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

We propose and analyse optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern & Peterseim (2015, Analysis-suitable adaptive T-mesh refinement with linear complexity. 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 Bramble-Pasciak-Xu (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.