Local multigrid solvers for adaptive isogeometric analysis in hierarchical spline spaces

We propose local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical B-splines ((T)HB-splines). Smoothing is only performed in or near the refinement areas on each level, leading to a computationally efficient solving strategy. We prove robust convergence of the proposed solvers with respect to the number of levels and the mesh sizes of the hierarchical discretization space under the assumption that the hierarchical mesh satisfies an admissibility condition, i.e., the number of interacting mesh levels is uniformly bounded. We also provide several numerical experiments. The main analytical tools are quasi-interpolators for THB-splines and the abstract convergence theory of subspace correction methods.