BPX Preconditioners for Isogeometric Analysis Using (Truncated) Hierarchical B-splines

TL;DR

This paper develops BPX preconditioners tailored for isogeometric analysis with hierarchical B-splines, improving computational efficiency and ensuring condition number bounds independent of mesh levels.

Contribution

It introduces a novel multilevel preconditioner exploiting hierarchical spline locality, reducing computational effort and providing theoretical condition number bounds.

Findings

Condition number bounded independently of levels

Numerical results confirm theoretical efficiency

Reduced computational effort on each level

Abstract

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.