Applied Mathematics and Mechanics >
Two-level stabilized finite element method for Stokes eigenvalue problem
Received date: 2011-05-04
Revised date: 2012-02-10
Online published: 2012-05-10
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 10901131, 10971166, and 10961024), the National High Technology Research and Development Program of China (No. 2009AA01A135), and the Natural Science Foundation of Xinjiang Uygur Autonomous Region (No. 2010211B04)
A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered. This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H2), which can still maintain the asymptotically optimal accuracy. It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution, which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h. Hence, the two-level stabilized finite element method can save a large amount of computational time. Moreover, numerical tests confirm the theoretical results of the present method.
Peng-zhan HUANG;Yin-nian HE;Xin-longFENG . Two-level stabilized finite element method for Stokes eigenvalue problem[J]. Applied Mathematics and Mechanics, 2012 , 33(5) : 621 -630 . DOI: 10.1007/s10483-012-1575-7
[1] Babuska, I. and Osborn, J. E. Eigenvalue problems. Handbook of Numerical Analysis, Vol. II,Finite Element Method (Part I) (eds. Ciarlet, P. G. and Lions, J. L.), North-Holland, Amsterdam,641-787 (1991)
[2] Babuska, I. and Osborn, J. E. Finite element-Galerkin approximation of the eigenvalues andeigenvectors of selfadjoint problems. Math. Comp., 52, 275-297 (1989)
[3] Lin, Q. and Xie, H. Asymptotic error expansion and Richardson extrapolation of eigenvalue approximationsfor second order elliptic problems by the mixed finite element method. Appl. Numer.Math., 59, 1884-1893 (2009)
[4] Lin, Q. Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners.Numer. Math., 58, 631-640 (1991)
[5] Jia, S., Xie, H., Yin, X., and Gao, S. Approximation and eigenvalue extrapolation of Stokeseigenvalue problem by nonconforming finite element methods. Appl. Math., 54, 1-15 (2009)
[6] Chen, H., Jia, S. H., and Xie, H. Postprocessing and higher order convergence for the mixed finiteelement approximations of the eigenvalue problem. Appl. Numer. Math., 61, 615-629 (2011)
[7] Chen, H., Jia, S., and Xie, H. Postprocessing and higher order convergence for the mixed finiteelement approximations of the Stokes eigenvalue problems. Appl. Math., 54, 237-250 (2009)
[8] Huang, P. Z., He, Y. N., and Feng, X. L. Numerical investigations on several stabilized finiteelement methods for the Stokes eigenvalue problem. Math. Probl. Eng., 2011, 1-14 (2011)
[9] Chen, W. and Lin, Q. Approximation of an eigenvalue problem associated with the Stokes problemby the stream function-vorticity-pressure method. Appl. Math., 51, 73-88 (2006)
[10] Mercier, B., Osborn, J., Rappaz, J., and Raviart, P. A. Eigenvalue approximation by mixed andhybrid methods. Math. Comput., 36, 427-453 (1981)
[11] Xu, J. and Zhou, A. H. A two-grid discretization scheme for eigenvalue problems. Math. Comput.,70, 17-25 (2009)
[12] Yin, X., Xie, H., Jia, S., and Gao, S. Asymptotic expansions and extrapolations of eigenvaluesfor the Stokes problem by mixed finite element methods. J. Comput. Appl. Math., 215, 127-141(2008)
[13] Lovadina, C., Lyly, M., and Stenberg, R. A posteriori estimates for the Stokes eigenvalue problem.Numerical Methods for Partial Differential Equations, 25, 244-257 (2009)
[14] Luo, F., Lin, Q., and Xie, H. Computing the lower and upper bounds of Laplace eigenvalueproblem: by combining conforming and nonconforming finite element methods. Preprint athttp://arxiv.org/abs/1109.5977 (2011)
[15] Hu, J., Huang, Y., and Lin, Q. The lower bounds for eigenvalues of elliptic operators: by nonconformingfinite element methods. Preprint at http://arxiv.org.abs/1112.1145 (2011)
[16] Bochev, P., Dohrmann, C. R., and Gunzburger, M. D. Stabilization of low-order mixed finiteelements for the Stokes equations. SIAM J. Numer. Anal., 44, 82-101 (2006)
[17] Li, J. and He, Y. N. A stabilized finite element method based on two local Gauss integrations forthe Stokes equations. J. Comput. Appl. Math., 214, 58-65 (2008)
[18] Li, J. and Chen, Z. A new local stabilized nonconforming finite element method for the Stokesequations. Computing, 82, 157-170 (2008)
[19] Li, J., He, Y. N., and Chen, Z. X. A new stabilized finite element method for the transientNavier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 197, 22-35 (2007)
[20] Li, J. Investigations on two kinds of two-level stabilized finite element methods for the stationaryNavier-Stokes equations. Appl. Math. Comput., 182, 1470-1481 (2006)
[21] Huang, P. Z., Zhang, T., and Si, Z. Y. A stabilized Oseen iterative finite element method forstationary conduction-convection equations. Math. Meth. Appl. Sci., 35, 103-118 (2012)
[22] Xu, J. A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput., 15,231-237 (1994)
[23] Xu, J. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal.,33, 1759-1778 (1996)
[24] Layton,W. and Tobiska, L. A two-level method with backtracking for the Navier-Stokes equations.SIAM J. Numer. Anal., 35, 2035-2054 (1998)
[25] Ma, F. Y., Ma, Y. C., and Wo, W. F. Local and parallel finite element algorithms based on twogriddiscretization for steady Navier-Stokes equations. Appl. Math. Mech. -Engl. Ed., 28(1), 27-35(2007) DOI 10.1007/s10483-007-0104-x
[26] Qin, X. Q., Ma, Y. C., and Zhang, Y. Two-grid method for characteristics finite-element solutionof 2D nonlinear convection-dominated diffusion problem. Appl. Math. Mech. -Engl. Ed., 26(11),1506-1514 (2005) DOI 10.1007/BF03246258
[27] Wang, C., Huang, Z. P., and Li, L. K. Two-grid partition of unity method for second order ellipticproblems. Appl. Math. Mech. -Engl. Ed., 29(4), 527-533 (2008) DOI 10.1007/s10483-008-0411-x
[28] Zhang, Y. and He, Y. N. A two-level finite element method for the stationary Navier-Stokesequations based on a stabilized local projection. Numer. Meth. Part. Differ. Equ., 27, 460-477(2011)
[29] Ervin, V., Layton, W., and Maubach, J. A posteriori error estimators for a two-level finite elementmethod for the Navier-Stokes equations. Numer. Meth. Part. Differ. Equ., 12, 333-346 (1996)
[30] He, Y. N. and Li, K. T. Two-level stabilized finite element methods for the steady Navier-Stokesproblem. Computing, 74, 337-351 (2005)
[31] He, Y. N. and Wang, A. W. A simplified two-level method for the steady Navier-Stokes equations.Comput. Methods Appl. Mech. Engrg., 197, 1568-1576 (2008)
[32] Shang, Y. Q. and Luo, Z. D. A parallel two-level finite element method for the Navier-Stokesequations. Appl. Math. Mech. -Engl. Ed., 31(11), 1429-1438 (2010) DOI 10.1007/s10483-010-1373-7
[33] Becker, R. and Hansbo, P. A simple pressure stabilization method for the Stokes equation. Commun.Numer. Meth. Engrg., 24, 1421-1430 (2008)
[34] Hecht, F., Pironneau, O., Hyaric, A. L., and Ohtsuka, K. FreeFEM++, Version 2.3-3 (2008)Software avaible at http://www.freefem.org
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