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Conforming Stokes Elements Yielding Divergence-Free Approximations on Quadrilateral Meshes

Sap, Duygu (2017) Conforming Stokes Elements Yielding Divergence-Free Approximations on Quadrilateral Meshes. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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In this dissertation, we propose conforming finite element methods that yield divergence-free velocity approximations for the steady Stokes problem on cubical and quadrilateral meshes. In the first part, we construct the finite element spaces for the two-dimensional problem on rectangular grids. Then in the second part, we extend these spaces to n-dimensional spaces. We use discrete differential forms and smooth de Rham complexes to verify the stability and the conformity of the proposed methods, and the solenoidality of the velocity approximations. In the third part, we shift our focus from a dimensionwise extension to a meshwise improvement by introducing macro elements on general shape-regular quadrilateral meshes. By utilizing a smooth de Rham complex, we prove that the macro finite element method yields divergence-free velocity solutions, and with the construction of a Fortin operator, we validate the stability of the method. To improve the pressure approximation properties, we compute a post-processed pressure solution locally. In addition, we describe the implementation process of the (velocity) macro finite element. We show that the methods developed in this dissertation yield optimal convergence rates and present numerical experiments which are supportive of the theoretical results. Moreover, we provide experimental results of our method for the Navier-Stokes equations and show that the convergence rates are preserved.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Sap, Duygudus8@pitt.edudus8
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairNeilan, Michael J.neilan@pitt.eduneilan
Committee MemberLayton, William J.wjl@pitt.eduwjl
Committee MemberTrenchea, Catalintrenchea@pitt.edutrenchea
Committee MemberWalkington, Noel
Date: 28 September 2017
Date Type: Publication
Defense Date: 6 July 2017
Approval Date: 28 September 2017
Submission Date: 14 August 2017
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 90
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 Method, Stokes Flow, Navier-Stokes Equations, Fluid Dynamics, Quadrilateral Meshes
Date Deposited: 29 Sep 2017 00:21
Last Modified: 29 Sep 2017 00:21


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