The Research Unit of Numerical Analysis consists of the following 5 workgroups:
- Computational Mathematics (Prof. Dr. Markus MELENK)
- Computational PDEs (Prof. Dr. Michael FEISCHL)
- Numerics of Evolution Equations, Stiff and Singular Problems (Prof. Dr. Winfried AUZINGER, Prof. Dr. Gaby SCHRANZ-KIRLINGER, Prof. Dr. Ewa WEINMÜLLER (retired))
- Numerical Optimization (Dr. Kevin STURM)
- Numerics of PDEs (Prof. Dr. Dirk PRAETORIUS)
Many problems arising in Science and Engineering are mathematically described by differential equations, or, more generally, stochastic differential equations that also account for noise and data uncertainty. Since differential equations typically cannot be solved in closed form in situations of practical interest, numerical computer-based simulations have to be employed for quantitative answers. This makes numerical methods an indispensable pillar of modern science and technology, and it is an aim of the Research Unit of Numerical Analysis to contribute to the technological advancement as well as the quantitative understanding in science and technology.
A common goal of the scientists of the Research Unit of Numerical Analysis is to develop and analyze new algorithms and methods for solving numerically differential equations so that the subsequent computer-based simulation can provide approximate solutions that are accurate up to some user prescribed tolerance. Developing and analyzing efficient and convergent numerical schemes requires a variety of mathematical tools ranging from functional analysis and PDE theory to numerical linear algebra to computer science.
Besides elliptic and parabolic equations arising in structural and fluid mechanics, the members of the research unit study time-dependent and nonlinear equations used in the modelling of wave phenomena, in computational micromagnetics, in shape optimization, in the context of Schrödinger-type equations, in self-similar reaction-diffusion systems, and shallow water phenomena.
The Research Unit has expertise in many fields of numerical analysis for ODEs and PDEs including
- finite element methods (FEM) and high order methods
- adaptivity (for differential and integral equations)
- numerical optimization
- integral equation techniques (boundary element methods, matrix compression)
- uncertainty quantification (UQ) and stochastic Galerkin methods
- evolution equations (splitting methods, geometric integrators)
- numerical methods for ODEs and DAEs (singular and stiff problems)