10. Numerical methods for non-local operators
Non-local operators arise for example in the modelling of exterior domains with transparent/absorbing boundary conditions or in the computation of stray fields in micromagnetic calculations. One efficient technique to treat discretizations of non-local operator such as integral operators areH -matrices recently introduced by W. Hackbusch (MPI Leipzig). These operators allow for (almost) optimal compression of densely populated matrices as well as (almost) optimal complexity of operations such as matrix-vector multiplication and solution of linear systems. The group of Jens Markus Melenk, which focuses on numerical methods for elliptic partial differential equations and the efficient numerical treatment of non-local operators, could bring in expertise in this field.
Additionally, the group has expertise in both finite element methods (FEM) and boundary element methods (BEM). For problems posed in unbounded domains, these two techniques are often coupled: the BEM is used to model the unbounded exterior domain, and the FEM is employed to treat complicated phenomena in a bounded domain. The group can contribute with expertise in both FEM and BEM covering many aspects such as adaptivity and error estimation, anisotropic mesh refinement, high order methods, efficient solution techniques such as multilevel preconditioning, and questions of coupling different discretizations (e.g., mortar methods).
Commencing fall 2008, some aspects of coupling FEM and BEM will be explored in PhD work (jointly supervised with Dieter Suess, E138) funded by the TU-Doktoratskolleg “partial differential equations in technical systems” for a model arising in micromagnetics.