## 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. 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.

Original language | English |
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Pages (from-to) | 2285-2300 |

Number of pages | 16 |

Journal | Computers & Mathematics with Applications |

Volume | 71 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1 Jun 2016 |

Event | Conference on Advances in Scientific Computing and Applied Mathematics - Las Vegas, United States Duration: 9 Oct 2015 → 12 Oct 2016 http://www.csm.ornl.gov/workshops/ASCAM/index.html |

## Keywords

- Least-squares
- Mass conservation
- Mimetic methods
- Spectral element method