  We present a novel stabilized isogeometric formulation for the Stokes
problem, where the geometry of interest is obtained via overlapping NURBS
(non-uniform rational B-spline) patches, i.e., one patch on top of another in
an arbitrary but predefined hierarchical order. All the visible regions
constitute the computational domain, whereas independent patches are coupled
through visible interfaces using Nitsche's formulation. Such a geometric
representation inevitably involves trimming, which may yield trimmed elements
of extremely small measures (referred to as bad elements) and thus lead to the
instability issue. Motivated by the minimal stabilization method that
rigorously guarantees stability for trimmed geometries [1], in this work we
generalize it to the Stokes problem on overlapping patches. Central to our
method is the distinct treatments for the pressure and velocity spaces:
Stabilization for velocity is carried out for the flux terms on interfaces,
whereas pressure is stabilized in all the bad elements. We provide a priori
error estimates with a comprehensive theoretical study. Through a suite of
numerical tests, we first show that optimal convergence rates are achieved,
which consistently agrees with our theoretical findings. Second, we show that
the accuracy of pressure is significantly improved by several orders using the
proposed stabilization method, compared to the results without stabilization.
Finally, we also demonstrate the flexibility and efficiency of the proposed
method in capturing local features in the solution field.
