MATHEON Multiscale Seminar

Abstracts

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.