Volume 50, Number 2, March-April 2016
|Page(s)||361 - 380|
|Published online||19 February 2016|
Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations
Waseda Institute for Advanced Study, Waseda
University, 3-4-1, Ohkubo,
2 Department of Mathematics, Waseda University, 3-4-1, Ohkubo, Shinjuku, 169-8555, Tokyo, Japan
Received: 16 February 2015
Revised: 26 May 2015
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.
Mathematics Subject Classification: 65M12 / 65M25 / 65M60 / 76D05 / 76M10
Key words: Error estimates / the finite element method / the Lagrange−Galerkin method / pressure-stabilization / the Navier−Stokes equations
© EDP Sciences, SMAI 2016
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