Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Professoren und Mitarbeiter aus allen zugehörigen Lehrstühlen, insbesondere Angewandte Funktionalanalysis, Numerische Lineare Algebra, und Partielle Differentialgleichungen, besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

• Kolloquium ModNumDiff WS 2012/13 [1]
• Kolloquium ModNumDiff SS 2012 [2]
• Kolloquium ModNumDiff WS 2011/12 [3]
• Kolloquium ModNumDiff SS 2011 [4]

Hamiltonian splitting for Vlasov equations and long time behavior problems
Dienstag, den 16.04.2013, 16.15 Uhr in MA 313
Abstract:

We consider Vlasov equations and first discuss their geometrical structures and long time behavior properties. We will then consider their numerical discretization by semi-Lagrangian methods. We will show how it is possible to build high order in time methods that preserve the Hamiltonian structure of the equation. We will discuss convergence results, and also show that for Vlasov-Poisson equations, specific properties of the system allow to reduce the number of space interpolation to be done at each step. This is a joint work with F. Casas, N. Crouseilles and M. Mehrenberger.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

BEM-based FEM: A Non-Standard Finite Element Method based on Boundary Integral Operators
Dienstag, den 28.05.2013, 16.15 Uhr in MA 313
Abstract:

We present and analyze a new non-standard finite element method based on element-local boundary integral operators that permits polyhedral element shapes as well as meshes with hanging nodes. The method employs elementwise PDE-harmonic trial functions and can thus be interpreted as a local Trefftz method. In fact, in his classical paper, Trefftz (1926) already proposed to use PDE-harmonic trial or basis functions in a domain decomposition framework. Our construction principle requires the explicit knowledge of the fundamental solution of the partial differential operator, but only locally, in every polyhedral element. This allows us to solve PDEs with elementwise constant coefficients. This construction technique via boundary integral operator representations of the Poincarè - Steklov operator is borrowed from the boundary element domain decomposition method proposed by Wendland and Hsiao (1991) and then used by many authors.

In this talk we mainly consider the diffusion equation as a model problem, but the method can be generalized to convection-diffusion-reaction problems and to systems of PDEs like the linear elasticity system and the time-harmonic Maxwell equations with elementwise constant or, at least, elementwise smooth coefficients. We provide a rigorous H^1- and L_2-error analysis of the method for smooth and non-smooth solutions under quite general assumptions on the geometric properties of the polyhedral elements in the case of the diffusion equation. It turns out that, for the convection-diffusion problems, the exact version (without approximation of the fluxes) of our scheme is the best possible SUPG scheme in the sense as discussed by Brezzi, Franca, Hughes and Russo (1997), Hughes, Feijo, Mazzei and Quincy (1998), and Hughes and Sangalli (2007a). Finally, we discuss fast solvers for these non-standard finite element schemes. Numerical results confirm our theoretical predictions.

This talk is based on a joint work with Clemens Hofreither and Clemens Pechstein within the Doctoral Program "Computational Mathematics" supported by the Austrian Science Fund FWF under the grant W1214, project DK4. Earlier results were obtained in joint work with Dylan Copeland and David Pusch within the FWF project P19255.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Sigma-Delta quantization for compressed sensing
Dienstag, den 11.06.2013, 16.15 Uhr in MA 313
Abstract:

Compressed sensing is a novel paradigm in signal processing concerning the recovery of approximately sparse signals from undersampled linear measurements. Recently, this approach has been combined with coarse quantization allowing the reconstruction of the signal from quantized Gaussian measurements with polynomial accuracy.

In the talk, we will discuss a new approach for the error analysis of such schemes linking them to the Restricted Isometry Property and present generalizations of the result to more general classes of measurements.

This is joint work with Joe-Mei Feng, Rayan Saab, and Özgür Yılmaz.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Robust Discretization and Precondition for Coupled PDE Systems
Montag, den 17.06.2013, 17.00 Uhr in MA 313
Abstract:

In this talk, I will present some recent works on discretization and preconditioning techniques for coupled PDE systems such as complex fluids, FSI and MHD that involves Stokes and/or Maxwell equations. Both conforming and discontinuous Galerkin methods will be discussed. One particular issue to be addressed is the role played by the divergence-free condition in the design of both discretization and preconditioning methods.

Preservation of positivity and invariance of sets with numerical methods for IVPs
Dienstag, den 18.06.2013, 16.15 Uhr in MA 313
Abstract:

Modeling of many time dependent physical processes - such as fluid flow, heat transfer, chemical reactions, combustion, traffic flow - we often introduce initial value problems (IVPs) where the state variables belong forward in time to particluar sets defined by inequality constraints (e.g. by nonnegativity or box-constraints). Naturally, numerical simulations should respect these constraints as well.

In this talk we investigate discrete positive invariance of convex, closed sets under time discretization methods including, among others the families of Runge-Kutta, Linear Multistep, Rosenbrock-type, Linearly Implicit and Peer methods. We give a computable formula for the time step size thresholds that guarantee numerical preservation of the invariance. Although the formula for the step size threshold for arbitrary convex, closed sets is sharp in many important cases of applications, it gives very pessimistic result for some smooth initial vectors and higher order methods. So, in addition, we derive a related formula for some sets appearing smooth functions defined by the help of Fourier series exploiting methodology and results of the theory of positive trigonometric series. We show that, in accordance to numerical experiments, this analysis provides a larger threshold for invariance.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Structure-preserving model reduction of port-Hamiltonian systems
Dienstag, den 25.06.2013, 16.15 Uhr in MA 313
Abstract:

Port-Hamiltonian models arise from first principles network modeling of multi-physics systems. They are useful for analysis and control since they reflect the underlying physical characteristics of the systems. In case of large-scale systems, or in case of systems arising from structure-preserving discretization of port-Hamiltonian distributed parameter systems, there is a clear need for model reduction methods which preserve the port-Hamiltonian structure of the models.

After giving a general introduction to port-Hamiltonian systems theory we will outline in this talk a number of structure-preserving model reduction approaches for (linear) port-Hamiltonian systems, illustrate them on some examples, and discuss open problems.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Generalized FEM for wave propagation in periodic structures
Dienstag, den 9.07.2013, 16.15 Uhr in MA 313
Abstract:

The generalized finite element method (g-FEM) is an extension of standard polynomial FEM that uses piecewise basis functions on a mesh, but those basis functions can be arbitrary functions. The hope is that if problem-adapted basis functions are chosen, the method is more efficient than by using polynomials. This allows to solve problems that are otherwise computationally infeasible. The basis functions are often not known analytically and need to be computed numerically. In periodic structures those basis functions are the Bloch modes, and they are much easier to compute than the solution for the whole device.

In the talk we introduce a g-FEM, show the steps to develop suitable basis functions and how to implement a g-FEM that is able to compute solutions for periodic structures with the same computational cost and accuracy independent of their size. Finally, we present the error analysis of this g-FEM including best-approximation bounds and inf-sup constants.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Kato's square root theorem as a basis for relative estimation theory of eigenvalue approximations
Freitag, den 12.07.2013, 15.30 Uhr in MA 313
Abstract:

We present new residual estimates based on Kato's square root theorem for spectral approximations of diagonalizable non-self-adjoint differential operators of diffusion-convection-reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations. We present a posteriori error estimates both for eigenvalues as well as eigenfunctions and prove that they are reliable. We demonstrate the efficiency of the proposed approach on a collection of benchmark examples.

This is a joint work with Stefano Giani, Agnieszka Miedlar and Jeff Oval.