01.03.2017–28.02.2026
Project Part of SFB65 "Taming Complexity in Partial Differential Systems"
Project leader: Joachim SCHÖBERL

In this project we consider various physical models and the coupling of them. Typical multiphysics problems are electric machines, where magnetic fields induce forces and therefore movement, or the deformation of elastic bodies floating in a fluid flow. Such problems can be simulated on a
computer by means of the finite element method.

We develop the finite element software Netgen/NGSolve. With NGSolve one can formulate such coupled field problems in the mathematical language of variational formulations. These formulations are represented as expression trees, from which NGSolve generates automatically proper finite element discretizations. It generates code, which is optimized for various modern computer architectures.

This software package is freely available as Python module, and is intensively used in academic as well as industrial environment. If you like to try it out online just now, then you can do so from the site  https://www.tuwien.at/en/mg/asc/cme/interactive-gallery.

[Translate to English:] Grafische Darstellung einer Wellengleichung, das in der Form einem Kleeblatt gleicht

01.03.2017–28.02.2026
FWF Special Research Programme
Project leader: Anton ARNOLD

Temporal evolution problems describe a variety of dynamic processes in technology and the natural sciences and are often described by ordinary or partial differential equations, sometimes also by integro-differential equations. In a thermodynamic context, many of these systems can be expected to reach equilibrium, i.e. a time-independent steady state, after a “long time”.

In the FWF-funded SFB project part “Large-time behavior of continuous dissipative systems”, the long-term behavior of solutions of such systems is investigated, in particular, mostly exponential convergence towards equilibrium is to be proven by constructing suitable Lyapunov functionals.

A central aspect of the project are degenerate dissipative systems, i.e. systems whose symmetric part of the system matrix or the generator is not uniformly dissipative. Such systems are called hypocoercive, and there the trend towards equilibrium typically sets in with a time delay. In the course of the project, this time delay could be characterized very precisely with a newly introduced “hypocoercivity index”.

Project members: Stefan EGGER, Franz ACHLEITNER, Eduard NIGSCH, Artur STEPHAN

Translated with DeepL.com (free version)

01.03.2017–28.02.2026
Project Part of SFB65 "Taming Complexity in Partial Differential Systems"
Project leader: Jens Markus MELENK (E101-02)

An established tool to describe physical phenomena are Partial Differential Operators, which are are local in nature and only suitable to account for certain short-range effects. A classical example are diffusion processes modelled by the Laplace operator. Recent developments in science such as in material science, financial mathematics, or image processing point at the necessity to account for long-range interaction. The resulting operators are then non-local. A prototype are fractional diffusion processes modelled by the fractional Laplace operator. Equations involving such operators typically cannot be solved explicitly so that numerical methods have to be brought to bear.

The project "non-local operators" in the SFB 65, 'taming complexity in partial differential systems', develops fast and efficient numerical method for equations involving non-local operators. Efficiency can be achieved in several ways. One focus of the project is to use high order methods with their potential of high accuracy with only a small number of discretization parameters. Another way to obtain an efficient method  is to exploit matrix compression techniques to reduce the storage requirement when numerically realizing the discretization of the non-local operator.
 

01.03.2017–28.02.2026
Project Part of SFB65 "Taming Complexity in Partial Differential Systems"
Project leader Dirk PRAETORIUS (E101-02)

 

01.03.2017–28.02.2026
Project Part of SFB65 "Taming Complexity in Partial Differential Systems"
Project leader: Elisa DAVOLI (E101-01)

01.03.2017–28.02.2026
Project Part of SFB65 "Taming Complexity in Partial Differential Systems"
Project leader: Ansgar JÜNGEL (E101-01)

Dissipative systems are characterized by the fact that they are far from thermodynamic equilibrium and exchange energy or matter with the environment. For long time, the physical quantities converge towards equilibrium if there are no additional forces. This phenomenon is described by the so-called entropy. The aim of the project is the mathematical proof of the convergence towards equilibrium for discrete equations that play a role in numerical simulations. The difficulty is that many mathematical tools cannot be used for discrete structures. This problem will to be overcome by developing new discrete entropy methods. Various mathematical theories (differential equations, stochastics, numerics) will be linked together. The theory will be applied to equations of cell biology and micromagnetism. The project is supposed to help to improve the stability of numerical methods.

Project team members: Katharina SCHUH, Peter HIRVONEN, Sara XHAHYSA