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.
DOI : 10.1051/m2an/2015047
Keywords: Error estimates, the finite element method, the Lagrange−Galerkin method, pressure-stabilization, the Navier−Stokes equations
Notsu, Hirofumi 1 ; Tabata, Masahisa 2
@article{M2AN_2016__50_2_361_0,
author = {Notsu, Hirofumi and Tabata, Masahisa},
title = {Error estimates of a stabilized {Lagrange\ensuremath{-}Galerkin} scheme for the {Navier\ensuremath{-}Stokes} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {361--380},
year = {2016},
publisher = {EDP Sciences},
volume = {50},
number = {2},
doi = {10.1051/m2an/2015047},
mrnumber = {3482547},
zbl = {1381.76192},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2015047/}
}
TY - JOUR AU - Notsu, Hirofumi AU - Tabata, Masahisa TI - Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 361 EP - 380 VL - 50 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2015047/ DO - 10.1051/m2an/2015047 LA - en ID - M2AN_2016__50_2_361_0 ER -
%0 Journal Article %A Notsu, Hirofumi %A Tabata, Masahisa %T Error estimates of a stabilized Lagrange−Galerkin scheme for the Navier−Stokes equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 361-380 %V 50 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2015047/ %R 10.1051/m2an/2015047 %G en %F M2AN_2016__50_2_361_0
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
and , Convergence analysis of a finite element projection/Lagrange–Galerkin method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 37 (2000) 799–826. | MR | Zbl | DOI
R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1994). | MR | Zbl
and , A second order characteristics finite element scheme for natural convection problems. J. Comput. Appl. Math. 235 (2011) 3270–3284. | MR | Zbl | DOI
and , Modified Lagrange–Galerkin methods of first and second order in time for convection-diffusion problems. Numer. Math. 120 (2012) 601–638. | MR | Zbl | DOI
, and , Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. part II: fully discretized scheme and quadrature formulas. SIAM J. Numer. Anal. 44 (2006) 1854–1876. | MR | Zbl | DOI
, , and , A high-order characteristics/finite element method for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 25 (1997) 1421–1454. | MR | Zbl | DOI
and , Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225–235. | MR | Zbl | DOI
F. Brezzi and J. Pitkäranta, On the Stabilization of Finite Element Approximations of the Stokes Equations, In Efficient Solutions of Elliptic Systems, edited by W. Hackbusch. Vieweg, Wiesbaden (1984) 11–19. | MR | Zbl
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
and , Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19 (1982), 871–885. | MR | Zbl | DOI
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer, Berlin (1976). | MR | Zbl
R.E. Ewing and T.F. Russell, Multistep Galerkin methods along characteristics for convection-diffusion problems, In Advances in Computer Methods for Partial Differential Equations, edited by R. Vichnevetsky and R.S. Stepleman. IMACS IV (1981) 28–36.
and , Error analysis of some Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680–1697. | MR | Zbl | DOI
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986). | MR | Zbl
, and , Characteristic stabilized finite element method for the transient Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 199 (2010) 2996–3004. | MR | Zbl | DOI
, and , Stability of the Lagrange–Galerkin method with non-exact integration. Model. Math. Anal. Num. 22 (1988) 625–653. | MR | Zbl | Numdam | DOI
H. Notsu, Numerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme. Trans. Japan Soc. Comput. Eng. Sci. (2008) 20080032.
and , A combined finite element scheme with a pressure stabilization and a characteristic-curve method for the Navier–Stokes equations. Trans. Japan Soc. Ind. Appl. Math. 18 (2008) 427–445 (in Japanese).
and , A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier–Stokes equations. J. Scientific Comput. 38 (2009) 1–14. | MR | Zbl | DOI
and , Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. J. Scientific Comput. 65 (2015) 940–955. | MR | Zbl | DOI
, On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math. 38 (1982) 309–332. | MR | Zbl | DOI
and , Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type. Int. J. Numer. Methods Fluids 64 (2010) 1240–1253. | MR | Zbl | DOI
, Exact projections and the Lagrange–Galerkin method: a realistic alternative to quadrature. J. Comput. Phys. 112 (1994) 316–333. | MR | Zbl | DOI
and , A second order characteristic finite element scheme for convection-diffusion problems, Numer. Math. 92 (2002) 161–177. | MR | Zbl | DOI
Y. Saad, Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003). | MR | Zbl
A.H. Stroud, Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs, New Jersey (1971). | MR | Zbl
, Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math. 53 (1988) 459–483. | MR | Zbl | DOI
, Discrepancy between theory and real computation on the stability of some finite element schemes. J. Comput. Appl. Math. 199 (2007) 424–431. | MR | Zbl | DOI
M. Tabata and S. Fujima, Robustness of a Characteristic Finite Element Scheme of Second Order in Time Increment, In Computational Fluid Dynamics 2004. Edited by C. Groth and D.W. Zingg. Springer (2006) 177–182.
M. Tabata and S. Uchiumi, A genuinely stable Lagrange–Galerkin scheme for convection-diffusion problems. Preprint [math.NA]. | arXiv | MR
M. Tabata and S. Uchiumi, A Lagrange–Galerkin scheme with a locally linearized velocity for the Navier–Stokes equations. Preprint [math.NA]. | arXiv
, and , A characteristic finite element method using the exact integration. Annual Report of Research Institute for Information Technology, Kyushu University 2 (2002) 11–18 (in Japanese).
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