This thesis is devoted to the generalization of the Dual-Primal Finite
Element Tearing and Interconnecting (FETI-DP) method to linear algebraic
systems arising from the Isogemetric Analysis (IgA) of linear elliptic
boundary value problems, like stationary diffusion or heat conduction
problems. This IgA version of the FETI-DP method is called Dual-Primal
Isogeometric Tearing and Interconnect (IETI-DP) method. The FETI-DP method
is well established as parallel solver for large-scale systems of finite
element equations, especially, in the case of heterogeneous coefficients
having jumps across subdomain interfaces. These methods belong to the class
of non-overlapping domain decomposition methods.
In practise, a complicated domain can often not be represented by a
single patch, instead a collection of patches is used to represent the
computational domain, called multi-patch domains. Regarding the solver, it
is a natural idea to use this already available decomposition into patches
directly for the construction of a robust and parallel solver. We
investigate the cases where the IgA spaces are continuous or even
discontinuous across the patch interfaces, but smooth within the patches.
In the latter case, a stable formulation is obtained by means of
discontinuous Galerkin (dG) techniques. Such formulations are important for
various reasons, e.g, if the IgA spaces are not matching across patch
interfaces (different mesh-sizes, different spline degrees) or if the
patches are not matching (gap and overlapping regions).
Using ideas from dG-FETI-DP methods, we extend IETI-DP methods in such a
way that they can efficiently solve multi-patch dG-IgA schemes. This thesis
also provides a theoretical foundation of IETI-DP methods. We prove the
quasi-optimal dependence of the convergence behaviour on the mesh-size for
both versions. Moreover, the numerical experiments indicate robustness of
these methods with respect to jumps in the coefficient and a weak dependence
on the spline degree. All algorithms are implemented in the C++ library
Finally, this thesis investigates space-time methods for linear parabolic
initial-boundary value problems, like instationary diffusion or heat
conduction problems. The focus is again on efficient solution techniques.
The aim is the development of solvers which are on one hand robust with
respect to certain parameters and on the other hand parallelizeable in space
and time. We develop special block smoothers that lead to robust and
efficient time-parallel multigrid solvers. The parallelization in space is
again achieved by means of IETI-DP methods.