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Seminar Numerische Mathematik / Numerical mathematics seminars
aktuelles Programm / current program

Donnerstag, 26. 11. 2009, 14:00 Uhr (ESH)

A. Khrabustovski   (B. Verkin Institute for Low Temperature Physics and Engineering, Ukraine)
Analysis of a general linear reaction-diffusion-system by means of homogenization methods

In the talk we discuss some important qualitative properties of an initial boundary value problem for a complicated linear reaction-diffusion system. Such type systems describe the transport of particles of various species in a medium and the transformation of the particles into each other. The transport can be forced by local (diffusion) and nonlocal interaction (jumps) of the particles with the medium, moreover, the transport and the reactions can be nonlocal in time (memory effects). We show that the solution to this problem preserves positivity (if the initial vector-function is positive) and converges exponentially to some explicitly calculated constant as time goes to infinity. For the proof of this results we developed a new method based on the possibility to approximated the original system by a simple diffusion equation on a specially constructed Riemannian manifold depending on a small parameter. In another words, we -Y´replace¡ our original complicated system on a simple domain by a simple equation on a complicated manifold, i.e. our original system will be the result of the homogenization of the diffusion equation on a complicated manifold. This method gives a microscopic interpretation of the terms of the system and allows us to "guess" easily the limit constant.

Montag, 23. 11. 2009, 14:00 Uhr (ESH)

A. Caiazzo   (INRIA Rocquencourt, France)
Complex Automata models for in-stent restenosis and FE schemes for blood flows in stented aneurysms

In the first part, a multiscale model for in-stent restenosis, a coronary artery disease resulting in an abnormal tissue growth inside blood vessels, will be described. The model is realized coupling a lattice Boltzmann scheme for blood flow, an agent based model for the biological tissue growth, and finite difference scheme for drug diffusion. In practice, the model is realized as a multiscale Complex Automata, i.e. a graph of single-scale models connected through a CxA-dedicated coupling library.
The second part of the talk focuses on blood flow through porous stents in Abdominal Aortic Aneurysms (AAA), investigating different FE formulations to take into account the effect of a porous interface on an incompressible fluid. Theoretical properties of the schemes will be discussed, and numerical validations (with simple benchmarks and realistic geometries, including Blood-Vessel interaction) will be shown.

Montag, 23. 11. 2009, 10:00 Uhr (ESH)

Dr. V. Dymkou   (Coventry University, UK)
MHD turbulence and spectral methods: Simulation, stabilization and control

The proposed work is devoted to the theoretical and numerical study of the liquid metal flows of an electrically conducting fluid in the presence of external magnetic field. Such magnetohydrodynamic (MHD) flows are important in different contexts of theoretical fluid mechanics, mathematical physics or of nuclear fusion reactors, MHD turbulent flows play a big role in transporting the reactor heat, which is crucial for the efficiency of the device. Since existing numerical models cannot be efficiently applied in many realistic MHD flows due to the complex geometries or demands for significant computational resources, new developments are required. This shall be where we concentrate our efforts using the original numerical method based on the least dissipative modes and low-dimensional models, which present the advantage of mimicking the flow. Moreover, since the behavior of the flow in MHD liquid metal experiments can be externally forced by injecting electric current though metallic electrodes embedded in insulating walls, we design a flow control that eliminates undesired fluctuations and stabilizes the flow, which has an important practical applications.

Donnerstag, 19. 11. 2009, 14:00 Uhr (ESH)

E. Hoarau - Andra   (Universite Blaise Pascal, Clermont-Ferrand)
Finite-volume simulation of carbon steel corrosion in a nuclear waste deep repository

In France, the solution explored to manage the high and intermediate level radioactive waste is to store them in a deep geological repository. The waste packages for the high level waste are mainly made of carbon steel and are designed to confine the waste for about 1000 years. An important effort is dedicated to study the long term degradation of this materiel, particularly the corrosion processes. In this context, predictive models have to be built to evaluate the confining duration of the waste packages. A model has been developed to represent the corrosion processes for carbon steel under anaerobic conditions. It is partly based on observations made on small scales experiments and archaeological analogous. The DPCM model (Diffusion Poisson Coupled Model), developed in Andra, is formulated in terms of conservation equations governing the mass and charge transport, coupled with a Poisson equation for the electric potential. The particularity of the proposed model, compared to those published before, is to obtain the potential directly by solving the Poisson equation (in literature potential is often assumed to be linear) and to consider the unsteady dimension of the problem. This leads to a set of non linear partial differential equations (derive-diffusion equations for the species), closed to those used for semi-conductors. The main difficulties are the high coupling between all the equations, the particular form of the boundary conditions (Robin condition) and the high differences in terms of characteristic time for species transport. This talk presents the based equation of the model, the numerical methods implemented to solve the set of equations and numerical results obtained. The numerical model has been built with finite volume formulation and it takes into account the change of domain thickness due to the formation of corrosion products. The moving boundaries have been implemented using two different methods: (i) remeshing techniques at each time step (ii) introduction of a fixed spatial domain by transforming the variables. The corrosion rates obtained with the model for several set of parameters will be shown. Some results will be presented to illustrate the capacity of the model to describe corrosion processes.

Mittwoch, 11. 11. 2009, 15:15 Uhr (ESH)

Prof. R. Eymard   (Universite Paris Est, Departement de mathematiques)
Application of discrete functional analysis to the convergence study of a finite volume scheme for nonlinear elliptic/parabloic problems

In the continuous setting, the use of Minty and Leray-Lions tricks are well-known for getting the existence of a solution to some nonlinear elliptic problems, such as the p-Laplacian diffusion problem. We show how these tools have a discrete counterpart, applicable in two settings (p-Laplacian, regularized mean curvature motion level set equation).

Donnerstag, 6. 10. 2009, 14:15 Uhr (ESH)

Prof. Ph. Angot   (Universite de Provence,~France)
On the well-posed coupling between free fluid and porous flows

We present a new fictitious domain model for the Brinkman or Stokes/Brinkman problems in order to handle general embedded jump boundary conditions (E.B.C.) on an immersed interface. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface separating two subdomains. It is issued from a generalization to vector problems of a previous model for scalar elliptic problems with jump boundary conditions. The proposed model is first proved to be well-posed. A class of fictitious domain methods can be then derived within the same unified formulation which provides various interface or boundary conditions, e.g. a given-traction of Neumann type or a velocity Dirichlet condition. In particular, we prove the consistency of the given-traction E.B.C. method including the so-called do nothing outflow boundary condition. In the second part, we discuss the global solvability of some models with stress or velocity jump boundary conditions for the coupling of free fluid and porous flows. When the porous flow is governed by the Brinkman equation, the Ochoa-Tapia and Whitaker stress jump boundary condition with a continuous velocity was derived with volume averaging techniques in for the momentum transport at the fluid-porous interface, instead of the usual stress and velocity continuity at the interface. This condition appears to fall into our general framework. Hence, as a by-product of our work, this latter jump boundary condition yields a well-posed Stokes/Brinkman problem. Then, we show by an asymptotic analysis the well-posed-Ness of the Stokes/Darcy problem with the Beavers and Joseph boundary condition, more recently derived by homogenization techniques. Our result does hold not only for the \textit{Saffman's} simplication of it in the case of a porous bulk with a small permeability. This was not already clearly stated up to our knowledge.

Montag, 27. 07. 2009, 10:00 Uhr (ESH)

Prof. B. Zaltzmann   (Ben Gurion University of the Negev, Jacob Blaustein Institute for Desert Research)
Probing the extended space charge by harmonic disturbances

We assess the possibility of probing the diffuse electric double layer at a permeable charge-selective interface, such as a non-blocking electrode or ion-exchange membrane, under a finite steady-state current/voltage bias by small harmonic high frequency current/voltage disturbances. Our main conclusion is that for a finite under-limiting bias, the electric double layer at such an interface is not amenable to this kind of probe; the high-frequency response of the entire system is dominated by the quasi-electro-neutral bulk. This is similar to the equilibrium zero-bias case. On the other hand, the extended space charge in such double layers may be probed in this way both by the linear and non-linear response, correspondingly by the method of electric impedance spectroscopy and by the previously described anomalous rectification effect. The latter appears preferable over the former as a potential experimental tool for the study of the extended space charge of a non-equilibrium electric double-layer.

Donnerstag, 16. 07. 2009, 14:00 Uhr (ESH)

Dr. A. Nemet   (ALSTOM, Switzerland Ltd.)
Gas turbines for efficient, flexible and clean power generation - How ALPEG/BOP supports mastering the challenges in development

Mittwoch, 1. 07. 2009, 15:15 Uhr (ESH)

Dr. C. Zheng   (Tsinghua University Beijing, China)
Gaussian beam method for the boundary value problem of Helmholtz equation in the high frequency regime

Donnerstag, 18. 06. 2009, 14:00 Uhr (ESH)

Dr. G. Carini   (Brookhaven National Laboratory, National Synchrotron Light Source, New York)
Photon science at Brookhaven National Laboratory: Light sources, detectors and applications

Mittwoch, 29. 04. 2009, 15:15 Uhr (ESH)

Prof. D. Bothe   (Technische Universität Darmstadt)
Nonlinear evolution equations with applications to reaction-diffusion systems

Montag, 27. 04. 2009, 17:00 Uhr (ESH)

Prof. D. Bothe   (Technische Universität Darmstadt)
Modelling and Analysis of transport processes at fluidic interfaces

Mittwoch, 08. 04. 2009, 15:15 Uhr (ESH)

Prof. C. Besse   (Université Lille 1, France)
Open boundary conditions and computational schemes for Schrödinger equations with general potentials and nonlinearities

This talk will address the construction of absorbing boundary conditions for the one-dimensional Schrödinger equation with a general variable repulsive potential or with a cubic nonlinearity. Semi-discrete time schemes, based on Crank-Nicolson approximations, will be presented for the associated initial boundary value problems. Finally, some numerical simulations will give a comparison of the various absorbing boundary conditions to analyse their accuracy and efficiency.

Donnerstag, 12. 02. 2009, 14:00 Uhr (ESH)

Prof. Messoud A. Efendiev   (Helmholtz Zentrum München, GmbH)
On a new class of degenerate parabolic systems arising in the modelling of biofilms

We deal with a new class of equations arising in the modelling of biofilms. Based on the experimental observations we will derive these equations. The peculiarity of these equations is that they are comprises two kind of degeneracy:porous medium as well as fast diffusion. Long time dynamics based on the global attractors of such equations as well as biological interpetations is discussed. Some open problems will be discussed.

Mittwoch, 28. 01. 2009, 15:15 Uhr (ESH)

PD Dr. D. Ševčovič   (Comenius University Bratislava, Slovak Republic)
Higher order estimates for the curvature and nonlinear stability of stationary solutions for curvature flow with triple junction

We are interested in the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. We derive higher order energy estimates yielding a priori estimates for the H2-norm of the curvature of moving arcs. As a consequence of these estimates we will be able to prove exponential decay of the H2-norm of the curvature. As a consequence, we will show that a linear stability criterion due to Ikota and Yanagida [2] is also sufficient for nonlinear stability of stationary solutions for curvature flow with triple junction. This is a joint work with Harald Garcke and Yoshihito Kohsaka [1].
[1] H. Garcke, Y. Kohsaka and D. Sevcovic: Nonlinear stability of stationary solutions for curvature flow with triple junction, submitted. arXiv:0802.3036
[2] Ikota R. and Yanagida E.: A stability criterion for stationary curves to the curvature-driven motion with a triple junction, Differential Integral Equations 16 (2003), 707--726.

Montag, 12. 01. 2009, 10:00 Uhr (ESH)

Prof. Dr. M. H. Gutknecht   (ETH Zürich und TU Berlin)
IDR in variations

The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was first developed by Sonneveld around 1979 and documented on three and a half pages of a 1980 proceedings paper by Wesseling and Sonneveld. Soon after IDR, Sonneveld introduced his widely applied Conjugate Gradient Squared (CGS) algorithm. Then, in 1990, van der Vorst suggested Bi-CGSTAB that he claimed to improve both those methods. Bi-CGSTAB has become a method of choice for nonsymmetric linear systems, and it has been generalized in various ways in the hope of further improving its reliability and speed. Among these generalizations there is the ML(k)BiCGSTAB method of Yeung and Chan, which in the framework of block Lanczos methods can be understood as a variation of Bi-CGSTAB with right-hand side block size 1 and left-hand side block size k. In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR is equally fast but preferable to Bi-CGSTAB, and that IDR(s) may be much faster than IDR = IDR(1). It turned out that IDR(s) is closely related to BiCGSTAB if s = 1 and to ML(s)BiCGSTAB if s > 1. In 2008, a new, particularly ingenious and elegant variant of IDR(s) has been proposed by the same authors.
In this talk we first try to explain the basic, seemingly quite general IDR approach, which differs completely from traditional approaches to Krylov space methods. Then we compare the basic properties of the above mentioned methods and discuss some of their connections.