Robust Parallel Solvers for discontinuous Galerkin Space-Time Isogeometric Analysis of Parabolic Evolution Problems

MSc Christoph Hofer

March 20, 2018, 4:15 p.m. S2 416-1

In this talk, we construct and investigate fast solvers for
large-scale linear systems of algebraic equations arising from the
application of isogeometric analysis (IgA) to parabolic diffusion
problems. We consider decompositions of the space-time cylinder into
time slabs, where each slab is again decomposed into several space-time
patches. We use dG techniques to provide information transfer between
the time slabs, whereas the patches within a time slab are coupled in a
conforming way. The considered discretization is based on a time-upwind
scheme and gives optimal convergence rates for sufficiently smooth
solutions, see \cite{HLM:ref1}.

The aim of the talk is to investigate fast solvers, which are based on
the time parallel multigrid method, developed in \cite{HLM:ref2}. We
present two strategies for an efficient implementation of the smoother,
which is the most costly part. One is directly based on the
Isogeometric Tearing and Interconnecting (IETI-DP) method, a
non-overlapping domain decomposition method utilizing the multipatch
structure of the time slabs. This method has an excellent parallel
scalability. For symmetric and positive definite problems it is proven
that the iteration numbers are quasi-optimal with respect to the
mesh-size. Numerical experiments also show the same behavior for the
non-symmetric space-time matrices. Furthermore, we construct a
preconditioner for approximating the smoother in such a way that only a
preconditioner for the spatial problem is required. The main idea is a
decomposition of the space-time matrix into a series of spatial problems
via an eigendecomposition. The proposed algorithms are well suited for
parallelization in time as well as in space. We conclude the talk with
numerical experiments, confirming the theoretical results. Moreover, we
present scalability studies, having parallelization in space and time
simultaneously, up to several hundreds of cores on the RADON1 located at
RICAM, Linz, Austria.

\begin{thebibliography}{2}
\bibitem{HLM:ref1}
C. Hofer, M. Neum\"uller, U. Langer, and I. Toulopoulos,
\newblock ``Time-Multipatch Discontinuous Galerkin Space-Time
Isogeometric Analysis of Parabolic Evolution Problems'',
\newblock \textit{DK Computational Mathematics Linz Report Series},
Report No. 2017-\textbf{05}, (2017).

\bibitem{HLM:ref2}
M.J. Gander and, M. Neum\"uller,
\newblock ``Analysis of a new space-time parallel multigrid algorithm
for parabolic problems'',
\newblock \textit{SIAM Journal on Scientific Computing}, \textbf{38},
A2173--A2208 (2016).
\end{thebibliography}