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  [ 17 ] to incompressible flow problems, we theoretically prove that, combined with the Raviart-Thomas isogeometric element , we are able to recover the well-posedness of the formulation and, consequently, optimal a priori error estimates. Numerical results corroborating the theory and extending it for the case of the isogeometric N\'ed\'elec and Taylor-Hood elements are provided.
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.
A
~
A
~
R
17
16
R
isogeometric element
, the N\'ed\'elec, and the Taylor-Hood (in this case, only when the isogeometric map is the identity) isogeometric elements
D
and extending it for the case of the isogeometric N\'ed\'elec and Taylor-Hood elements