Computational Methods for PDEs - Research Topics

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.