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A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  10 November 2015

Deepjyoti Goswami  and
Pedro D. Damázio
Deepjyoti Goswami*
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam -784028, India
Pedro D. Damázio
Affiliation:
Department of Mathematics, Universidade Federal do Paraná, Centro Politécnico, Curitiba, Cx.P: 19081, CEP: 81531-990, PR, Brazil
*
*Corresponding author
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Abstract

We analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size H and solving a Stokes problem on a fine grid of size h, h « H. This method gives optimal convergence for velocity in H1-norm and for pressure in L2-norm. The analysis mainly focuses on the loss of regularity of the solution at t = 0 of the Navier-Stokes equations.

Keywords

Two-gridfinite elementNavier-Stokes equationsnon-smooth initial dataerror estimates

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M12: Stability and convergence of numerical methods 65M15: Error bounds
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 549 - 581
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Abboud, H., Girault, V., and Sayah, T., A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme, Numer. Math., vol. 114, no. 2 (2009), pp. 189–231.CrossRefGoogle Scholar
[2]Abboud, H., and Sayah, T., A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme, M2AN Math. Model. Numer. Anal., vol. 42, no. 1 (2008), pp. 141–174.CrossRefGoogle Scholar
[3]Brezzi, F., and Fortin, M., Mixed and Hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York, 1991. x+350 pp.Google Scholar
[4]Bercovier, M., and Pironneau, O., Error estimates for finite element solution of the Stokes problem in the primitive variables, Numer. Math., vol. 33, no. 2 (1979), pp. 211–224.CrossRefGoogle Scholar
[5]Chen, C., and Liu, W., A two-grid method for finite element solutions of nonlinear parabolic equations, Abstr. Appl. Anal. (2012), Art. ID 391918, 11 pp.Google Scholar
[6]Chen, C., Liu, W., and Zhao, X., A two-grid finite element method for a second-order nonlinear hyperbolic equation, Abstr. Appl. Anal. (2014), Art. ID 803615, 6 pp.Google Scholar
[7]De Frutos, J., García-archilla, B., and Novo, J., The postprocessed mixed finite-element method for the Navier-Stokes equations: refined error bounds, SIAM J. Numer. Anal., vol. 46, no. 1 (2008), pp. 201–230.CrossRefGoogle Scholar
[8]De Frutos, J., García-archilla, B., and Novo, J., A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations, J. Comput. Appl. Maths., vol. 236 (2011), pp. 1103–1122.CrossRefGoogle Scholar
[9]De Frutos, J., García-archilla, B., and Novo, J., Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations, Appl. Math. Comput., vol. 218, no. 13 (2012), pp. 7034–7051.Google Scholar
[10]De Frutos, J., García-archilla, B., and Novo, J., Static two-grid mixed finite-element approximations to the Navier-Stokes equations, J. Sci. Comput., vol. 52 (2012), pp. 619–637.CrossRefGoogle Scholar
[11]Giraul, V., and Lions, J.-L., Two-grid finite-element schemes for the transient Navier-Stokes problem, M2AN Math. Model. Numer. Anal., vol. 35, no. 5 (2001), pp. 945–980.CrossRefGoogle Scholar
[12]Girault, V., and Raviart, P. A., Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, 749. Springer-Verlag, Berlin-New York, 1980. vii+200pp.Google Scholar
[13]He, Y., A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations. I. Spatial discretization, J. Comput. Math., vol. 22, no. 1 (2004), pp. 21–32.Google Scholar
[14]He, Y., and Li, J., Two-level methods based on three corrections for the 2D/3D steady Navier-Stokes equations, Int. J. Numer. Anal. Model., Ser. B, vol. 2, no. 1 (2011), pp. 42–56.Google Scholar
[15]He, Y., Miao, H., and Ren, C., A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations. II. Time discretization, J. Comput. Math., vol. 22, no. 1 (2004), pp. 33–54.Google Scholar
[16]Heywood, J. G., and Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem: I. Regularity of solutions and second order error estimates for spatial discretization, SIAM J. Numer. Anal., vol. 19, no. 2 (1982), pp. 275–311.CrossRefGoogle Scholar
[17]Heywood, J. G., and Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem: I. Error analysis for second-order time discretization, SIAM J. Numer. Anal., vol. 27, no. 2 (1990), pp. 353–384.CrossRefGoogle Scholar
[18]Layton, W., A two-level discretization method for the Navier-Stokes equations, Comput. Math. Appl., vol. 26, no. 2 (1993), pp. 33–38.CrossRefGoogle Scholar
[19]Layton, W., and Lenferink, W., Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput., vol. 69, no. 2-3 (1995), pp. 263–274.Google Scholar
[20]Layton, W., and Tobiska, L., A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal., vol. 35, no. 5 (1998), pp. 2035–2054 (electronic).CrossRefGoogle Scholar
[21]Liu, Q., and Hou, Y., A two-level finite element method for the Navier-Stokes equations based on a new projection, Appl. Math. Model., vol. 87, no. 2 (2010), pp. 383–399.CrossRefGoogle Scholar
[22]Liu, Q., and Hou, Y., A two-level correction method in space and time based on Crank-Nicolson scheme for Navier-Stokes equations, Int. J. Comput. Math., vol. 87, no. 11 (2010), pp. 2520–2532.CrossRefGoogle Scholar
[23]Liu, Q., Hou, Y., and Liu, Q., A two-level method in time and space for solving the Navier-Stokes equations based on Newton iteration, Comput. Math. Appl., vol. 64, no. 11 (2012), pp. 3569–3579.CrossRefGoogle Scholar
[24]Margolin, L. G., Titi, E. S. and Wynne, S., The postprocessing Galerkin and nonlinear Galerkin methods-A truncation analysis point of view, SIAM J. Numer. Anal., vol. 41 (2003), pp. 695–714.CrossRefGoogle Scholar
[25]Olshanskii, M. A., Two-level method and some a priori estimates in unsteady Navier-Stokes calculations, J. Comput. Appl. Math., vol. 104, no. 2 (1999), pp. 173–191.CrossRefGoogle Scholar
[26]Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, Third Edition. Studies in Mathematics and its Application 2. North-Holland Publishing Co., Amsterdam, 1984. xii+526 pp.Google Scholar
[27]Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1983. xi+141 pp.Google Scholar
[28]Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, Second Edition, Applied Mathematical Sciences, 68. Springer-Verlag New York, 1997. xxii+648 pp.Google Scholar
[29]Xu, J., A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., vol. 15, no. 1 (1994), pp. 231–237.CrossRefGoogle Scholar
[30]Xu, J., Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., vol. 33, no. 5 (1996), pp. 1759–1777.CrossRefGoogle Scholar