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A Finite Volume Method Based on the Constrained Nonconforming Rotated Q1-Constant Element for the Stokes Problem

Published online by Cambridge University Press:  03 June 2015

Jing Qi ,
Wanfu Tian  and
Yonghai Li
Jing Qi*
Affiliation:
Jilin University, Changchun 130012, China
Wanfu Tian*
Affiliation:
School of Science, Shenyang Aerospace University, Shenyang 110136, China
Yonghai Li*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, China
*
Email: qijing08@mails.jlu.edu.cn
Email: yonghai@jlu.edu.cn
Corresponding author. Email: tianwf@163.com
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Abstract

We construct a finite volume element method based on the constrained nonconforming rotated Q1-constant element (CNRQ1-P0) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ1 elements and the pressure by piecewise constants. The errors for the velocity in the H1 norm and for the pressure in the L2 norm are O(h) and the error for the velocity in the L2 norm is O(h2). Numerical experiments are presented to support our theoretical results.

Keywords

65N0865N1565N3076D05Stokes problemfinite volume methodconstrained nonconforming rotated Q1 element
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 1 , February 2012 , pp. 46 - 71
Copyright
Copyright © Global-Science Press 2012

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References

[1] Bochev, P. B., Dohrmann, C. R. and Gunzburger, M. D., Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal., 44(1) (2006), pp. 82–101.Google Scholar
[2] Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, Springer, New York, third edition, 2008.Google Scholar
[3] Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Volume 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991.Google Scholar
[4] Cai, Z. Q., Douglas, J. Jr., and Ye, X., A stable nonconforming quadrilateral finite element method for stationary Stokes and Navier-Stokes equations, Calcolo, 36(4) (1999), pp. 215–232.Google Scholar
[5] Chen, Z. X., Finite Element Methods and Their Applications, Scientific Computation, Springer-Verlag, Berlin, 2005.Google Scholar
[6] Chen, Z. Y., Li, R. H. and Zhou, A. H., A note on the optimal L2-estimate of the finite volume element method, Adv. Comput. Math., 16(4) (2002), pp. 291–303.Google Scholar
[7] Chou, S.-H. and Kwak, D. Y., A covolume method based on rotated bilinears for the generalized Stokes problem, SIAM J. Numer. Anal., 35(2) (1998), pp. 494–507.Google Scholar
[8] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam, 1978, Studies in Mathematics and Its Applications, Vol. 4.Google Scholar
[9] Cui, M. and Ye, X., Superconvergence of finite volume methods for the Stokes equations, Numer. Meth. Part. D. E., 25(5) (2009), pp. 1212–1230.Google Scholar
[10] Girault, V. and Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986.Google Scholar
[11] Hu, J., Man, H. Y. and Shi, Z. C., Constrained nonconforming rotated Q1 element for Stokes flow and planar elasticity, Math. Numer. Sinica, 27(3) (2005), pp. 311–324.Google Scholar
[12] Hu, J. and Shi, Z. C., Constrained quadrilateral nonconforming rotated Q1 element, J. Com-put. Math., 23(5) (2005), pp. 561–586.Google Scholar
[13] Kang, K. S. and Kwak, D. Y., Covolume method for new nonconforming rectangular element for the Stokes problem, Comput. Math. Appl., 43(8-9) (2002), pp. 1063–1078.Google Scholar
[14] Li, J. and He, Y. N., A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math., 214(1) (2008), pp. 58–65.Google Scholar
[15] Li, R. H., Chen, Z. Y. and Wu, W., Generalized Difference Methods for Differential Equations (Numerical Analysis of Finite Volume Methods), Volume 226 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2000.Google Scholar
[16] Li, Y. H. and Li, R. H., Generalized difference methods on arbitrary quadrilateral networks, J. Comput. Math., 17 (1999), pp. 653–672.Google Scholar
[17] Liu, H. P. and Yan, N. N., Global superconvergence for optimal control problems governed by Stokes equations, Int. J. Numer. Anal. Model., 3(3) (2006), pp. 283–302.Google Scholar
[18] Liu, H. P. and Yan, N. N., Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations, Adv. Comput. Math., 29(4) (2008), pp. 375–392.Google Scholar
[19] Lv, J. L. and Li, Y. H., L2 error estimate of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math., 33 (2010), pp. 129–148.Google Scholar
[20] Man, H. Y. and Shi, Z. C., P1-nonconforming quadrilateral finite volume element method and its casadic multigrid algorithm for elliptic problems, J. Comput. Math., 24 (2006), pp. 59–80.Google Scholar
[21] Mao, S. P. and Chen, S. C., Convergence and superconvergence of a nonconforming finite element method for the Stokes problem, J. Numer. Math., 14(2) (2006), pp. 83–101.CrossRefGoogle Scholar
[22] Mao, S. P. and Chen, S. C., A quadrilateral nonconforming finite element for linear elasticity problem, Adv. Comput. Math., 28(1) (2008), pp. 81–100.Google Scholar
[23] Park, C. and Sheen, D., P1-nonconforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal., 41(2) (2003), pp. 624–640.Google Scholar
[24] Pitkaranta, J. and Strnberg, R., Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, in The Mathematics of Finite Elements and Applications V, ed. Whiteman, J., Academic Press, London, pp. 325–334, 1985.Google Scholar
[25] Rannacher, R. and Turek, S., Simple nonconforming quadrilateral stokes element, Numer. Meth. Part. D. E., 8(2) (1992), pp. 97–111.Google Scholar
[26] Stenberg, R., Analysis of mixed finite element methods for the Stokes problem: a unified approach, Math. Comput., 42(165) (1984), pp. 9–23.Google Scholar
[27] Tian, W. F., Mixed Finite Volume Element Method, PhD thesis, Jilin University, Changchun, 2010.Google Scholar
[28] Yang, M., A second-order finite volume element method on quadrilateral meshes for elliptic equations, ESAIM: M2AN, 40(6) (2006), pp. 1053–1067.Google Scholar
[29] Ye, X. and Zikatanov, L., Some observations on Babusška and Brezzi theories, Numer. Math., 94(1) (2003), pp. 195–202.Google Scholar