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

We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection of numerical examples validates the theoretical results and the performance of the preconditioner. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper introduces the construction of additive multilevel preconditioners, known as BPX preconditioners, for solving the linear system in isogeometric adaptive schemes with hierarchical B-splines. The locality of hierarchical spline functions is exploited to design efficient multilevel decompositions, reducing computational effort on each level. The condition number of the preconditioned system is shown to be bounded independently of the number of levels for suitably graded hierarchical meshes, with numerical examples validating theoretical results and performance.