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FINITE ELEMENT METHODS FOR AXISYMMETRIC PDES AND DIVERGENCE FREE FINITE ELEMENT PAIRS ON PARTICULAR MESH REFINEMENTS

Zytoon, Ahmed (2021) FINITE ELEMENT METHODS FOR AXISYMMETRIC PDES AND DIVERGENCE FREE FINITE ELEMENT PAIRS ON PARTICULAR MESH REFINEMENTS. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

This dissertation discusses the following two main topics.

1) Finite element approximation for
Partial Differential Equations (PDEs)
defined on axisymmetric domains:

We introduce the Darcy equations on axisymmetric domains
and we show the stability of a low--order Raviart-Thomas
element pair. We provide numerical experiments to support our theoretical results.

Also, we introduce the Stokes equations on axisymmetric domains and show that the axisymmetric Stokes equations can fit within a commutative de Rham complex.

2) Connection between the grad-div stabilized and divergence-free Stokes finite element pairs and low--order divergence-free elements on particular mesh refinements:

We introduce the most recent results that connect
the grad-div stabilized Taylor--Hood (TH) finite element pair and divergence-free Scott--Vogelius (SV) finite element pairs, and we use these results to extend and generalize this connection to other Stokes finite element pairs.

Finally, we provide numerical examples for low order divergence-free Stokes finite element pairs defined on particular mesh refinements. This research is focused on the numerical implementation aspects of these finite element pairs.

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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Zytoon, Ahmedamz56@pitt.eduamz56@pitt.edu
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairNeilan, Michaelneilan@pitt.eduneilan@pitt.edu
Committee MemberLayton, Williamwjl@pitt.eduwjl@pitt.edu
Committee MemberYotov, Ivanyotov@pitt.eduyotov@pitt.edu
Committee MemberTrenchea, Catalintrenchea@pitt.edutrenchea@pitt.edu
Committee MemberErvin, Vincentvjervin@clemson.eduN/A
Date: 8 October 2021
Date Type: Publication
Defense Date: 25 June 2021
Approval Date: 8 October 2021
Submission Date: 6 September 2021
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 108
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, Divergence Free Pairs, Stokes Equations, Darcy Equations, Axisymmetric Domains.
Date Deposited: 08 Oct 2021 19:41
Last Modified: 08 Oct 2021 19:41
URI: http://d-scholarship.pitt.edu/id/eprint/41767

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