numerical tools to solve free boundary problems corresponding to the motion of interfaces. A body of prior literature suggests that interface motion via surface diffusion is the long-time, sharp interface limit of microscopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function. Contrary to this conventional wisdom, we show via a careful asymptotic analysis involving the matching of exponential terms that the the long-time behaviour of a degenerate Cahn-Hilliard equation with a polynomial free energy leads to a sharp interface model that couples bulk and surface diffusion, thus permitting coarsening.

### Patrik Marschalik

**Fundamental concepts of the methods of
matched asymptotic approximations and multiple scales
expansions**

Mittwoch, 15. Juli 2015, 10.35 Uhr

There are mainly two distinct approaches to construct approximate
solutions of multiple scale problems: First the *method of matched
asymptotic approximations* and second the *method of multiple
scale expansions*.

The first approach seeks for
approximations which are uniformly valid in certain subdomains and
then *matches* these approximations to a composite
approximation which is uniformly valid on the whole domain. There are
two main matching techniques. First the *intermediate matching*
based on *extension theorems* and the *overlap
hypothesis*, and secondly the matching based on Van Dyke's
*asymptotic matching principles*. Wiktor Eckhaus' elegant
notation allows us to consistently formulate these matching
principles.

For the second approach several new independent
variables are introduced. In this way, the original problem is
embedded in a higher-dimensional space of independent variables, the
new coordinates representing asymptotically rescaled spacio-temporal
dependencies. In the following process one finds an expansion that is
not uniformly valid on the whole domain per se. In order to obtain
uniform validity one needs a rule, the *non-secularity
condition*, that provides an appropriate guide in eliminating
certain indeterminacies in the expansions so that uniform validity is
achieved.

In this talk the concepts behind these two
approaches are discussed. We will see how the two matching techniques
interrelate and that one has to be very careful in stating the
*non-secularity condition* when utilizing multiples scales
expansions. Finally we will discuss differences between the two
approaches. The focus of the talk is on the concepts rather than
applications.

### Sina Reichelt

**Two-scale homogenization of nonlinear
reaction-diffusion systems involving different diffusion length
scales**

Mittwoch, 3. Dezember 2014, 9.15 Uhr

Many reaction-diffusion processes arising in civil engineering, biology, or chemistry take place in strongly heterogeneous media, for instance concrete carbonation or the spread-out of substances in biological tissues. Letting the heterogenities be periodically distributed with microscopic period length ε>0, we are facing difficulties with respect to numerical simulations and the study of pattern formation. It is therefore our aim to rigorously derive effective equations for the limit ε→0.

The system under consideration comprises one species with
characteristic diffusion length of order O(1) and another with
diffusion length O(ε), whereas both species are coupled via nonlinear
reaction terms. The slow diffusion of the second species leads to
degenerating gradient bounds and hence a lack of compactness, which in
turn prevents a straight forward convergence of the nonlinear reaction
terms. To overcome the complication of missing compactness and
nonlinearities, which was not done before, we employ the method of
periodic unfolding and we prove strong two-scale convergence of the
slow diffusive species. Finally, we obtain a novel system of coupled
reaction-diffusion equations in the two-scale space, which consists of
the macroscopic domain and the microscopic unit cell attached to each
macroscopic point. Moreover, for smooth given data, we have
convergence rates of order O(ε^{-1/2}).

### Anastasia Thöns-Zueva

**Asymptotic expansion
for nonlinear viscous acoustic equations close to rigid
wall**

Mittwoch, 3. Dezember 2014, 10.35 Uhr

In this study we continue to investigate the acoustic model of the compressible Navier-Stokes equations without mean flow and heat flux taking into account nonlinear advection term. For gases the (dynamic) viscosity η is very small and leads to viscosity boundary layers close to walls. Current work is constrained to the case of the acoustic source with a small amplitude of order O(η). We are going to derive a complete asymptotic expansion using the method of multiscale analysis, which separates velocity and pressure into far field and correcting near field. With this approach the far field solution in addition to absorption inside the boundary layer takes into account the advection term and gives highly accurate description of the pressure or velocity in the domain outside a small layer close to the boundary.

### Sergiy Nesenenko

**Homogenization in elasto-plasticity via a
phase-shift technique**

Montag, 7. Juli 2014, 9.15
Uhr

The goal of this talk is to present a homogenization method based
on a phase-shift technique for the quasistatic initial-boundary value
problem with internal variables modelling an inelastic solid body at
small strain. We start our exposition from the formal derivation of
the homogenized equations using the standard two-scale asymptotic
ansatz. After discussing the difficulties arising in the justification
of the homogenized model derived by the asymptotic ansatz, we present
the shift-phase method and show that the solutions of the microscopic
problem converges towards the solutions of the homogenized problem in
an averaged sense over phase shifts of the microstructure. Based on
this result we construct an asymptotic solution, which converges to
the solution of the microscopic problem in the L^{2}–norm,
thus avoiding the averaging.

### Barbara Wagner

**Unsteady non-uniform base states and their
stability**

Montag, 7. Juli 2014, 10.35 Uhr

In this talk we consider several pattern forming systems, ranging from phase separation of polymer blends, self-assembly of crystalline films to dewetting of polymer films. These systems all have unsteady non-uniform base states. We develop asymptotic techniques to analyse their associated linear stability problems and derive expressions for predicting the dominant wave-length of the pattern.

### Alfonso Caiazzo

**Multiscale modeling
of weakly compressible elastic materials in harmonic
regime**

Mittwoch, 23. April 2014, 16.00 Uhr

This talk focuses on the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive.

First, we extend to the time harmonic regime a recently proposed homogenized model [Baffico et al. SIAM MMS, 2008] which describes the solid-gas mixture as a compressible material in terms of an effective elasticity tensor. As next, we derive and validate numerically analytical approximations for the effective elastic coefﬁcients in terms of macroscopic parameters only. This simpliﬁed description is used to to set up an inverse problem for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

### Antonin Novotny

**Discrete relative entropy and error
estimates for some finite volume/finite element schemes to
compressible Navier-Stokes equations**

Mittwoch, 23. April
2014, 17.00 Uhr

We will talk about several issues related to the notions of weak solutions, dissipative solutions and stability properties to the compressible Navier-Stokes system aiming applications in the error analysis of some numerical approximations to these equations.

### Stefan Neukamm

**Quantitative results in stochastic
homogenization**

Donnerstag, 27. Juni 2013 um 9.00 Uhr in
Raum MA 415

I will present recent quantitative results for the stochastic homogenization of linear elliptic equations with random coefficients in a discrete setting. Classical qualitative homogenization theory states that on large length scales the random coefficients can be replaced by homogenized coefficients that are deterministic and constant in space. The homogenized coefficients are characterized by a formula that involves the solution to the so called "corrector problem". In contrast to periodic homogenization, in the stochastic setting the corrector problem is a highly degenerate elliptic equation on a probability space. In this talk I will explain how to obtain various optimal estimates on the corrector, on approximations of the homogenized coefficients and on the homogenization error based on a quantification of ergodicity that in particular covers the case of independent and identically distributed coefficients. The approach is mainly based on elliptic and parabolic regularity theory combined with some elements of statistical mechanics and probability theory. The talk is based on joint work with A. Gloria (Université Libre de Bruxelles) and F. Otto (MPI Leipzig).

### Maria Bruna

**Diffusion of finite-size
particles: multiple species and confined geometries**

Donnerstag, 27. Juni 2013 um 10.30 Uhr in MA 415

We
discuss nonlinear Fokker-Planck models describing diffusion processes
with

particle interactions. These models are motivated by the
study of systems in biology and ecology composed of many interacting
individuals, and arise as the population-level description of a
stochastic particle-based model. In particular, we consider a system
finite-sized hard-core interacting Brownian particles and use the
method of matched asymptotic expansions to obtain a systematic model
reduction. The result is a nonlinear Fokker-Planck equation, with the
nonlinear term accounting for the size-exclusion interactions. We will
present two applications: the diffusion of heterogeneous species (e.g.
two types of cell populations), and the diffusion in confined domains
(e.g. ion transport in channels).

### Adrien Semin

**Construction and analysis of improved
Kirchhoff conditions for acoustic wave propagation in a junction of
thin slots**

Donnerstag, 27. Juni 2013 um 11.30 Uhr in MA
415

We study the acoustic wave propagation in a network
of thin slots. As "thin slots" we consider structures
whose transverse direction is much smaller than the wavelength, and we
focus on what happens in a junction of thin slots. We study here a
family of problems where the transverse cross sections of each slot
scales with the factor ε of a given reference cross section, and
describe the corresponding solutions when ε tends to 0 via the
solution of some approximated model defined on the 1D limit geometry.
We recall the limit conditions we obtain at the junction (the
so-called Kirchhoff conditions) and we show how to improve these
conditions. More explicitly we discuss the conditions for a junction
of two slots with an angle.

### Alexander Mielke

**Evolutionary Gamma convergence and
amplitude equations**

Montag, 8. April 2013, 14.15 Uhr in
MA 313

We consider the spatially homogeneous Swift-Hohenberg equation as a
prototype of a pattern-forming system. Close to the threshold of
instability the solutions behave locally as a periodic solution that
is modulated on a larger spatial scale. This modulation is described
by the so-called amplitude equation also called envelope equation,
which in this case is the real Ginzburg-Landau equation.

We
consider the amplitude equation as an effective equation for the
multiscale system. While first proofs of this multiscale limit were
given in the early 1990, we provide a new proof that relies on the
gradient structure of the Swift-Hohenberg equation. The general theory
of evolutionary Gamma convergence provides sharper results for the
convergence theory and highlights the underlying structural properties
of the system.

### Carsten Hartmann

**Optimal control of multiscale
diffusions**Montag, 8. April 2013, in Anschluß an
Vortrag um 14.15 Uhr in MA 313

Stochastic differential equations with multiple time scales appear
in various fields of applications, e.g. biomolecular dynamics,
material sciences or climate modelling. The separation between the
fastest and the slowest relevant timescales poses severe difficulties
for control and simulation of such systems. If fast and slow scales
are well separated, however, asymptotic techniques for diffusion
processes are a means to derive simplified reduced order models that
are easier to simulate and control.

In certain situation, the
limit theorems of averaging and homogenization theory provide bounds
on the approximation error, e.g. for the relevant slow degrees of
freedom. The situation becomes more difficult if the system is subject
to additional control variables that are chosen so as to maximize or
minimize a given cost functional. One of the questions that arise here
is whether the optimal (feedback) control computed from a reduced
model is a reasonable approximation of the optimal control obtained
from the full system, the computation of which is often infeasible. It
turns out that very few reduced models are "backward stable"
in the aforementioned sense, even though they are forward stable, in
that they give good approximations when the control is known in
advance. This talk tries to shed light on this issue. To this end we
review the "standard" asymptotic theory for uncontrolled
stochastic differential equations, along with illustrating examples
from physics, engineering and biology, and discuss the problem of
backward stability.

### Maciek Korzec

**Multiple scales in silicon type
microstructure growth**

Donnerstag, 24. Januar 2013, 9.30
Uhr in MA 415

Multiple scales are intrinsically present in continuum models for growth of thin crystalline silicon type layers that are incorporated in modern solar cells. The bravais lattices on atomic level lead to continuous surface energy density formulas used on a larger scale or to a strain energy density given due to a lattice mismatch. Moving grain boundaries in amorphous materials are very thin in comparison to the extent of the bulk material. Once the re-crystallization is complete the time-scale is significantly decreased. Coarsening of quantum dots in surface diffusion based models slows down in time. For the long-time behavior, or completely equilibrated states, a very large time-scale needs to be considered. While the general evolution may be slow, topological changes in an Ostwald ripening fashion - i.e. the vanishing of a quantum dot or of a grain - may happen fast, so that adaptivity in time is sought in numerical schemes. In this talk I present various aspects of modeling, analysis and simulation of evolution equations aimed for understanding and improving microstructure growth for application in photovoltaics. Therefore one has to properly cope with the different scales.

### Thomas Petzold

**Modelling and simulation of
multi-frequency induction hardening of steel parts**

Donnerstag, 24. Januar 2013, im Anschluss an den Vortrag um 9.30
Uhr in MA 415

Induction hardening is a modern method for the heat treatment of steel parts. A well directed heating by electromagnetic waves and subsequent quenching of the workpiece increases the hardness of the surface layer. The process is very fast and energy efficient and plays a big role in modern manufacturing facilities in many industrial application areas.

In the talk, a model for induction hardening of steel parts is presented. It consist of a system of partial differential equations including Maxwell's equations and the heat equation. Both of these equations live on different time scales. Due to the use of multiple frequencies, also different time scales occur within Maxwell's equations. The finite element method is used to perform numerical simulations in 3D. This requires a suitable discretization of Maxwell's equations in space, using edge-finite-elements, and in time. Further challenges when solving applied industrial problems, e.g. arising from nonlinear material data, will be addressed and simulation results will be presented.

### Daniel Peterseim

**A New Multiscale Method for (Semi-)Linear
Elliptic Problems**

Freitag, 30. November 2012, 9.30 Uhr
in MA 415

We propose and analyze a new multiscale method for solving (semi-)linear elliptic problems with heterogeneous and highly variable coefficients. For this purpose we construct a generalized finite element basis that spans a low dimensional global multiscale space based on some coarse mesh. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of order H|log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations we give a rigorous proof for the linear convergence of the energy error with respect to the coarse mesh size without any pre-asymptotic effects. Moreover, we show that the discretized operator captures small eigenvalues of the partial differential operator very accurately (in a superconvergent way). The results are illustrated in numerical experiments.

### Ludwig Gauckler

**Modulated Fourier expansion: Multiscale
expansions for analysing oscillatory Hamiltonian systems**

Freitag, 30. November 2012, im Anschluss an den Vortrag um 9.30 Uhr
in MA 415

Modulated Fourier expansions are multiscale expansions in time for analysing weakly nonlinear oscillatory systems over long times, both continuous and discrete systems, in finite and infinite dimensions. In the talk we will consider a finite dimensional oscillatory Hamiltonian system coupled to a slow motion as a model problem. We will discuss the exchange of energy between the fast (oscillatory) and the slow system, and we will explain how modulated Fourier expansions can be used to explain the lack of any energy exchange on long time intervals.

### Rupert Klein

**A three-scale asymptotic problem in
atmospheric flows**

Donnerstag, 21. Juni 2012, 9.30 Uhr in
MA 313

The Euler and Navier-Stokes equations for incompressible flow can be justified as low Mach number asymptotic limiting models for flows on engineering length and time scales. Atmospheric flows generally feature small Mach numbers as well but, as a consequence of their much larger characteristic scales, they are not "incompressible". In fact, today there remain several competing candidates for an atmospheric analogue of the engineers's incompressible flow equations. In this talk I will explain how this ambiguity is rooted in an asymptotic three time scale limit for atmospheric flows, and I will discuss recent steps towards a rigorous justification of associated "sound-proof" model equations.

### Kersten Schmidt

**High order asymptotic expansion for
viscous acoustic equations close to rigid walls**

Donnerstag, 21. Juni 2012, im Anschluss an den Vortrag um 9.30 Uhr
in MA 313

In this study we are investigating the acoustic equations as a
perturbation of the Navier-Stokes equations around a stagnant uniform
fluid and without heat flux. For gases the viscosities η and η'
are very small and lead to viscosity boundary layers close to walls.
We will restrict our attention on those viscosity boundary layers and
do not consider non-linear convection.

As a small factor η
comes out in front of the *curl curl* operator in the governing
equations, the system is singularly perturbed, i.e., first, its
formal limit η→ 0 does not provide a meaningful solution, and
secondly, a boundary layer close to the wall ∂Ω appears. The choice
of asymptotic expansion method seems to be the best adapted to this
case.

In this approach we separate the solution in far
field and correcting *near field*, where far field represents
the area away the wall and exhibits no boundary layer, at the same
time near field decays exponentially outside the zone of size O(√η)
from the boundary.

To complete the solution, effective
(impedance) boundary conditions are derived for the far
field.

## Adresse

**TU Berlin**

Institut für Mathematik

Sekr. MA 6-4

Straße des 17. Juni 136

D-10623 Berlin

## So finden Sie uns

Mathematikgebäude (MA)3. Obergeschoss

Räume 363, 365 u. 379

- Campusplan [41]
- Campusplan (pdf) [42]
- Anfahrt (Google Maps) [43]

## Links

- C++ Bibliothek "Concepts" [47]

mbers/rupert_klein.html

diff/fg_mathematische_methoden_in_der_photovoltaik/v_me

nue/mitarbeiter/wagner/parameter/de/

diff/nachwuchsgruppe_dr_kersten_schmidt/nachwuchsgruppe

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