# 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 renement.

Mesh renement 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 dierent 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.

REFERENCES

[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

(2017).

[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).