A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition
Abstract
Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximated by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.
- Authors:
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
- OSTI Identifier:
- 1631365
- Alternate Identifier(s):
- OSTI ID: 1259851; OSTI ID: 1351654
- Report Number(s):
- SAND-2015-7054J
Journal ID: ISSN 0898-1221; S0898122116300293; PII: S0898122116300293
- Grant/Contract Number:
- 14-017511; AC04-94AL85000
- Resource Type:
- Published Article
- Journal Name:
- Computers and Mathematics with Applications (Oxford)
- Additional Journal Information:
- Journal Name: Computers and Mathematics with Applications (Oxford) Journal Volume: 71 Journal Issue: 11; Journal ID: ISSN 0898-1221
- Publisher:
- Elsevier
- Country of Publication:
- United Kingdom
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; least-squares; mimetic methods; spectral element method; mass conservation
Citation Formats
Gerritsma, Marc, and Bochev, Pavel. A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition. United Kingdom: N. p., 2016.
Web. doi:10.1016/j.camwa.2016.01.033.
Gerritsma, Marc, & Bochev, Pavel. A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition. United Kingdom. https://doi.org/10.1016/j.camwa.2016.01.033
Gerritsma, Marc, and Bochev, Pavel. Wed .
"A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition". United Kingdom. https://doi.org/10.1016/j.camwa.2016.01.033.
@article{osti_1631365,
title = {A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition},
author = {Gerritsma, Marc and Bochev, Pavel},
abstractNote = {Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximated by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.},
doi = {10.1016/j.camwa.2016.01.033},
journal = {Computers and Mathematics with Applications (Oxford)},
number = 11,
volume = 71,
place = {United Kingdom},
year = {Wed Jun 01 00:00:00 EDT 2016},
month = {Wed Jun 01 00:00:00 EDT 2016}
}
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https://doi.org/10.1016/j.camwa.2016.01.033
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