### Abstract

Two stabilized Lagrange-Galerkin schemes for the Navier-Stokes equations are reviewed. The schemes are based on a combination of the Lagrange-Galerkin method and Brezzi-Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure,the numbers of degrees of freedom are much smaller than that of other typical elements for the equations,e.g.,P2/P1. Therefore,the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams-Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme,whose theoretical result is also recognized numerically.

Original language | English |
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Pages (from-to) | 331-343 |

Number of pages | 13 |

Journal | Modeling and Simulation in Science, Engineering and Technology |

DOIs | |

Publication status | Published - 2016 |

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### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Engineering(all)
- Computational Mathematics
- Modelling and Simulation

### Cite this

**Stabilized lagrange-galerkin schemes of first-and second-order in time for the navier-stokes equations.** / Notsu, Hirofumi; Tabata, Masahisa.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Stabilized lagrange-galerkin schemes of first-and second-order in time for the navier-stokes equations

AU - Notsu, Hirofumi

AU - Tabata, Masahisa

PY - 2016

Y1 - 2016

N2 - Two stabilized Lagrange-Galerkin schemes for the Navier-Stokes equations are reviewed. The schemes are based on a combination of the Lagrange-Galerkin method and Brezzi-Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure,the numbers of degrees of freedom are much smaller than that of other typical elements for the equations,e.g.,P2/P1. Therefore,the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams-Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme,whose theoretical result is also recognized numerically.

AB - Two stabilized Lagrange-Galerkin schemes for the Navier-Stokes equations are reviewed. The schemes are based on a combination of the Lagrange-Galerkin method and Brezzi-Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure,the numbers of degrees of freedom are much smaller than that of other typical elements for the equations,e.g.,P2/P1. Therefore,the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams-Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme,whose theoretical result is also recognized numerically.

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U2 - 10.1007/978-3-319-40827-9_26

DO - 10.1007/978-3-319-40827-9_26

M3 - Article

SP - 331

EP - 343

JO - Modeling and Simulation in Science, Engineering and Technology

JF - Modeling and Simulation in Science, Engineering and Technology

SN - 2164-3679

ER -