Isogeometric discretizations of the Stokes problem on trimmed geometries

We investigate the isogeometric approximation of the Stokes problem in a trimmed domain, where the underlying mesh is not fitted to the physical domain boundary, posing a challenge for enforcing essential boundary conditions. We introduce three families of isogeometric elements (Raviart-Thomas, Nédélec, and Taylor-Hood) and use them to discretize the problem. The widely used Nitsche method [1] is commonly employed to address this issue. However, we identify that the Nitsche method lacks stability in certain degenerate trimmed domain configurations, potentially polluting the computed solutions. Our remedy is twofold. On the one hand, we locally change the evaluation of the normal derivatives of the velocities in the weak formulation (generalizing the procedure introduced for the Poisson problem in [2]); on the other, we modify the space of the discrete pressures, eliminating the degrees of freedom associated with badly trimmed elements. We demonstrate that this approach restores the coercivity of the bilinear form for the velocities. Although numerical results show that our method restores the inf-sup stability of the discrete problem, a rigorous mathematical proof is still missing. We prove optimal a priori error estimates and provide numerical experiments to validate the theory, emphasizing the validation of the inf-sup stability of our method.