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A TWO-LEVEL DEFECT–CORRECTION METHOD FOR NAVIER–STOKES EQUATIONS

Part of: Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems

Published online by Cambridge University Press:  21 October 2009

QINGFANG LIU  and
YANREN HOU
QINGFANG LIU*
Affiliation:
School of Science, Xi’an Jiaotong University, Xi’an 710049, PR China (email: [email protected])
YANREN HOU
Affiliation:
School of Science, Xi’an Jiaotong University, Xi’an 710049, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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A two-level defect–correction method for the steady-state Navier–Stokes equations with a high Reynolds number is considered in this paper. The defect step is accomplished in a coarse-level subspace Hm by solving the standard Galerkin equation with an artificial viscosity parameter σ as a stability factor, and the correction step is performed in a fine-level subspace HM by solving a linear equation. H1 error estimates are derived for this two-level defect–correction method. Moreover, some numerical examples are presented to show that the two-level defect–correction method can reach the same accuracy as the standard Galerkin method in fine-level subspace HM. However, the two-level method will involve much less work than the one-level method.

Keywords

two-level methoddefect–correction methodspectral methodNavier–Stokes equations

MSC classification

Secondary: 65N35: Spectral, collocation and related methods 35Q30: Navier-Stokes equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 442 - 454
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

Subsidized by NSF of China (Grant No. 10471110, No. 10871156) and NCET.

References

[1]Ait Ou Ammi, A. and Marion, M., ‘Nonlinear Galerkin method and mixed finite elements: two-grid algorithms for the Navier–Stokes equations’, Numer. Math. 68 (1994), 189–213.CrossRefGoogle Scholar
[2]Axelsson, O. and Layton, W., ‘Defect–correction methods for convection dominated convection-diffusion equations’, RAIRO Math. Model. Numer. Anal. 24 (1990), 423–455.CrossRefGoogle Scholar
[3]Böhmer, K., Hemker, P. W. and Stetter, H. J., ‘The defect correction approach’, in: Defect Correction Methods: Theory and Applications, Computing Supplement, 5 (eds. Bohmer, K. and Stetter, H. J.) (Wien, Springer, Vienna, 1984), pp. 1–32.CrossRefGoogle Scholar
[4]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods in Fluid Dynamics (Springer, Berlin, 1987).Google Scholar
[5]Kaya, S., Layton, W. and Riviere, B., ‘Subgrid stabilized defect correction methods for the Navier–Stokes equations’, SIAM J. Numer. Anal. 44 (2006), 1639–1654.CrossRefGoogle Scholar
[6]Layton, W., ‘A two-level discretization method for the Navier–Stokes equations’, Comput. Math. Appl. 26 (1993), 33–38.CrossRefGoogle Scholar
[7]Layton, W., ‘Solution algorithm for incompressible viscous flows at high Reynolds number’, Vestnik Moskov Gos. Univ. Ser. 15 (1996), 25–35.Google Scholar
[8]Layton, W., Lee, H. K. and Perterson, J., ‘Numerical solution of the stationary Navier–Stokes equations using a multilevel method’, SIAM J. Sci. Comput. 20 (1998), 1–12.CrossRefGoogle Scholar
[9]Layton, W., Lee, H. K. and Peterson, J., ‘A defect–correction method for the incompressible Navier–Stokes equations’, Appl. Math. Comput. 129 (2002), 1–19.Google Scholar
[10]Layton, W. and Lenferink, W., ‘Two-level Picard and modified Picard methods for the Navier–Stokes equations’, Appl. Math. Comput. 69 (1995), 263–274.Google Scholar
[11]Stetter, H. J., ‘The defect–correction principle and discretization methods’, Numer. Math. 29 (1978), 425–443.CrossRefGoogle Scholar
[12]Temam, R., Navier–Stokes Equations and Nonlinear Functional Analysis, CBNS-NSF Regional Conference Series in Applied Mathematics, 66 (SIAM, Philadelphia, 1983).Google Scholar
[13]Temam, R., Navier–Stokes Equations, Theory and Numerical Analysis, 3rd edn (North-Holland, Amsterdam, 1984).Google Scholar
[14]Xu, J., ‘A novel two-grid method for semi-linear elliptic equations’, SIAM J. Sci. Comput. 15 (1994), 231–237.CrossRefGoogle Scholar
[15]Xu, J., ‘Two-grid discretization techniques for linear and nonlinear PDEs’, SIAM, J. Numer. Anal. 33 (1996), 1759–1777.CrossRefGoogle Scholar