MATHEON Multiscale Seminar

# MATHEON Multiscale Seminar

 Coordinators: Prof. Dr. Rupert Klein (FU Berlin), HomepageDr. Sergiy Nesenenko (TU Berlin), HomepageDr. Kersten Schmidt (TU Berlin), HomepageProf. Dr. Barbara Wagner (TU Berlin), Homepage Dates: 1-2 seminars per term with 2 talks each Content: Talks about recent work on partial differential equations with multiple scales
Dates
Time
Room
Speaker and title of talk
Poster
17.02.2016,
2.00 p.m.
TU Berlin, MA 313
Prof. Dr. Nicola Popovic (U Edinburgh)
A geometric analysis of fast-slow models for stochastic gene expression
(Abstract)

Dr. Martin Heida (WIAS Berlin)
On Homogenization of Rate-independent Systems
(Abstract)
Poster
11.12.2015, 9.15 a.m.
TU Berlin, MA 313
Prof. Dr. Ralf Kornhuber (FU Berlin)
Direct and iterative methods for numerical homogenization (Abstract)

Prof. Dr. Dirk Pauly (Universität Duisburg-Essen)
Low-frequency asymptotics for time-harmonic Maxwell equations in exterior domains (Abstract)
Poster
30.10.2015,
9.15 a.m.
TU Berlin, MA 313
Dr. Agnes Lamacz (TU Dortmund)
Effective Maxwell's equations in a geometry with flat split-rings (Abstract)

Dr. Adrien Semin (TU Berlin)
Error estimates for an Helmholtz transmission problem with a perforated thin structure and corner singularities (Abstract)
Poster
15.07.2015,
9.15 a.m.
TU Berlin, MA 313
Prof. Dr. Andreas Münch (University of Oxford, UK)
Asymptotic analysis of phase-field models involving surface diffusion (Abstract)

Dr. Patrik Marschalik (Universität Mainz)
Fundamental concepts of the methods of matched asymptotic approximations and multiple scales expansions (Abstract)
Poster
03.12.2014, 9.15 a.m.
TU Berlin, MA 415
Sina Reichelt (WIAS Berlin)
Two-scale homogenization of nonlinear reaction-diffusion systems involving different diffusion length scales (Abstract)

Dr. Anastasia Thöns-Zueva (TU Berlin)
Asymptotic expansion for nonlinear viscous acoustic equations close to rigid wall (Abstract)
Poster
07.07.2014, 9.15 a.m.
TU Berlin, MA 415
Dr. Sergiy Nesenenko (Universität Duisburg-Essen)
Homogenization in elasto-plasticity via a phase-shift technique
(Abstract)

Prof. Dr. Barbara Wagner (TU Berlin)
Unsteady non-uniform base states and their stability (Abstract)

Poster
23.04.2014,
4.00 p.m.
TU Berlin,
MA 313
Dr. Alfonso Caiazzo (WIAS Berlin)
Multiscale modeling of weakly compressible elastic materials in harmonic regime
(Abstract)

Prof. Dr. Antonin Novotny (University of Toulon, France)
Discrete relative entropy and error estimates for some finite volume/finite element schemes to compressible Navier-Stokes equations
(Abstract)

Poster
27.06.2013,
9.00 a.m.
TU Berlin,
MA 415
Dr. Stefan Neukamm (WIAS Berlin)
Quantitative results in stochastic homogenization (Abstract)

Dr. Maria Bruna (University of Oxford, UK)
Diffusion of finite-size particles: multiple species and confined geometries (Abstract)

Dr. Adrien Semin (TU Berlin)
Construction and analysis of improved Kirchhoff conditions for acoustic wave propagation in a junction of thin slots (Abstract)

Poster
08.04.2013,
2.15 p.m.
TU Berlin,
MA 313
Prof. Dr. Alexander Mielke (WIAS / HU Berlin)
Evolutionary Gamma convergence and amplitude equations (Abstract)

Prof. Dr. Carsten Hartmann (FU Berlin)
Optimal control of multiscale diffusion (Abstract)

Poster
24.01.2013,
9.30 a.m.
TU Berlin,
MA 415
Dr. Maciek Korzec (TU Berlin)
Multiple scales in silicon type microstructure growth (Abstract)

Thomas Petzold (WIAS Berlin)
Modelling and simulation of multi-frequency induction hardening of steel parts (Abstract)

Poster
30.11.2012,
9.30 a.m.
TU Berlin,
MA 415
Dr. Daniel Peterseim (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
21.06.2012,
9.30 a.m.
TU Berlin,
MA 313
Prof. Dr. Rupert Klein (FU Berlin)
A three-scale asymptotic problem in atmospheric flows (Abstract)

Dr. Kersten Schmidt (TU Berlin)
High order asymptotic expansion for viscous acoustic equations close to rigid walls (Abstract)

Poster

## Abstracts

### Nicola Popovic

A geometric analysis of fast-slow models for stochastic gene expression
Wednesday, Februar 17th 2016, 2.00 p.m.

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
Wedesnay, Februar 17th 2016, 3.20 p.m.

We study stochastic homogenization problems of the form
$0\in\partial\Psi_{\epsilon}(\partial_{t}u^\epsilon)+D{\cal E}_{\epsilon}(t,u^\epsilon)$,
where ${\cal E}_{\epsilon}: [0,T]\times B_{\epsilon}\to\overline{\mathbb R}$
is a proper, quadratic functional and $\Psi_{\epsilon}: B_{\epsilon}\to\overline{\mathbb R}$ 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
Friday, December 11th, 2015, 9.15 a.m.

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
Friday, December 11th, 2015, 10.35 a.m.

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
Friday, October 30th, 2015, 9.15 a.m.

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 η1), thin (order η2) and highly conductive (order η-3) metallic split-rings are distributed in a domain Ω⊂R3. 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 μ0>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.

Error estimates for an Helmholtz transmission problem with a perforated thin structure and corner singularities
Friday, October 30th, 2015, 10.35 a.m.

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 δα, 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
Wednesday, July 15th, 2015, 9.15 a.m.

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
Wednesday, July 15th, 2015, 10.35 a.m.

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
Wednesday, December 3rd, 2014, 9.15 a.m.

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
Wednesday, December 3rd, 2014, 10.35 a.m.

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
Monday, July 7th, 2014, 9.15 am

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 L2–norm, thus avoiding the averaging.

### Barbara Wagner

Unsteady non-uniform base states and their stability
Monday, July 7th, 2014, 10.35 am

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
Wednesday, April 23rd, 2014, 4.00pm

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
Wednesday, April 23rd, 2014, 5.00pm

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
Thursday, June 27th, 2013 at 9.00 am in room 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
Thursday, June 27th, 2013 at 10.30 am 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).

Construction and analysis of improved Kirchhoff conditions for acoustic wave propagation in a junction of thin slots
Thursday, June 27th, 2013 at 11.30 am 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
Monday, April 8th 2013, 2.15 pm 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
Monday, April 8th 2013, following the talk at 14.15 pm 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
Thursday, January 24th 2013, 9.30 am 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
Thursday, January 24th 2013, following the talk at 9.30 am 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
Friday, November 30th 2012, 9.30 am 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
Friday, November 30th 2012, following the talk at 9.30 am 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
Thursday, June 21st, 2012, 9.30 am 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
Thursday, June 21st, 2012, following the talk at 9.30 am 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.