Geodesic
Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations
Notsu, Hirofumi1 ; Tabata, Masahisa2
1 Waseda Institute for Advanced Study, Waseda University, 3-4-1, Ohkubo, Shinjuku, 169-8555, Tokyo, Japan
2 Department of Mathematics, Waseda University, 3-4-1, Ohkubo, Shinjuku, 169-8555, Tokyo, Japan
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 361-380

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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−Pitkäranta’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.

Reçu le :
DOI : 10.1051/m2an/2015047
Classification : 65M12, 65M25, 65M60, 76D05, 76M10
Keywords: Error estimates, the finite element method, the Lagrange−Galerkin method, pressure-stabilization, the Navier−Stokes equations

Notsu, Hirofumi 1 ; Tabata, Masahisa 2

1 Waseda Institute for Advanced Study, Waseda University, 3-4-1, Ohkubo, Shinjuku, 169-8555, Tokyo, Japan
2 Department of Mathematics, Waseda University, 3-4-1, Ohkubo, Shinjuku, 169-8555, Tokyo, Japan 
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     title = {Error estimates of a stabilized {Lagrange\ensuremath{-}Galerkin} scheme for the {Navier\ensuremath{-}Stokes} equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
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Notsu, Hirofumi; Tabata, Masahisa. Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 361-380. doi: 10.1051/m2an/2015047

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