L2-projected least-squares finite element methods for the Stokes equations

Huo-yuan Duan, Ping Lin, P. Saikrishnan, Roger C. E. Tan

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    Abstract

    Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition. A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results
    Original languageEnglish
    Pages (from-to)732-752
    Number of pages21
    JournalSIAM Journal on Numerical Analysis
    Volume44
    Issue number2
    DOIs
    Publication statusPublished - 2006

    Fingerprint

    Least-squares Finite Element Method
    Stokes Equations
    Finite element method
    Vorticity
    Least Square Method
    Interpolation
    Interpolate
    Finite Element
    Optimal Error Estimates
    Order of Approximation
    Stokes Problem
    First-order System
    Projector
    Unequal
    Dirichlet
    Momentum
    Exact Solution
    Regularity
    Numerical Experiment
    Boundary conditions

    Keywords

    • Stokes equation
    • Velocity-pressure-vorticity least-squares finite element method
    • L2 projection
    • Mass-lumping

    Cite this

    Duan, Huo-yuan ; Lin, Ping ; Saikrishnan, P. ; Tan, Roger C. E. / L2-projected least-squares finite element methods for the Stokes equations. In: SIAM Journal on Numerical Analysis. 2006 ; Vol. 44, No. 2. pp. 732-752.
    @article{c9912dc83d0243c2bb7d9cb6cc7462b3,
    title = "L2-projected least-squares finite element methods for the Stokes equations",
    abstract = "Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition. A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results",
    keywords = "Stokes equation, Velocity-pressure-vorticity least-squares finite element method, L2 projection, Mass-lumping",
    author = "Huo-yuan Duan and Ping Lin and P. Saikrishnan and Tan, {Roger C. E.}",
    note = "dc.publisher: Society for Industrial and Applied Mathematics",
    year = "2006",
    doi = "10.1137/040613573",
    language = "English",
    volume = "44",
    pages = "732--752",
    journal = "SIAM Journal on Numerical Analysis",
    issn = "0036-1429",
    publisher = "Society for Industrial and Applied Mathematics",
    number = "2",

    }

    L2-projected least-squares finite element methods for the Stokes equations. / Duan, Huo-yuan; Lin, Ping; Saikrishnan, P.; Tan, Roger C. E.

    In: SIAM Journal on Numerical Analysis, Vol. 44, No. 2, 2006, p. 732-752.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - L2-projected least-squares finite element methods for the Stokes equations

    AU - Duan, Huo-yuan

    AU - Lin, Ping

    AU - Saikrishnan, P.

    AU - Tan, Roger C. E.

    N1 - dc.publisher: Society for Industrial and Applied Mathematics

    PY - 2006

    Y1 - 2006

    N2 - Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition. A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results

    AB - Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition. A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results

    KW - Stokes equation

    KW - Velocity-pressure-vorticity least-squares finite element method

    KW - L2 projection

    KW - Mass-lumping

    U2 - 10.1137/040613573

    DO - 10.1137/040613573

    M3 - Article

    VL - 44

    SP - 732

    EP - 752

    JO - SIAM Journal on Numerical Analysis

    JF - SIAM Journal on Numerical Analysis

    SN - 0036-1429

    IS - 2

    ER -