  The $k$-method is the isogeometric method based on splines (or NURBS, etc.)
with maximum regularity. When implemented following the paradigms of classical
finite element methods, the computational resources required by the $k-$method
are prohibitive even for moderate degree. In order to address this issue, we
propose a matrix-free strategy combined with weighted quadrature, which is an
ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free
weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more
important, greatly reduces memory consumption. Our strategy also requires an
efficient preconditioner for the linear system iterative solver. In this work
we deal with an elliptic model problem, and adopt a preconditioner based on the
Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our
numerical tests show that the isogeometric solver based on MF-WQ is faster than
standard approaches (where the main cost is the matrix formation by standard
Gaussian quadrature) even for low degree. But the main achievement is that,
with MF-WQ, the $k$-method gets orders of magnitude faster by increasing the
degree, given a target accuracy. Therefore, we are able to show the
superiority, in terms of computational efficiency, of the high-degree
$k$-method with respect to low-degree isogeometric discretizations. What we
present here is applicable to more complex and realistic differential problems,
but its effectiveness will depend on the preconditioner stage, which is as
always problem-dependent. This situation is typical of modern high-order
methods: the overall performance is mainly related to the quality of the
preconditioner.
