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G. Gantner, D. Haberlik, D. Praetorius:
"Axioms of adaptivity revisited: Optimal adaptive IGAFEM";
Talk: WONAPDE 2019 - Sixth Chilean Workshop on Numerical Analysis of Partial Differential Equations, Concepción (invited); 01-21-2019 - 01-25-2019.

The axioms of adaptivity from [C. Carstensen, M. Feischl, M. Page, D. Praetorius. Axioms of adaptivity. Computers & Mathematics with Applications, 67, 1195-1253, 2014] analyze under which assumptions on the a posteriori error estimator and the mesh-refinement strategy, a mesh re fining adaptive algorithm yields convergence with optimal algebraic rates. In our talk, which is based on our recent work [G. Gantner, D. Haberlik, D. Praetorius. Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines. Mathematical Models and Methods in Applied Sciences, 27, 2631-2674, 2017], we now address the question which properties of the FEM are sufficient to ensure that the usual weighted-residual error estimator is well-defined and satisfies the axioms of adaptivity. In particular, our analysis covers conforming FEM in the framework of isogeometric analysis with hierarchical splines.