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

Dr. Svetlana Matculevich

April 10, 2018, 2:15 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 effcient 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 effcient technique for minimizing the majorant leading to extremely accurate, guaranteed upper bounds of the norm of the error with effciency 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 refinement. Mesh refinement in IgA is more involved than in the finite 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 effciency of the presented a posterior error estimates for IGA solutions to several examples exhibiting different 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 first 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 (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).