# Vorträge in 2017

# Kolloquiumsvortrag am Freitag, d. 21.7.2017

### A Family of Crouzeix-Raviart Non-Conforming Finite Elements in Two- and Three Spatial Dimensions

**Stefan Sauter**(Universität Zürich, Schweiz)

In this talk we will present a family of non-conforming "Crouzeix-Raviart" type finite elements in two and three dimensions. They consist of local polynomials of maximal degree **p** on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices.

We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions goes back to the seminal paper of Crouzeix and Raviart in 1973. However, the definition is implicit and the derivation of an explicit representation of the local basis functions for general p in 3D was an open problem.

We present explicit representations for these functions by developing some theoretical tools for fully symmetric and reflection symmetric orthogonal polynomials on triangles and their representation.

Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space. This talk comprises joint work with P. Ciarlet Jr., ENSTA, Paris and Charles F. Dunkl, Virginia Tech.

**Zeit und Ort: **14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2

# Fakultätskolloquium am Di., d. 18.07.2017

**Maria Lopez Fernandez**(Sapienza University of Rome, Italy)

Lubich's Convolution Quadrature is nowadays a well established method for the time discretization of retarded potentials associated to wave equations. It has been very much developed in the last decade, both from the theoretical and the algorithmic point of view. However, despite its nice properties, the Convolution Quadrature is strictly restricted to the use of fixed time steps. In this talk I will present the "generalized Convolution Quadrature", a new family of methods designed to overcome the strong restriction to uniform temporal grids. I will show stability and convergence estimates and numerical results illustrating the good behaviour of the new method. I will also outline the current limitations in the implementation of the generalized Convolution Quadrature and future possibilities of development.;

**Zeit und Ort: **11 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2

# Kolloquiumsvortrag am Freitag, d. 2.6.2017

### On multilevel quadrature for elliptic partial differential equations with random input

**Helmut Harbrecht**(Universität Basel, Schweiz)

This talk is dedicated to multilevel quadrature methods for the rapid solution of partial differential equations with a random input parameter. The key idea of such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution’s moments with focus on the mean and the variance in case of second order elliptic boundary value problems with random diffusion. In particular, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments.

**Zeit und Ort: **14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2

# Kolloquiumsvortrag am Freitag, d. 12.5.2017

### Error estimates and convergence rates for filtered back projection

**Armin Iske**(Universität Hamburg)

We consider the approximation of target functions from fractional Sobolev spaces by the method of filtered back projection (FBP), which gives an inversion of the Radon transform. To this end, we analyze the intrinsic FBP approximation error which is incurred by the use of a low-pass filter with finite bandwidth, before we prove **L ^{2}**-error estimates on Sobolev spaces of fractional order. The obtained error bounds are affine-linear with respect to the distance between the filter's window function and the constant function

**1**in the

**L**norm. With assuming more regularity for the window function, we refine the error estimates to prove convergence for the FBP approximation in the

^{∞}**L**-norm as the filter's bandwidth goes to infinity. We finally give asymptotic convergence rates in terms of the bandwidth of the low-pass filter and the smoothness of the target function.

^{2}**Zeit und Ort: **14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2