Fast Multipatch Isogeometric Analysis Solvers

MSc Christoph Hofer

June 26, 2018, 3:30 p.m. S2 416-1

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 version. 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 G+Smo.

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 the 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.