# February 7 – Gregor Gantner: Optimal adaptivity for isogeometric finite and boundary element methods

Gregor Gantner: Thursday 7 February at 11 am, A415 Inria Paris.

The CAD standard for spline representation in 2D or 3D relies on tensor-product splines. To allow for adaptive refinement, several extensions have emerged, e.g., analysis-suitable T-splines, hierarchical splines, or LR-splines. All these concepts have been studied via numerical experiments, but there exists only little literature concerning the thorough analysis of adaptive isogeometric methods.
The work [1] investigates linear convergence of the weighted-residual error estimator (or equivalently: energy error plus data oscillations) of an isogeometric finite element method (IGAFEM) with truncated hierarchical B-splines. Optimal convergence was indepen- dently proved in [2, 3]. In [3], we employ hierarchical B-splines and propose a refinement strategy to generate a sequence of refined meshes and corresponding discrete solutions.
Usually, CAD provides only a parametrization of the boundary ∂Ω instead of the domain Ω itself. The boundary element method, which we consider in the second part of the talk, circumvents this difficulty by working only on the CAD provided boundary mesh. In 2D, our adaptive algorithm steers the mesh-refinement and the local smoothness of the ansatz functions. We proved linear convergence of the employed weighted-residual estimator at optimal algebraic rate in [5, 6]. In 3D, we consider an adaptive IGABEM with hierarchical splines and prove linear convergence of the estimator at optimal rate; see [6].

REFERENCES
[1] A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Math. Mod. Meth. Appl. S., Vol. 26, 2016.
[2] A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates. Math. Mod. Meth. in Appl. S., Vol. 27, 2017.
[3] G. Gantner, D. Haberlik, and Dirk Praetorius, Adaptive IGAFEM with optimal con- vergence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl. S., Vol. 27, 2017.
[4] G. Gantner, Adaptive isogeometric BEM, Master’s thesis, TU Wien, 2014.
[5] M. Feischl, G. Gantner, A. Haberl, and D. Praetorius, Optimal convergence for adap- tive IGA boundary element methods for weakly-singular integral equations. Numer. Math., Vol. 136, 2017
[6] G. Gantner, Optimal adaptivity for splines in finite and boundary element methods, PhD thesis, TU Wien, 2017.

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