  In the context of isogeometric analysis, we consider two discretization
approaches that make the resulting stiffness matrix nonsymmetric even if the
differential operator is self-adjoint. These are the collocation method and the
recently-developed weighted quadrature for the Galerkin discretization. In this
paper, we are interested in the solution of the linear systems arising from the
discretization of the Poisson problem using one of these approaches. In [SIAM
J. Sci. Comput. 38(6) (2016) pp. A3644--A3671], a well-established direct
solver for linear systems with tensor structure was used as a preconditioner in
the context of Galerkin isogeometric analysis, yielding promising results. In
particular, this preconditioner is robust with respect to the mesh size $h$ and
the spline degree $p$. In the present work, we discuss how a similar approach
can applied to the considered nonsymmetric linear systems. The efficiency of
the proposed preconditioning strategy is assessed with numerical experiments on
two-dimensional and three-dimensional problems.
