Alumni, Faculty, Graduate Students, Postdocs, Residents & Fellows
Dissertation Defense: Unfitted Finite Element Methods for the Stokes Problem using the Scott-Vogelius pair
In this thesis, we construct and analyze two unfitted finite element methods for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. For both methods, for $k\geq d$, where $d$ is the dimension of the space, the velocity space consists of continuous piecewise polynomials of degree $k$, and the pressure space consists of piecewise polynomials of degree $k-1$ without continuity constraints. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain.
The first unfitted finite element method we propose is a finite element method with boundary correction for the Stokes problem on 2D domains. We introduce a Lagrange multiplier space consisting of continuous piecewise polynomials of degree $k$ with respect to the boundary partition to enforce the boundary condition as well as to mitigate the lack of pressure robustness. We show the well-posedness of the method by proving several inf-sup conditions. In addition, we show this method has optimal order convergence rate and yields a divergenece-free velocity approximation.
The second unfitted finite element method we propose is a CutFEM for the Stokes problem on both 2D and 3D domains. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization. We ensure the stability with respect to small and anisotropic cuts of the bulk elements by adding local ghost penalty stabilization terms. We show the method is well-posed and possesses 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 around the boundary, we introduce local grad-div stablization. Through the error analysis, we show 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.
Advisor: Dr. Michael Neilan
Wednesday, July 20 at 1:00 p.m. to 4:00 p.m.
Benedum Hall, 226
3700 O'Hara Street, Pittsburgh, PA 15261
Dissertation Defense: Unfitted Finite Element Methods for the Stokes Problem using the Scott-Vogelius pair
In this thesis, we construct and analyze two unfitted finite element methods for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. For both methods, for $k\geq d$, where $d$ is the dimension of the space, the velocity space consists of continuous piecewise polynomials of degree $k$, and the pressure space consists of piecewise polynomials of degree $k-1$ without continuity constraints. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain.
The first unfitted finite element method we propose is a finite element method with boundary correction for the Stokes problem on 2D domains. We introduce a Lagrange multiplier space consisting of continuous piecewise polynomials of degree $k$ with respect to the boundary partition to enforce the boundary condition as well as to mitigate the lack of pressure robustness. We show the well-posedness of the method by proving several inf-sup conditions. In addition, we show this method has optimal order convergence rate and yields a divergenece-free velocity approximation.
The second unfitted finite element method we propose is a CutFEM for the Stokes problem on both 2D and 3D domains. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization. We ensure the stability with respect to small and anisotropic cuts of the bulk elements by adding local ghost penalty stabilization terms. We show the method is well-posed and possesses 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 around the boundary, we introduce local grad-div stablization. Through the error analysis, we show 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.
Advisor: Dr. Michael Neilan
Wednesday, July 20 at 1:00 p.m. to 4:00 p.m.
Benedum Hall, 226
3700 O'Hara Street, Pittsburgh, PA 15261
Alumni, Faculty, Graduate Students, Postdocs, Residents & Fellows