  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.
