Prof. Dr. Ernst P. Stephan

Forschungsschwerpunkte

• Numerical Methods for PDEs / Finite Elements (FEM)
• Numerical Methods for Boundary Integral Equations / Boundary Elements (BEM)
• FEM-BEM Coupling
• h-, p- and hp-Versions
• Efficient Solvers (Preconditioners) for Large Systems
• Adapive Methods

Forschungsfelder

• Numerical Analysis for partial differential equations and boundary integral equations: Finite element and boundary element methods. Error analysis for Galerkin, collocation and quadrature methods, numerical and theoretical aspects, adaptive schemes, h-, p- and hp- versions of the boundary and finite element methods. Coupling of finite elements and boundary elements. Wavelets, Mixed methods, Least squares methods. Mortar methods

• Scientific Computing: Efficient solvers/preconditioners for FE/BE discretisations (multigrid, multilevel and domain decomposition methods, Schwarz methods). Implementation of finite and boundary element methods (software development).

• Mathematical Modelling for problems arising in physics and engineering: Partial Differential Equations and Boundary Integral Equations (Pseudodifferential Equations) describing e.g. electromagnetic scattering problems or problems in solid mechanics (linear/nonlinear elasticity, plasticity). Regularity of elliptic boundary value problems in domains with corners and edges.

• FE/BE- Applications in Physics, Engineering and Industry: Elastoplastic, viscoplastic multibody contact problems arising in e.g. metal forming, metal chipping, high speed drilling. Sound radiation and contact for tires.

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.

My main research topics currently are:

• fem/bem for frictional contact problems
• bem for transient problems (retarded potentials, sound radiation, space/time discretization)
• development of fast solvers (multilevel (additive and multiplicative) Schwarz preconditioners) for bem and fem/bem coupling
• development of efficient and robust numerical methods for integral/differential equations: h-p version of bem, exponential convergence, adaptive bem, coupling of fem/bem for nonlinear problems
• application of (mixed) finite and boundary element methods to real world problems: elastoplastic materials, metal forming processes, scattering problems of acoustic and electromagnetic waves etc