A new perspective on adaptive hierarchical B-splines

PEDRO MORIN

Viena

Congreso; Workshop Interplay of geometric processing, mo- delling, and adaptivity in Galerkin methods; 2018

Erwin Schrodinger International Institute for Mathematics and Physics

We introduce a type of hierarchical spline spaces based on a parent-children relation, with two main features. First, the construction and handling is convenient for implementation and secondly, they are well suited for the theoretical analysis of adaptive isogeometric methods. The framework that we provide makes it simple to create hierarchical basis with control on the overlapping. Linear independence is always desired for the well posedness of the linear systems, and to avoid redundancy. The control on the overlapping of basis functions from different levels is necessary to close theoretical arguments in the proofs of optimality of adaptive methods. In order to guarantee linear independence, and to control the overlapping of the basis functions, some basis functions additional to those initially marked must be refined. However, with our framework and refinement procedures, the complexity of the resulting bases is under control. More precisely, if we construct hierarchical bases {Hk}k through subsequent calls to Hk+1=Refine(Hk,Mk), where Mk⊂Hk denotes the set of marked functions, we obtain#HR−#H0≤C∑k=0R−1#Mk,with a constant C independent of R.