Doktorandenseminar des WIAS


Forschungsgruppe ''Numerische Mathematik und Wissenschaftliches Rechnen''


2012     (2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004, 2003, 2002)

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

Donnerstag, 27. 06. 2013, 14:00 Uhr (ESH)

St. Moore   (RICAM, Linz, Austria)
Discontinuous Galerkin isogeometric analysis for elliptic PDEs on manifolds

Isogeometric analysis uses the same class of basis functions for both, representing the geometry of the computational domain and approximating the solution. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This patch representation corresponds to a domain decomposition. In this talk, we will present a Discontinuous Galerkin (DG) Method that allows for discontinuities only along the subdomain (patch) boundaries. The required smoothness is obtained by the DG terms associated with the boundary of the subdomains. The construction and corresponding discretization error analysis of such DG scheme will be presented for Elliptic PDEs living in plane domains as well as on open and closed surfaces.
This is a joint talk with Ulrich Langer (RICAM, Linz, Austria).

Freitag, 24. 05. 2013, 10:30 Uhr (ESH)

Prof. H.-J. Starkloff   (Westsächsische Hochschule Zwickau)
On some questions related to generalized polynomial chaos expansions

Generalized polynomial chaos expansions are series expansions of random variables or random functions using orthogonal polynomials in certain basic random variables. They are a versatile tool for efficiently working with stochastic quantities. So, recently they are widely used for the investigation of stochastic models, for uncertainty quantification methods or stochastic simulation purposes. In particular, recent approaches for the solution of ordinary or partial differential equations with random parameters often use certain versions of generalized polynomial chaos expansions. In the talk basic definitions and properties of generalized polynomial chaos expansions are presented. Furthermore some problems are considered which are related to convergence properties or algebraic transformations of generalized polynomial chaos expansions.
The talk is based on joint work with A. Mugler (BTU Cottbus) and O. Ernst and B. Sprungk (TU Chemnitz).

Dienstag, 23. 04. 2013, 11:00 Uhr (HVP 11a, 4.13)

M. Rother   (Otto-von-Guericke-Universität Magdeburg)
Eine Reise ins Ich und zurück in 20 Minuten

Der vorgesehene Vortrag soll den Bewerber vorstellen und das Aufgabengebiet seiner Tätigkeit während der ersten Arbeitsstelle nach dem Studium umreißen. Im ersten Teil der Präsentation stellt sich der Bewerber kurz vor und beschreibt persönliche Interessen für fachliche Vertiefungsgebiete. Der fachliche und zweite Vortragsteil wird sich mit der mathematischen Handhabung von nicht-lokalen Evolutionen von Populationsbilanzen beschäftigen. Solche Gleichungen findet man zum Beispiel in der Biolaborforschung. Dort kann man verteilte Größen messen, die wiederum modellhaft die Virusausinfektion in Zellkulturen beschreiben. Das Hauptproblem liegt in der Entwicklung einer stabilen und effizienten numerischen Methode zur Approximation der Gleichungen in mehr als 3 Dimensionen (angestrebt sind 6 bis 10).

und 13:30 Uhr (ESH)

Prof. J. Wensch   (TU Dresden)
Efficient time integration methods for the compressible Euler equations in atmospheric dynamics

The simulation of atmospheric dynamics is an important issue in Numerical Weather Prediction. It relies on the numerical solution of the Euler equations. These equations exhibit phenomena on different temporal scales. In the lower troposphere sound waves propagate approximately ten times faster than the advective waves. We use a spatial discretization based on finite volumes. An approach to overcome the CFL restriction caused by sound waves are split-explicit methods. By multirate techniques the terms relevant for sound waves are integrated by small time steps with a cheap time integration procedure, whereas the slow processes are solved by an underlying Runge-Kutta method using a larger macro step size. The analysis of these methods is based on the interpretation as an exponential or Lie group integrator. We construct methods based on TVD-RK schemes and discuss order and stability properties. Numerical simulation results for established benchmark problems are given.

Donnerstag, 18. 04. 2013, 14:00 Uhr (ESH)

M. Schmuck   (Imperial College London)
Upscaling of ionic transport equations in strongly heterogeneous media and finite element approximations

We consider the well-accepted Nernst-Planck-Poisson equations [6] for the description of ionic transport and electrokinetic phenomena such as electro-phoresis and -osmosis. Applications range from designing microfluidic devices, energy storage devices, and semiconductors to emulating communication in biological cells by synthetic nanopores. Based on this classical description, we derive a new effective macroscopic set of equations [1,2,3] which describe binary symmetric electrolytes in porous and strongly heterogeneous media. Heterogeneous materials naturally induce corrected transport parameters which we call ''material tensors''. Our systematic and well-accepted upscaling strategy by homogenization gives a reliable understanding on the influence of the micro-geometry on the macroscale. The new equations provide an essential computational advantage by a strong reduction of the degrees of freedom in comparison to the classical description which requires a full resolution of the pore geometry and hence a high-dimensional resolution of the problem. For the classical descritpion, we briefly present a linear finite element scheme [4,5] that is reliable in the sense that discrete solutions conserve mass, are non-negative and bounded, and converge towards weak solutions. Finally, we can qualitatively and quantitatively characterize the suitability of the new upscaled equations for specific applications by error estimates [2] which compare the exact microscopic solution with the solution of the new effective transport equations.
References:
[1] M. Schmuck, A new upscaled Poisson-Nernst-Planck system for strongly oscillating potentials, J. Math. Phys. 54:021504 (2013).
[2] M. Schmuck, First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations, Z. angew. Math. Mech. 92:304-319 (2012).
[3] M. Schmuck and P. Berg, Homogenization of a catalyst layer model for periodically distributed pore geometries in PEM fuel cells, Appl. Math. Res. Express. 2013(1):57-78 (2012).
[4] A. Prohl and M. Schmuck, Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system, ESAIM, Math. Model. Numer. Anal. 44(3):531-571 (2010).
[5] A. Prohl, and M. Schmuck, Convergent discretizations for the Nernst-Planck-Poisson system, Num. Math. 111 (4):591-630 (2009).
[6] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Mod. Meth. Appl. S. 19(6):993- 1015 (2009).

und 15:30 Uhr (Raum: 406)

Dr. L. Kamenski (The University of Kansas, USA)
Anisotrope Gitter: Den Zusammenhang zwischen Gittern und Eigenschaften der Diskretisierungsverfahren richtig verstehen

Dienstag, 02.04.2013, 9:00 Uhr (ESH)

G. Müller   (TU Berlin)
Beispiele numerischer Verfahren für PDEs und Optimalsteuerungsprobleme

und 12:30 Uhr (ESH)

Ch. Merdon   (HU Berlin)
Garantierte Fehlerkontrolle bei PDGLen mit aktuellen Equilibrierungstechniken

und 14:00 Uhr (ESH)

R. Richter   (Bundesanstalt für Materialforschung und -prüfung (BAM) )
Numerische Verfahren für die aktive Thermografie zur Untersuchung von Rückwandgeometrien und Singularitäten einer geometrischen Evolutionsgleichung (MCF)

Donnerstag, 14. 02. 2013, 14:00 Uhr (ESH)

Dr. L. Kamenski   (The University of Kansas)
Adaptive finite elements with anisotropic meshes

In this talk I will try to give an overview over anisotropic mesh adaptation for the finite element method, advantages and disadvantages of anisotropic meshes and how to construct them. The commonly used methods to obtain much needed directional information are either using error estimates or approximating derivatives of the true solution (post-processing). Both methods perform well in practice although they both still have open theoretical questions. I will try to give an explanation on how error estimates can be used for the mesh adaptation and why an inaccurate Hessian recovery is still good enough for the mesh adaptation.
I will also give some recent results on estimating the conditioning of finite element equations with anisotropic meshes. Interestingly, the conditioning with non-isotropic meshes is not as bad as generally assumed. In particular, we will see that, if the number of anisotropic elements is relatively small, the condition number of the preconditioned system is basically the same as with uniform meshes.

Donnerstag, 07. 02. 2013, 14:00 Uhr (Raum: 406)

Prof. G. Barrenechea   (University of Strathclude, UK)
Eigenvalue enclosures for the Maxwell operator

We propose a strategy which allows computing eigenvalue enclosures for the Maxwell operator by means of the finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements (nodal or otherwise). In the remaining part of the talk we formulate a general framework which is free from spectral pollution, predicts correct multiplicities and allows estimation of eigenfunctions. We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

und 15:00 Uhr (Raum: 406)

Dr. P. Knobloch   (Charles University, Institute of Numerical Mathematics, Czech Republic)
Mesh optimization for convection-diffusion problems

Numerical solution of convection-dominated problems requires the use of layer-adapted anisotropic meshes. Since a priori construction of such meshes is difficult for complex problems, it is proposed to generate them in an adaptive way by moving the node positions in the mesh such that an a posteriori error estimator of the overall error of the approximate solution is reduced. This approach is formulated for a SUPG finite element discretisation of a stationary convection-diffusion problem defined in a two-dimensional polygonal domain. The optimisation procedure is based on the discrete adjoint technique and a SQP method using the BFGS update. The optimisation of node positions is applied to a coarse grid only and the resulting anisotropic mesh is then refined by standard adaptive red-greed refinement. Four error estimators based on the solution of local Dirichlet problems are tested and it is demonstrated that an L2 norm based error estimator is the most robust one. The efficiency of the proposed approach is demonstrated on several model problems whose solutions contain typical boundary and interior layers.

Donnerstag, 31. 01. 2013, 14:00 Uhr (ESH)

M. Bessemoulin-Chatard   (Universite Blaise Pascal, Clermont-Ferrand II)
A finite volume scheme for a Patlak-Keller-Segel model with cross-diffusion

In this talk, I will analyse a finite volume scheme for a 2D Keller-Segel model with cross-diffusion, studied by S. Hittmeir et A. Jüngel in an article written in 2011. We consider the parabolic-elliptic model with an additional cross-diffusion term in the elliptic equation. This diffusion term avoids the blow-up and leads to the global-in-time existence of weak solutions. We consider an implicit in time and finite volume in space scheme. After proving the existence of a solution to the implicit scheme, we obtain an entropy inequality by using discrete versions of Sobolev inequalities. Then we can deduce some a priori estimates which allow to obtain the convergence of the discrete solution to the continuous one when the approximation parameters tend to zero. If the cross-diffusion parameter is sufficiently large, the approximate solution converges to the homogeneous steady-state. Thanks to a discrete logarithmic Sobolev inequality, we obtain an estimate of the rate of convergence.

Donnerstag, 24. 01. 2013, 14:00 Uhr (ESH)

S. Ullmann   (TU Darmstadt)
POD-Galerkin reduced-order modeling of flow problems with stochastic boundary conditions

The stochastic collocation method on sparse grids is a method to solve PDAE problems with uncertainty in the parameters or boundary conditions. The main computational effort consists of solving a large number of deterministic PDAEs, one for each collocation point. POD-Galerkin reduced-order models are obtained by projecting the deterministic equations on a subspace spanned by a small number of POD basis functions. In the talk it will be shown how POD-Galerkin models can be used to reduce the computational cost of the stochastic collocation method. A natural convection problem with a stochastic temperature boundary condition will be presented as an example.