### Abstract

We consider the penalty method for the stationary Navier–Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate (Formula presented.) in (Formula presented.)-norm, where (Formula presented.) is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate (Formula presented.) for the non-reduced-integration scheme with (Formula presented.), and the reduced-integration scheme with (Formula presented.), where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with (Formula presented.), we prove the convergence order (Formula presented.). The theoretical results are verified by numerical experiments.

Original language | English |
---|---|

Pages (from-to) | 1-36 |

Number of pages | 36 |

Journal | Journal of Scientific Computing |

DOIs | |

Publication status | Accepted/In press - 2015 Nov 27 |

### Fingerprint

### Keywords

- Finite element method
- Penalty method
- Slip boundary condition
- The Navier–Stokes equations

### ASJC Scopus subject areas

- Software
- Computational Theory and Mathematics
- Theoretical Computer Science
- Engineering(all)

### Cite this

*Journal of Scientific Computing*, 1-36. https://doi.org/10.1007/s10915-015-0142-0

**Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition.** / Zhou, Guanyu; Kashiwabara, Takahito; Oikawa, Issei.

Research output: Contribution to journal › Article

*Journal of Scientific Computing*, pp. 1-36. https://doi.org/10.1007/s10915-015-0142-0

}

TY - JOUR

T1 - Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

AU - Zhou, Guanyu

AU - Kashiwabara, Takahito

AU - Oikawa, Issei

PY - 2015/11/27

Y1 - 2015/11/27

N2 - We consider the penalty method for the stationary Navier–Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate (Formula presented.) in (Formula presented.)-norm, where (Formula presented.) is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate (Formula presented.) for the non-reduced-integration scheme with (Formula presented.), and the reduced-integration scheme with (Formula presented.), where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with (Formula presented.), we prove the convergence order (Formula presented.). The theoretical results are verified by numerical experiments.

AB - We consider the penalty method for the stationary Navier–Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate (Formula presented.) in (Formula presented.)-norm, where (Formula presented.) is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate (Formula presented.) for the non-reduced-integration scheme with (Formula presented.), and the reduced-integration scheme with (Formula presented.), where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with (Formula presented.), we prove the convergence order (Formula presented.). The theoretical results are verified by numerical experiments.

KW - Finite element method

KW - Penalty method

KW - Slip boundary condition

KW - The Navier–Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=84948692166&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948692166&partnerID=8YFLogxK

U2 - 10.1007/s10915-015-0142-0

DO - 10.1007/s10915-015-0142-0

M3 - Article

AN - SCOPUS:84948692166

SP - 1

EP - 36

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

ER -