BPX preconditioners for isogeometric analysis using (truncated) hierarchical B-splines

We present the construction of additive multilevel preconditioners, also known as BPX for solution linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. show that locality spline functions, naturally defined on a structure, can be suitably exploited to design and analyze efficient decompositions. By obtaining smaller subspaces respect standard tensor-product B-splines, computational effort each level is reduced. prove that, graded meshes, condition number preconditioned bounded independently levels. A selection numerical examples validates theoretical results performance preconditioner.