led by Ulrich Langer

The "Computational Mathematics Group" (CMG) will mainly focus on the development, analysis and implementation of novel computational methods for complicated multiple field problems (e.g., but not exclusively, modelled by systems of partial differential equations) arising in classical and novel application fields like computational fluid mechanics, structural mechanics, electro-magnetics, computational physics and chemistry, computational finance and, in the future, computational life sciences. Methods currently studied include domain decomposition and multigrid methods with special emphasis on adaptivity. Multi-level approximation techniques are among the most powerful tools for constructing fast and efficient (mostly parallel) methods for solving discretised field problems of high complexity. The development of the numerical methods is usually driven by hard applications. The numerical handling of hard applications requires some kind of Scientific Problem Solver Environment (PSE) that contains a set of interacting scientific computing tools such as:

• Geometry and problem editor
• Mesh generator and mesh handler
• Parallelisation tools
• Solver tools (adaptive discretisation and solution techniques)
• Optimisation tools
• Visualisation

Topics where this group plans to do research includes:

Numerical Methods: Construction -Analysis -Parallelisation

At the Radon Institute, the CMG will focus its main methodological research activities on the following advanced numerical methods with different fields of applications:

• Robust Adaptive Multilevel FEM Methods:
  Multigrid and multilevel methods are at least potentially the most powerful methods for solving large scale systems of finite element equations. The weak point of the standard Geometrical Multigrid (GMG) methods is the lack of the robustness that makes the direct methods so attractive to commercial users. To make multigrid convergence rate robust to bad parameters typical for some class of applications (e.g. elasticity propels for almost incompressible materials, thin plate or shell problems, anisotropies etc.), one needs deep theoretical insight into the interplay of the multigrid components. The construction and the analysis of robust GMG methods for different clauses of problems is our first main research direction.
• Algebraic Multigrid Methods:
  The CMG has investigated AMG methods mainly for SPD problems where we have developed a special technique for constructing the AMG components from an auxiliary problem. This approach was especially successful for Maxwell equations discretised by edge elements. At the Radon Institute, we will look for successful applications of these techniques including symmetric and indefinite problems as well as non-symmetric system of algebraic equations which are typically arising from the discretisation of Navier-Stokes equations.
• Domain Decomposition (DD) Methods:
  DD techniques are not only the basic parallelisation tool but also the basic method for handling different discretisation techniques in one scheme and multifield problems. The CMG has a lot of experience in handling the BEM-FEM coupling via DD techniques. We will develop DD techniques especially for surface coupled multifield problems as well as for coupling of different discretisation techniques.
• Solvers for Discretised Optimisation Problems:
  The construction of fast so-called All-At-Once solvers for large-scale optimisation problems arising typically from the finite element discretisation of optimisation problems with PDE constrains is a hot topic in the research with enormous potential use for a lot of practical applications. A successful work on these problems requires the combination of optimisation methods, regularisation techniques and preconditioning techniques. All the methods mentioned above (i.e. GMG, AMG and DD methods) can be used for constructing preconditioners for the KKT system or for parts (blocks) of the KKT system. In close co-operation with the Inverse Problems Group we will continue research in this direction, that we have already started in the SFB "Numerical and Symbolic Scientific Computing".
• Parallelisation Techniques:
  Our parallelisation approach is based on a special calculus with distributed data that is an formalisation of the data distribution and the corresponding operation that we used earlier in the non-overlapping domain decomposition method, This parallelisation strategy was used for developing a parallel geometrical multigrid 3D Maxwell solvers.

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The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)

Medieninhaber:
Österreichische Akademie der Wissenschaften
Juristische Person öffentlichen Rechts (BGBl 569/1921 idF BGBl I 130/2003)
Dr. Ignaz Seipel-Platz 2, 1010 Wien
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