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G. Gantner, D. Haberlik, D. Praetorius:
"Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines";
Talk: 5th International Conference on Isogeometric Analysis, Pavia (invited); 09-11-2017 - 09-13-2017.

The CAD standard for spline representation in 2D or 3D relies on tensor-product B-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 mathematical analysis of adaptive isogeometric finite element methods (IGAFEM).
For standard FEM with globally continuous piecewise polynomials, adaptivity is well understood; see, e.g., [3] for symmetric elliptic PDEs.
In this talk, we consider an IGAFEM for elliptic (possibly non-symmetric) second-order PDEs in arbitrary space dimension.
We employ hierarchical B-splines of arbitrary degree and different order of smoothness.
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 (resp. the sum of energy error plus data oscillations) with optimal algebraic rates.