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.
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.
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.
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".
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.
IgA is a novel numerical technique that uses splines or NURBS not only for the geometrical representation of the
computational domain as in Computer Aided Design (CAD) but also for the discretization of the PDEs which are living
in the computational domain. This new technique was introduced by Prof. T. Hughes (UT Austin, USA) and his co-workers
in 2005 and has successfully been applied to many problems in different fields. Since 2012 the CMG participates in
the National Research Network (NFN) „Geometry + Simulation” supported by the Austrian Science Fund FWF under the
grant S117. The CMG contributes to the NFN via the research project S117-03 "Discontinuous Galerkin Domain
Decomposition Methods in IgA" and the Software Project G+SMO.