Isogeometric discretizations of the Stokes problem on trimmed geometries

The isogeometric approximation of the Stokes problem in a trimmed domain is studied. This setting is characterized by an underlying mesh unfitted with the boundary of the physical domain making the imposition of the essential boundary conditions a challenging problem. A very popular strategy is to rely on the so-called Nitsche method~[22]. We show with numerically examples that in some degenerate trimmed domain configurations there is a lack of stability of the formulation, potentially polluting the computed solutions. After extending the stabilization procedure of~[16] to incompressible flow problems, we theoretically prove that, combined with the Raviart-Thomas, the N\'ed\'elec, and the Taylor-Hood (in this case, only when the isogeometric map is the identity) isogeometric elements, we are able to recover the well-posedness of the formulation and, consequently, optimal a priori error estimates. Numerical results corroborating the theory are provided.