  The design of fast solvers for isogeometric analysis is receiving a lot of
attention due to the challenge that offers to find an algorithm with a robust
convergence with respect to the spline degree. Here, we analyze the application
of geometric multigrid methods to this type of discretizations, and we propose
a multigrid approach based on overlapping multiplicative Schwarz methods as
smoothers. The size of the blocks considered within these relaxation procedures
is adapted to the spline degree. A simple multigrid V-cycle with only one step
of pre-smoothing results to be a very efficient algorithm, whose convergence is
independent on the spline degree and the spatial discretization parameter.
Local Fourier analysis is shown to be very useful for the understanding of the
problems encountered in the design of a robust multigrid method for IGA, and it
is performed to support the good convergence properties of the proposed solver.
In fact, an analysis for any spline degree and an arbitrary size of the blocks
within the Schwarz smoother is presented for the one-dimensional case. The
efficiency of the solver is also demonstrated through several numerical
experiments, including a two-dimensional problem on a nontrivial computational
domain.
