Volume 57, Number 1, January-February 2023
|Page(s)||143 - 165|
|Published online||12 January 2023|
A cutFEM divergence–free discretization for the stokes problem
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
2 Department of Mathematics, University of Houston, Houston, TX 77204, USA
* Corresponding author: email@example.com
Accepted: 8 August 2022
We construct and analyze 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. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O(h) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O(h−1), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.
Mathematics Subject Classification: 65N30 / 65N12 / 76D07
Key words: CutFEM / Stokes / Divergence—free / Convergence analysis
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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