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


Authors: Masahisa Tabata and Shinya Uchiumi
Journal: Math. Comp. 87 (2018), 39-67
MSC (2010): Primary 65M12, 65M25, 65M60, 76D05, 76M10
DOI: https://doi.org/10.1090/mcom/3222
Published electronically: May 11, 2017
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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 $ \mathrm {P}_{2}/\mathrm {P}_{1}$- and $ \mathrm {P_{1}+}/\mathrm {P}_{1}$-finite elements optimal error estimates are proved in $ \ell ^\infty (H^1)\times \ell ^2(L^2)$ 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.


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  • [1] Y. Achdou and J.-L. Guermond, Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal. 37 (2000), no. 3, 799–826. MR 1740383
  • [2] D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337–344 (1985). MR 799997
  • [3] M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math. 33 (1979), no. 2, 211–224. MR 549450
  • [4] R. Bermejo, P. Galán del Sastre, and L. Saavedra, A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal. 50 (2012), no. 6, 3084–3109. MR 3022255
  • [5] R. Bermejo and L. Saavedra, Modified Lagrange-Galerkin methods to integrate time dependent incompressible Navier-Stokes equations, SIAM J. Sci. Comput. 37 (2015), no. 6, B779–B803. MR 3418229
  • [6] K. Boukir, Y. Maday, B. Métivet, and E. Razafindrakoto, A high-order characteristics/finite element method for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 25 (1997), no. 12, 1421–1454. MR 1601529
  • [7] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
    Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132
  • [8] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
  • [9] Jim Douglas Jr. and Thomas F. Russell, 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), no. 5, 871–885. MR 672564
  • [10] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • [11] P. C. Hammer, O. J. Marlowe, and A. H. Stroud, Numerical integration over simplexes and cones, Math. Tables Aids Comput. 10 (1956), 130–137. MR 0086389, https://doi.org/10.1090/S0025-5718-1956-0086389-6
  • [12] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640
  • [13] R. O. Jack, Stability of the Lagrange-Galerkin method: The performance of quadrature in theory and practice, Tech. Report 88/15, The Mathematical Institute, University of Oxford, 1988.
  • [14] M. E. Laursen and M. Gellert, Some criteria for numerically integrated matrices and quadrature formulas for triangles, International Journal for Numerical Methods in Engineering 12 (1978), no. 1, 67-76.
  • [15] R. Löhner and J. Ambrosiano, A vectorized particle tracer for unstructured grids, Journal of Computational Physics 91 (1990), no. 1, 22-31.
  • [16] K. W. Morton, A. Priestley, and E. Süli, Stability of the Lagrange-Galerkin method with nonexact integration, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 4, 625–653 (English, with French summary). MR 974291
  • [17] Hirofumi Notsu and Masahisa Tabata, A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations, J. Sci. Comput. 38 (2009), no. 1, 1–14. MR 2472216
  • [18] Hirofumi Notsu and Masahisa Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 2, 361–380. MR 3482547
  • [19] H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme of second-order in time for the Navier-Stokes equations, to appear in Mathematical Fluid Dynamics, Present and Future (Y. Shibata and Y. Suzuki, eds.), Springer Proceedings in Mathematics & Statistics, Springer.
  • [20] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math. 38 (1981/82), no. 3, 309–332. MR 654100
  • [21] O. Pironneau and M. Tabata, Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type, Internat. J. Numer. Methods Fluids 64 (2010), no. 10-12, 1240–1253. MR 2767202
  • [22] A. Priestley, Exact projections and the Lagrange-Galerkin method: a realistic alternative to quadrature, J. Comput. Phys. 112 (1994), no. 2, 316–333. MR 1277281
  • [23] Hongxing Rui and Masahisa Tabata, A second order characteristic finite element scheme for convection-diffusion problems, Numer. Math. 92 (2002), no. 1, 161–177. MR 1917369
  • [24] Endre Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988), no. 4, 459–483. MR 951325
  • [25] Masahisa Tabata, Discrepancy between theory and real computation on the stability of some finite element schemes, J. Comput. Appl. Math. 199 (2007), no. 2, 424–431. MR 2269529
  • [26] M. Tabata and S. Fujima, Robustness of a Characteristic Finite Element Scheme of Second Order in Time Increment, Computational Fluid Dynamics 2004, Springer, 2006, pp. 177-182.
  • [27] Masahisa Tabata and Shinya Uchiumi, A genuinely stable Lagrange-Galerkin scheme for convection-diffusion problems, Jpn. J. Ind. Appl. Math. 33 (2016), no. 1, 121–143. MR 3459282
  • [28] K. Tanaka, A. Suzuki, and M. Tabata, A characteristic finite element method using the exact integration, Annual Report of Research Institute for Information Technology of Kyushu University 2 (2002), 11-18, (Japanese).
  • [29] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO Anal. Numér. 18 (1984), no. 2, 175–182. MR 743884

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Additional Information

Masahisa Tabata
Affiliation: Department of Mathematics, Waseda University, 3-4-1, Ohkubo, Shinjuku, Tokyo 169-8555, Japan
Email: tabata@waseda.jp

Shinya Uchiumi
Affiliation: Research Fellow of Japan Society for the Promotion of Science and Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1, Ohkubo, Shinjuku, Tokyo 169-8555, Japan
Email: su48@fuji.waseda.jp

DOI: https://doi.org/10.1090/mcom/3222
Keywords: Lagrange--Galerkin scheme, finite element method, Navier--Stokes equations, exact computation.
Received by editor(s): September 4, 2015
Received by editor(s) in revised form: August 2, 2016
Published electronically: May 11, 2017
Article copyright: © Copyright 2017 American Mathematical Society