OTUS, MUHARREM BARIS
(2022)
DIVERGENCE-FREE FINITE ELEMENT METHODS FOR THE STOKES PROBLEM ON DOMAINS WITH CURVED BOUNDARY.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
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.
Share
| Citation/Export: |
|
| Social Networking: |
|
Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
| Creators | Email | Pitt Username | ORCID  |
|---|
| OTUS, MUHARREM BARIS | mbo13@pitt.edu | mbo13 | |
|
| ETD Committee: |
|
| 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 |
Metrics
Monthly Views for the past 3 years
Plum Analytics
Actions (login required)
 |
View Item |
![[feed]](/images/feed-icon-32x32.png){kind=icon}