We consider large linear systems arising from the isogeometric discretization
of the Poisson problem on a single-patch domain. The numerical solution of such
systems is considered a challenging task, particularly when the degree of the
splines employed as basis functions is high. We consider a preconditioning
strategy which is based on the solution of a Sylvester-like equation at each
step of an iterative solver. We show that this strategy, which fully exploits
the tensor structure that underlies isogeometric problems, is robust with
respect to both mesh size and spline degree, although it may suffer from the
presence of complicated geometry or coefficients. We consider two popular
solvers for the Sylvester equation, a direct one and an iterative one, and we
discuss in detail their implementation and efficiency for 2D and 3D problems on
single-patch or conforming multi-patch NURBS geometries. Numerical experiments
for problems with different domain geometries are presented, which demonstrate
the potential of this approach
