  Finite element based simulation of phenomena governed by partial differential
equations is a standard tool in many engineering workflows today. However, the
simulation of complex geometries is computationally expensive. Many engineering
workflows require multiple simulations with small, non parametric changes in
between. The use of localized model order reduction for subsequent simulations
of geometries with localized changes is very promising. It produces lots of
computational tasks with little dependencies and thus parallelizes well.
Furthermore, the possibility to reuse intermediary results in the subsequent
simulations can lead to large computational savings. In this thesis, we
investigate different aspects of localized model order reduction and propose
various improvements. A simulation methodology named ArbiLoMod, comprising a
localized training, a localized a posteriori error estimator and an enrichment
procedure is proposed. A new localized a posteriori error estimator with
computable constants is presented and analyzed. A new training algorithm which
is based on a transfer operator is derived. It can be shown to converge nearly
as fast as the singular value decay of this operator. The transfer operator's
spectrum is observed to decay fast in electromagnetic simulations in printed
circuit boards. New online enrichment algorithms are proposed. All results are
supported by numerical experiments, for which the source code for reproduction
is provided.
