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G. Gantner, D. Haberlik, D. Praetorius:
"Adaptive isogeometric methods with optimal convergence rates";
Talk: ECCM-ECFD 2018, Glasgow (invited); 2018-06-11 - 2018-06-15.

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.
However, so far there exists only little literature concerning the thorough analysis of adaptive isogeometric finite element methods (IGAFEMs).
[Buffa, Giannelli, Math. Mod. Meth. Appl. S. 26 (2016)] investigates linear convergence of an IGAFEM with truncated hierarchical B-splines, where optimal convergence of the proposed algorithm was only recently proved in [Buffa, Giannelli, Math. Mod. Meth. Appl. S. 27 (2017)] .

This talk is based on our recent work [Gantner, Haberlik, Praetorius, Math. Mod. Meth. Appl. S. 27 (2017)] .
We consider an adaptive IGAFEM for second-order linear elliptic PDEs.
We employ hierarchical B-splines.
We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions, where adaptivity is driven by some weighted-residual a posteriori error estimator.
The adaptive algorithm guarantees linear convergence of the error estimator (or equivalently: energy error plus data oscillations) with optimal algebraic rates.
Unlike our strategy, the algorithm of [Buffa, Giannelli, Math. Mod. Meth. Appl. S. 26 (2016)] was designed for truncated hierarchical B-splines only and the use of hierarchical B-splines may lead to non-sparse Galerkin matrices.
Further, the analysis of [Buffa, Giannelli, Math. Mod. Meth. Appl. S. 27 (2017)] which appeared independently of [Gantner, Haberlik, Praetorius, Math. Mod. Meth. Appl. S. 27 (2017)] is restricted to symmetric PDEs.

Similar results were also obtained for an adaptive 3D boundary element method in [Gantner, PhD thesis, TU Wien (2017)].

isogeometric analysis, finite element method, boundary element method, adaptive refinement, optimal convergence