A spectral mimetic leastsquares method for the Stokes equations with noslip boundary condition
Formulation of locally conservative leastsquares finite element methods (LSFEMs) for the Stokes equations with the noslip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require nonstandard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the noslip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new nonstandard stability bound for the velocity–vorticity–pressure Stokes system augmented with a noslip boundary condition. This bound gives rise to a normequivalent leastsquares functional in which the velocity can be approximated by divconforming finite element spaces, thereby enabling a locallyconservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using highorder spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.
 Authors:

^{[1]};
^{[2]}
 Delft Univ. of Technology, Delft (The Netherlands)
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Publication Date:
 Report Number(s):
 SAND20157054J
Journal ID: ISSN 08981221; PII: S0898122116300293
 Grant/Contract Number:
 AC0494AL85000
 Type:
 Accepted Manuscript
 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 08981221
 Publisher:
 Elsevier
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; leastsquares; mimetic methods; spectral element method; mass conservation
 OSTI Identifier:
 1259851
 Alternate Identifier(s):
 OSTI ID: 1351654
Gerritsma, Marc, and Bochev, Pavel. A spectral mimetic leastsquares method for the Stokes equations with noslip boundary condition. United States: N. p.,
Web. doi:10.1016/j.camwa.2016.01.033.
Gerritsma, Marc, & Bochev, Pavel. A spectral mimetic leastsquares method for the Stokes equations with noslip boundary condition. United States. doi:10.1016/j.camwa.2016.01.033.
Gerritsma, Marc, and Bochev, Pavel. 2016.
"A spectral mimetic leastsquares method for the Stokes equations with noslip boundary condition". United States.
doi:10.1016/j.camwa.2016.01.033. https://www.osti.gov/servlets/purl/1259851.
@article{osti_1259851,
title = {A spectral mimetic leastsquares method for the Stokes equations with noslip boundary condition},
author = {Gerritsma, Marc and Bochev, Pavel},
abstractNote = {Formulation of locally conservative leastsquares finite element methods (LSFEMs) for the Stokes equations with the noslip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require nonstandard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the noslip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new nonstandard stability bound for the velocity–vorticity–pressure Stokes system augmented with a noslip boundary condition. This bound gives rise to a normequivalent leastsquares functional in which the velocity can be approximated by divconforming finite element spaces, thereby enabling a locallyconservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using highorder 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 States},
year = {2016},
month = {3}
}