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Adaptive isogeometric methods with hierarchical splines: an overview. (English) Zbl 1404.41002

Summary: We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [A. Buffa and C. Giannelli, Math. Models Methods Appl. Sci. 26, No. 1, 1–25 (2016; Zbl 1336.65181); Math. Models Methods Appl. Sci. 27, No. 14, 2781–2802 (2017; Zbl 1376.41004); A. Buffa et al., Comput. Aided Geom. Des. 47, 83–92 (2016; Zbl 1418.65011)], optimal convergence rates of the AIGM can be proved when suitable approximation classes are considered. This is in line with the theory of adaptive methods developed for finite elements, recently well reviewed in [R. H. Nochetto and A. Veeser, Lect. Notes Math. 2040, 125–225 (2012; Zbl 1252.65192)]. The important output of our analysis is the definition of classes of admissibility for meshes underlying hierarchical splines and the design of an optimal adaptive strategy based on these classes of meshes. The adaptivity analysis is validated on a selection of numerical tests. We also compare the results obtained with suitably graded meshes related to different classes of admissibility for 2D and 3D configurations.

MSC:

41A15 Spline approximation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

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