### Abstract

Error estimates with optimal convergence orders are proved for a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a combination of Lagrange-Galerkin method and Brezzi-Pitkaranta'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 theoretical convergence orders are recognized numerically by two- and three-dimensional computations.

Original language | English |
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Pages (from-to) | 361-380 |

Number of pages | 20 |

Journal | Mathematical Modelling and Numerical Analysis |

Volume | 50 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Mar 1 |

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### Keywords

- Error estimates
- Pressure-stabilization
- The finite element method
- The Lagrange-Galerkin method
- The Navier-Stokes equations

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics

### Cite this

*Mathematical Modelling and Numerical Analysis*,

*50*(2), 361-380. https://doi.org/10.1051/m2an/2015047

**Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations.** / Notsu, Hirofumi; Tabata, Masahisa.

Research output: Contribution to journal › Article

*Mathematical Modelling and Numerical Analysis*, vol. 50, no. 2, pp. 361-380. https://doi.org/10.1051/m2an/2015047

}

TY - JOUR

T1 - Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations

AU - Notsu, Hirofumi

AU - Tabata, Masahisa

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Error estimates with optimal convergence orders are proved for a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a combination of Lagrange-Galerkin method and Brezzi-Pitkaranta'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 theoretical convergence orders are recognized numerically by two- and three-dimensional computations.

AB - Error estimates with optimal convergence orders are proved for a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a combination of Lagrange-Galerkin method and Brezzi-Pitkaranta'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 theoretical convergence orders are recognized numerically by two- and three-dimensional computations.

KW - Error estimates

KW - Pressure-stabilization

KW - The finite element method

KW - The Lagrange-Galerkin method

KW - The Navier-Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=84975755186&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975755186&partnerID=8YFLogxK

U2 - 10.1051/m2an/2015047

DO - 10.1051/m2an/2015047

M3 - Article

VL - 50

SP - 361

EP - 380

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 2

ER -