  We define and analyze (local) multilevel diagonal preconditioners for
isogeometric boundary elements on locally refined meshes in two dimensions.
Hypersingular and weakly-singular integral equations are considered. We prove
that the condition number of the preconditioned systems of linear equations is
independent of the mesh-size and the refinement level. Therefore, the
computational complexity, when using appropriate iterative solvers, is optimal.
Our analysis is carried out for closed and open boundaries and numerical
examples confirm our theoretical results.
