A CutFEM divergence--free discretization for the Stokes problem
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The talk addresses construction and analysis of a CutFEM discretization for the Stokes problem based on the Scott-Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. Stability of the scheme as well as a divergence--free property of the discrete velocity outside an neighborhood of the boundary is shown. To mitigate the error caused by the violation of the divergence-free condition, a local grad-div stabilization is introduced. The error analysis shows that the grad-div parameter can scale like, allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.
