UNFITTED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM USING THE SCOTT-VOGELIUS PAIRLiu, Haoran (2022) UNFITTED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM USING THE SCOTT-VOGELIUS PAIR. Doctoral Dissertation, University of Pittsburgh. (Unpublished) This is the latest version of this item.
AbstractIn 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 Share
Details
Available Versions of this Item
MetricsMonthly Views for the past 3 yearsPlum AnalyticsActions (login required)
|