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Since boundary element methods (bem) have been one of my main research interests for years I would like to give a brief comment at this point. Boundary element methods are numerical procedures to solve boundary integral equations which result from reducing to the boundary the boundary value problems which are given in interior and exterior domains. These methods are used in a variety of applications in engineering (e.g. electromagnetic or acoustic scattering, in heat conduction, elasticity problems,...)

Many advantages of the boundary element methods over the finite element methods result from the fact that only the boundary of the domain has to be discretized and exterior problems in unbounded regions can be treated. But the boundary element method leads to dense matrices and therefore the development of efficient iterative methods to solve the resulting large linear systems is of vital importance. Especially one needs good preconditioniers (e.g. multigrid methods, (additive) multilevel methods, or domain decomposition methods) to apply appropriate iterative procedures such as the conjugate gradient method.

By using higher order elements (piecewise polynomials) together with mesh refinement, the so called hp-version, the resulting linear system can often be reduced in size.

Adaptive methods are necessary for the accurate computation of characteristic quantities such as the stress intensity factors in crack problems in elasticity. The development of adaptive methods based on reliable (and efficient, if possible) error indicators and a posteriori error estimates is crucial.

Another highly up-to-date research topic is the coupling of finite elements and boundary elements. This coupling method is applicable to interface problems with even nonlinear differential operators, e.g. describing non-linear materials in elastoplasticity and viscoplasticity. Here only the plastic region has to be discretized by finite elements whereas for the elastic region a boundary element discretization is used. This leads to efficient discretizations, e.g. in form of saddle point problems, which can be solved quickly by, e.g. the preconditioned minimum residual method and the semi-smooth Newton method..

My main research acitivites right now are centered on: