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

Hirofumi Notsu, Masahisa Tabata

    Research output: Contribution to journalArticle

    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 languageEnglish
    Pages (from-to)331-343
    Number of pages13
    JournalModeling and Simulation in Science, Engineering and Technology
    DOIs
    Publication statusPublished - 2016

    Fingerprint

    Lagrange
    Galerkin
    Navier Stokes equations
    Navier-Stokes Equations
    First-order
    Degrees of freedom (mechanics)
    Galerkin methods
    Linear equations
    Stabilization
    Lagrange Method
    System of Linear Equations
    Galerkin Method
    Convergence Analysis
    Convection
    Degree of freedom
    Finite Element
    Three-dimensional

    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.

    In: Modeling and Simulation in Science, Engineering and Technology, 2016, p. 331-343.

    Research output: Contribution to journalArticle

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    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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