### Abstract

Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitkäranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

Original language | English |
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Title of host publication | Mathematical Fluid Dynamics, Present and Future |

Publisher | Springer New York LLC |

Pages | 497-530 |

Number of pages | 34 |

Volume | 183 |

ISBN (Print) | 9784431564553 |

DOIs | |

Publication status | Published - 2016 |

Event | 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 - Tokyo, Japan Duration: 2014 Nov 11 → 2014 Nov 14 |

### Other

Other | 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 |
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Country | Japan |

City | Tokyo |

Period | 14/11/11 → 14/11/14 |

### Fingerprint

### Keywords

- Error estimates
- Navier-Strokes equations
- Second-order scheme
- Stabilized Lagrange-Galerkin scheme

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Fluid Dynamics, Present and Future*(Vol. 183, pp. 497-530). Springer New York LLC. https://doi.org/10.1007/978-4-431-56457-7_18

**Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations.** / Notsu, Hirofumi; Tabata, Masahisa.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Mathematical Fluid Dynamics, Present and Future.*vol. 183, Springer New York LLC, pp. 497-530, 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014, Tokyo, Japan, 14/11/11. https://doi.org/10.1007/978-4-431-56457-7_18

}

TY - GEN

T1 - Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations

AU - Notsu, Hirofumi

AU - Tabata, Masahisa

PY - 2016

Y1 - 2016

N2 - Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitkäranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

AB - Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitkäranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

KW - Error estimates

KW - Navier-Strokes equations

KW - Second-order scheme

KW - Stabilized Lagrange-Galerkin scheme

UR - http://www.scopus.com/inward/record.url?scp=85009786158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009786158&partnerID=8YFLogxK

U2 - 10.1007/978-4-431-56457-7_18

DO - 10.1007/978-4-431-56457-7_18

M3 - Conference contribution

AN - SCOPUS:85009786158

SN - 9784431564553

VL - 183

SP - 497

EP - 530

BT - Mathematical Fluid Dynamics, Present and Future

PB - Springer New York LLC

ER -