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UNFITTED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM USING THE SCOTT-VOGELIUS PAIR

Liu, Haoran (2022) UNFITTED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM USING THE SCOTT-VOGELIUS PAIR. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

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.


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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Liu, Haoranhal104@pitt.eduhal104
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairNeilan, Michaelneilan@pitt.edu
Committee MemberLayton, Williamwjl@pitt.edu
Committee MemberOlshanskii, Maximmaolshanskiy@uh.edu
Committee MemberYotov, Ivanyotov@math.pitt.edu
Date: 12 October 2022
Date Type: Publication
Defense Date: 20 July 2022
Approval Date: 12 October 2022
Submission Date: 31 July 2022
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 131
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Unfitted Finite Element Method, Divergnece-free
Date Deposited: 12 Oct 2022 15:10
Last Modified: 12 Oct 2022 15:55
URI: http://d-scholarship.pitt.edu/id/eprint/43584

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  • UNFITTED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM USING THE SCOTT-VOGELIUS PAIR. (deposited 12 Oct 2022 15:10) [Currently Displayed]

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