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DIVERGENCE-FREE FINITE ELEMENT METHODS FOR THE STOKES PROBLEM ON DOMAINS WITH CURVED BOUNDARY.

OTUS, MUHARREM BARIS (2022) DIVERGENCE-FREE FINITE ELEMENT METHODS FOR THE STOKES PROBLEM ON DOMAINS WITH CURVED BOUNDARY. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

In this thesis, we propose finite element methods that yield divergence-free velocity approximations for the two dimensional Stokes problem on domains with curved boundary. In the first part, we propose and analyze an isoparametric method that is globally $\bH(\text{div})$-conforming. The corresponding pair is defined by mapping the Scott-Vogelius
finite element space via a Piola transform. We use Stenberg's macro element technique to show that the method is stable and we also prove that the resulting method converges with optimal order, is divergence--free, and is pressure robust. In the second part, we build on our work from the first part and extend it to a globally $\bH^1$-conforming isoparametric method by considering an enriched local reference space. We show that the enrichment procedure respects stability, optimal order convergence as well as the divergence-free property of the discrete velocity solution. Here, we also discuss the implementation of the proposed enriched velocity space. In the third part, we construct and analyze a boundary correction finite element method for the Stokes problem
based on the Scott-Vogelius pair on Clough-Tocher splits. Here, we also introduce a Lagrange multiplier space to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions leading
to the well-posedness of the method. We also show that the resulting method converges with optimal order,
and the velocity approximation is divergence--free.


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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
OTUS, MUHARREM BARISmbo13@pitt.edumbo13
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairNeilan, Michaelneilan@pitt.edu
Committee MemberYotov, Ivanyotov@math.pitt.edu
Committee MemberTrenchea, Catalintrenchea@pitt.edu
Committee MemberGuzman, Johnnyjohnny_guzman@brown.edu
Date: 12 October 2022
Date Type: Publication
Defense Date: 21 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: 135
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: finite element methods, stokes problem, divergence-free, pressure-robust, mixed finite element methods.
Date Deposited: 12 Oct 2022 16:00
Last Modified: 12 Oct 2022 16:00
URI: http://d-scholarship.pitt.edu/id/eprint/43429

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