Space-Time Adaptive Isogeometric Analysis of Parabolic Initial-Boundary Value Problem

Dr. Svetlana Matculevich

April 10, 2018, 4:15 p.m. S2 416-1

We derive reliable space-time Isogeometric Analysis (IGA) schemes for parabolic initial-boundary
value problems. In particular, we deduce functional-type a posteriori error estimates, and show
their ecient implementation in space-time IGA. Since the derivation is based on purely functional
arguments, the estimates are valid for any approximation from the admissible (energy) class. They
imply a posteriori error estimates for mesh-dependent norms associated with stabilized spacetime
IGA approximations introduced in [2]. We propose an ecient technique for minimizing
the majorant leading to extremely accurate, guaranteed upper bounds of the norm of the error
with eciency indices close to 1. Since this upper bound is nothing but the sum of the local
contributions, these local values of the majorant can be used as error indicators for mesh re nement.
Mesh re nement in IgA is more involved than in the nite element method. In particular, we use
Truncated Hierarchical B-spline (THB-spline) for localized meshes in our fully unstructured spacetime
adaptive IGA scheme. Finally, we illustrate the reliability and eciency of the presented a
posterior error estimates for IGA solutions to several examples exhibiting di erent features. We
also report about the costs of computing the upper bound. In all our examples, this is only a
small portion of the time required for generating the IgA approximation. Last but not least, the
numerical examples show that the space-time THB-spline based adaptive procedure works very
well. The talk is partly based on the results published in our arXiv paper [1].
The work of the rst two authors has been supported by the Austrian Science Fund (FWF) via the
NFN S117-03 project. The implementation was done using the open-source C++ library G+Smo.
[1] U. Langer, S. Matculevich, and S. Repin, Guaranteed error control bounds for the stabilised
space-time IgA approximations to parabolic problems, arXiv:1712.06017 [math.NA], pp. 1-24
[2] U. Langer, S. Moore, and M. Neumuller, Space-time isogeometric analysis of parabolic evolution
equations, Comput. Methods Appl. Mech. Engrg., Vol. 306, pp. 342-363 (2016).