# BPX preconditioners for isogeometric analysis using analysis-suitable T-splines

## TL;DR

This paper introduces and analyzes two optimal multilevel BPX preconditioners for isogeometric analysis on locally refined T-meshes, demonstrating their efficiency and theoretical optimality through numerical experiments.

## Contribution

The paper develops two novel BPX preconditioners tailored for analysis-suitable T-splines on locally refined meshes, with proven optimal complexity.

## Key findings

- Both preconditioners achieve optimal complexity.
- Numerical experiments confirm theoretical efficiency.
- Preconditioners outperform existing methods in tests.

## Abstract

We propose and analyze optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern and Peterseim (2015, Comput. Aided Geom. Design, 34, 50-66) we can guarantee that the obtained T-meshes have a multilevel structure, and that the associated T-splines are analysis-suitable, for which we can define a dual basis and a stable projector. Taking advantage of the multilevel structure, we develop two BPX preconditioners: the first on the basis of local smoothing only for the functions affected by a newly added edge by bisection, and the second smoothing for all the functions affected after adding all the edges of the same level. We prove that both methods have optimal complexity, and present several numerical experiments to confirm our theoretical results, and also to compare the practical performance of the proposed preconditioners.