An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates

Masahisa Tabata, Shinya Uchiumi

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P2/P1- and P1+/P1-finite elements optimal error estimates are proved in ℓ(H1)×ℓ2(L2) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.

    Original languageEnglish
    Pages (from-to)39-67
    Number of pages29
    JournalMathematics of Computation
    Volume87
    Issue number309
    DOIs
    Publication statusPublished - 2018 Jan 1

    Fingerprint

    Lagrange
    Galerkin
    Navier Stokes equations
    Error Estimates
    Navier-Stokes Equations
    Galerkin methods
    Backward Euler Method
    Cavity Flow
    Lagrange Method
    Ordinary differential equations
    Composite function
    Numerical Quadrature
    Optimal Error Estimates
    Reynolds number
    Stability and Convergence
    Robust Stability
    Galerkin Method
    Ordinary differential equation
    Fluids
    Composite materials

    Keywords

    • Exact computation
    • Finite element method
    • Lagrange-Galerkin scheme
    • Navier-Stokes equations

    ASJC Scopus subject areas

    • Algebra and Number Theory
    • Computational Mathematics
    • Applied Mathematics

    Cite this

    An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates. / Tabata, Masahisa; Uchiumi, Shinya.

    In: Mathematics of Computation, Vol. 87, No. 309, 01.01.2018, p. 39-67.

    Research output: Contribution to journalArticle

    @article{41bbd8d487114e71aa00a88761169d27,
    title = "An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates",
    abstract = "We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P2/P1- and P1+/P1-finite elements optimal error estimates are proved in ℓ∞(H1)×ℓ2(L2) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.",
    keywords = "Exact computation, Finite element method, Lagrange-Galerkin scheme, Navier-Stokes equations",
    author = "Masahisa Tabata and Shinya Uchiumi",
    year = "2018",
    month = "1",
    day = "1",
    doi = "10.1090/mcom/3222",
    language = "English",
    volume = "87",
    pages = "39--67",
    journal = "Mathematics of Computation",
    issn = "0025-5718",
    publisher = "American Mathematical Society",
    number = "309",

    }

    TY - JOUR

    T1 - An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates

    AU - Tabata, Masahisa

    AU - Uchiumi, Shinya

    PY - 2018/1/1

    Y1 - 2018/1/1

    N2 - We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P2/P1- and P1+/P1-finite elements optimal error estimates are proved in ℓ∞(H1)×ℓ2(L2) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.

    AB - We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P2/P1- and P1+/P1-finite elements optimal error estimates are proved in ℓ∞(H1)×ℓ2(L2) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.

    KW - Exact computation

    KW - Finite element method

    KW - Lagrange-Galerkin scheme

    KW - Navier-Stokes equations

    UR - http://www.scopus.com/inward/record.url?scp=85038959574&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85038959574&partnerID=8YFLogxK

    U2 - 10.1090/mcom/3222

    DO - 10.1090/mcom/3222

    M3 - Article

    VL - 87

    SP - 39

    EP - 67

    JO - Mathematics of Computation

    JF - Mathematics of Computation

    SN - 0025-5718

    IS - 309

    ER -