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Autumn Semester 2021

Date / Time Speaker Title Location
13 October 2021
16:15-17:15 		Prof. Dr. Kirill Cherednichenko
University of Bath
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Zurich Colloquium in Applied and Computational Mathematics

Title Metamaterials, or media with time and spatial dispersion: a new approach to the analysis of composites with contrast
Speaker, Affiliation Prof. Dr. Kirill Cherednichenko, University of Bath
Date, Time 13 October 2021, 16:15-17:15
Location Zoom Meeting
Abstract I shall discuss a novel approach to the homogenisation of critical-contrast periodic PDEs, which yields an explicit construction of their norm-resolvent asymptotics. A practically relevant outcome of this result is that it interprets composite media with micro-resonators as a class of (temporally and spatially) dispersive media. This is joint work with Yulia Ershova and Alexander Kiselev.
Metamaterials, or media with time and spatial dispersion: a new approach to the analysis of composites with contrastread_more
Zoom Meeting
27 October 2021
16:15-17:15 		Wasilij Barsukow
MPI Munich, CNRS
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Zurich Colloquium in Applied and Computational Mathematics

Title New strategies for all-Mach number finite volume methods
Speaker, Affiliation Wasilij Barsukow, MPI Munich, CNRS
Date, Time 27 October 2021, 16:15-17:15
Location HG E 1.2
Zoom Meeting
Abstract In the limit of low Mach number, the compressible Euler equations become incompressible. In the compressible case, numerical stabilization (upwinding) for explicit methods for a long time has been inspired by Riemann problems, i.e. high-Mach compressible phenomena. It has been noticed that this kind of stabilization introduces strong numerical errors for low Mach number flow. Modifications of Riemann solvers have been proposed which allow usage of coarse grids in the low Mach number regime, but the modifications are ad hoc and generally affect stability. In the talk I will show new approaches to achieving all-Mach number methods which are stable, and can be derived from first principles. The methods are truly multi-dimensional, reflecting the fact that incompressible flow is only nontrivial in multiple spatial dimensions.
New strategies for all-Mach number finite volume methodsread_more
HG E 1.2
Zoom Meeting
3 November 2021
16:15-17:15 		Prof. Dr. Qiang Du
Columbia University, New York
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Zurich Colloquium in Applied and Computational Mathematics

Title Recent progress on nonlocal modeling, analysis and applications
Speaker, Affiliation Prof. Dr. Qiang Du, Columbia University, New York
Date, Time 3 November 2021, 16:15-17:15
Location Zoom Meeting
Abstract Nonlocality has become increasingly noticeable in nature. The modeling and simulation of its presence and impact motivate new development of mathematical theory. In this lecture, we focus on nonlocal models with a finite horizon of interactions, and illustrate their roles in the understanding of various phenomena involving anomalies, singularities and other effects due to nonlocal interactions. We also present some recent analytical studies concerning nonlocal operators and nonlocal function spaces. The theoretical advances are making nonlocal modeling and simulations more reliable, effective and robust for applications ranging from classical mechanics to traffic flows of autonomous and connected vehicles.
Recent progress on nonlocal modeling, analysis and applicationsread_more
Zoom Meeting
10 November 2021
16:15-17:15 		Dr. Chupeng Ma
Universität Heidelberg
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Zurich Colloquium in Applied and Computational Mathematics

Title Generalized FEMs based on locally optimal spectral approximations for Helmholtz equation with heterogeneous coefficient
Speaker, Affiliation Dr. Chupeng Ma, Universität Heidelberg
Date, Time 10 November 2021, 16:15-17:15
Location HG E 1.2
Zoom Meeting
Abstract In this talk, I will present generalized FEMs with optimal local approximation spaces for solving Helmholtz equation with heterogeneous coefficient and high wavenumber. The optimal local approximation spaces are constructed by eigenvectors of local eigenvalue problems involving a partition of unity function defined on generalized harmonic spaces. Nearly exponential convergent and wavenumber explicit local approximation errors are derived both at the continuous and fully discrete level. The method can be viewed as an extension of Trefftz methods to Helmholtz equation with heterogeneous coefficients. This is a joint work with Robert Scheichl.
Generalized FEMs based on locally optimal spectral approximations for Helmholtz equation with heterogeneous coefficientread_more
HG E 1.2
Zoom Meeting
24 November 2021
16:15-17:15 		Dr. Barbara Verfürth
Fakultät für Mathematik, KIT
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Zurich Colloquium in Applied and Computational Mathematics

Title Numerical homogenization for nonlinear multiscale problems
Speaker, Affiliation Dr. Barbara Verfürth , Fakultät für Mathematik, KIT
Date, Time 24 November 2021, 16:15-17:15
Location HG E 1.2
Abstract Many applications, such as geophysical flow problems or scattering from Kerr-type media, require the combination of nonlinear material laws and multiscale features, which together pose a huge computational challenge. In this talk, we discuss how to construct a problem-adapted multiscale basis in a linearized and localized fashion for nonlinear problems such as the quasilinear diffusion equation or the nonlinear Helmholtz equation. For this, we will adapt two different perspectives: (a) determining a fixed multiscale space for the nonlinear problem or (b) adaptively and iteratively updating the multiscale space during an iteration scheme for the nonlinear problem. We prove optimal error estimates for the corresponding generalized finite element methods. In particular, neither higher regularity of the exact solution nor structural properties of the coefficients such as scale separation or periodicity need to be assumed. Numerical examples show very promising results illustrating the theoretical convergence rates.
Numerical homogenization for nonlinear multiscale problemsread_more (CANCELLED)
HG E 1.2
1 December 2021
16:15-17:15 		Prof. Dr. Wolfgang Hackbusch
Max Planck Leipzig
Details

Zurich Colloquium in Applied and Computational Mathematics

Title On nonclosed tensor formats
Speaker, Affiliation Prof. Dr. Wolfgang Hackbusch, Max Planck Leipzig
Date, Time 1 December 2021, 16:15-17:15
Location Zoom Meeting
Abstract Since tensor spaces may have a huge dimension, it is often not possible to store tensors by all their entries. Instead one uses certain representations (also called 'formats'), which describe a subset of tensors. For some formats used in practice the set of representable tensors is not closed. This leads to an instability comparable with the cancellation effect in the case of numerical differentiation. Under general conditions we prove for the finite-dimensional case that there is some minimal strength of the instability. For the special case of the 2-term format a quantitative result can be proved. In the infinite-dimensional case with a tensor norm not weaker than the injective crossnorm, the same instability behaviour can be proved. Even the constants in the estimates are under control. As a result, it is sufficient to study the instability behaviour for finite-dimensional model spaces.
On nonclosed tensor formatsread_more
Zoom Meeting
8 December 2021
16:15-14:15 		Dr. Michael Dumbser
University of Trento, Italy
Details

Zurich Colloquium in Applied and Computational Mathematics

Title Numerical schemes for a unified first order hyperbolic system of continuum mechanics
Speaker, Affiliation Dr. Michael Dumbser, University of Trento, Italy
Date, Time 8 December 2021, 16:15-14:15
Location HG E 1.2
Abstract In the first part of this talk we present the unified first order hyperbolic formulation of Newtonian continuum mechanics proposed by Godunov, Peshkov and Romenski (GPR). The governing PDE system can be derived from a variational principle and belongs to the class of symmetric hyperbolic and thermodynamically compatible systems (SHTC), which have been studied for the first time by Godunov in 1961 and later in a series of papers by Godunov & Romenski. An important feature of the model is that the propagation speeds of all physical processes, including dissipative processes, are finite. The GPR model is a geometric approach to continuum mechanics that is able to describe the behavior of nonlinear elasto-plastic solids at large deformations, as well as viscous Newtonian and non-Newtonian fluids within one and the same governing PDE system. This is achieved via appropriate relaxation source terms in the evolution equations for the distortion field and the thermal impulse. It can be shown that the GPR model reduces to the compressible Navier-Stokes equations in the stiff relaxation limit, i.e. when the relaxation times tend to zero. The unified system is also able to describe material failure, such as crack generation and fatigue. In the absence of source terms, the homogeneous part of the GPR model is endowed with involutions, namely the distortion field A and the thermal impulse J need to remain curl-free. In the second part of the talk we therefore present a new structure-preserving scheme that is able to preserve the curl-free property of both fields exactly also on the discrete level. This is achieved via the definition of appropriate and compatible discrete gradient and curl operators on a judiciously chosen staggered grid. Furthermore, the pressure terms are discretized implicitly, in order to capture the low Mach number limit of the equations properly, while all other terms are discretized explicitly. In this manner, the resulting pressure system is symmetric and positive definite and can be solved with efficient iterative solvers like the conjugate gradient method. Last but not least, the new staggered semi-implicit scheme is asymptotic-preserving and thus also able to reproduce the stiff relaxation limit of the governing PDE system properly, recovering an appropriate discretization of the compressible Navier-Stokes equations and of the incompressible equations in the low Mach number limit. In the final part of the talk we present a new thermodynamically compatible finite volume scheme that is exactly compatible with the overdetermined structure of the model at the semi-discrete level, making use of a discrete form of the continuous formalism introduced by Godunov in 1961. A very particular feature of our new thermodynamically compatible finite volume scheme is the fact that it directly discretizes the entropy inequality, rather than the total energy conservation law. Energy conservation is instead achieved as a mere consequence of the scheme, thanks to the thermodynamically compatible discretization of all the other equations.
Numerical schemes for a unified first order hyperbolic system of continuum mechanics read_more
HG E 1.2

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