| Title:
|
Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods (English) |
| Author:
|
Jia, Shanghui |
| Author:
|
Xie, Hehu |
| Author:
|
Yin, Xiaobo |
| Author:
|
Gao, Shaoqin |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
1-15 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, $Q_1^{{\rm rot}}$ and $EQ_1^{{\rm rot}}$. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. (English) |
| Keyword:
|
Stokes eigenvalue problem |
| Keyword:
|
stream function-vorticity-pressure method |
| Keyword:
|
asymptotic expansion |
| Keyword:
|
extrapolation |
| Keyword:
|
a posteriori error estimates |
| Keyword:
|
nonconforming finite element methods |
| Keyword:
|
convergence |
| MSC:
|
35A35 |
| MSC:
|
35P05 |
| MSC:
|
35P15 |
| MSC:
|
35Q30 |
| MSC:
|
35Q35 |
| MSC:
|
65N12 |
| MSC:
|
65N15 |
| MSC:
|
65N25 |
| MSC:
|
65N30 |
| MSC:
|
76D07 |
| MSC:
|
76M10 |
| idZBL:
|
Zbl 1212.65434 |
| idMR:
|
MR2476018 |
| DOI:
|
10.1007/s10492-009-0001-0 |
| . |
| Date available:
|
2010-07-20T12:42:54Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140344 |
| . |
| Reference:
|
[1] Babuška, I., Osborn, J. F.: Estimate for the errors in eigenvalue and eigenvector approximation by Galerkin methods with particular attention to the case of multiple eigenvalue.SIAM J. Numer. Anal. 24 (1987), 1249-1276. MR 0917451, 10.1137/0724082 |
| Reference:
|
[2] Babuška, I., Osborn, J. F.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems.Math. Comput. 52 (1989), 275-297. MR 0962210, 10.1090/S0025-5718-1989-0962210-8 |
| Reference:
|
[3] Bercovier, M., Pironneau, O.: Error estimates for finite element method solution of the Stokes problem in the primitive variables.Numer. Math. 33 (1979), 211-224. Zbl 0423.65058, MR 0549450, 10.1007/BF01399555 |
| Reference:
|
[4] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol. 15.Springer New York (1991). MR 1115205, 10.1007/978-1-4612-3172-1_1 |
| Reference:
|
[5] Chen, W., Lin, Q.: Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method.Appl. Math. 51 (2006), 73-88. Zbl 1164.65489, MR 2197324, 10.1007/s10492-006-0006-x |
| Reference:
|
[6] Chen, W., Lin, Q.: Asymptotic expansion and extrapolation for the eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet-Raviart scheme.Adv. Comput. Math. 27 (2007), 95-106. Zbl 1122.65106, MR 2317923, 10.1007/s10444-007-9031-x |
| Reference:
|
[7] Chen, Z.: Finite Element Methods and Their Applications.Springer Berlin (2005). Zbl 1082.65118, MR 2158541 |
| Reference:
|
[8] Ciarlet, P.: The Finite Element Method for Elliptic Problems.North-Holland Amsterdam (1978). Zbl 0383.65058, MR 0520174 |
| Reference:
|
[9] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms.Springer Berlin (1986). Zbl 0585.65077, MR 0851383 |
| Reference:
|
[10] Glowinski, R., Pironneau, O.: On a mixed finite element approximation of the Stokes problem. I. Convergence of the approximate solution.Numer. Math. 33 (1979), 397-424. MR 0553350 |
| Reference:
|
[11] Han, H.: Nonconforming elements in the mixed finite element method.J. Comput. Math. 2 (1984), 223-233. Zbl 0573.65083, MR 0815417 |
| Reference:
|
[12] Jia, S., Xie, H., Yin, X., Gao, S.: Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods.Numer. Methods Partial Differ. Equations 24 (2008), 435-448. Zbl 1151.65086, MR 2382790, 10.1002/num.20268 |
| Reference:
|
[13] Křížek, M.: Conforming finite element approximation of the Stokes problem.Banach Cent. Publ. 24 (1990), 389-396. MR 1097422, 10.4064/-24-1-389-396 |
| Reference:
|
[14] Lin, Q., Huang, H., Li, Z.: New expansion of numerical eigenvalue for $-\Delta u=\lambda\rho u$ by nonconforming elements.Math. Comput. 77 (2008), 2061-2084. MR 2429874, 10.1090/S0025-5718-08-02098-X |
| Reference:
|
[15] Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement.China Sci. Press Beijing (2006). |
| Reference:
|
[16] Lin, Q., Lü, T.: Asymptotic expansions for finite element eigenvalues and finite element solution.Bonn. Math. Schrift 158 (1984), 1-10. MR 0793412 |
| Reference:
|
[17] Lin, Q., Yan, N.: The Construction and Analysis of High Efficiency Finite Element Methods.Hebei University Press Hebei (1996), Chinese. |
| Reference:
|
[18] Lin, Q., Zhang, S., Yan, N.: Extrapolation and defect correction for diffusion equations with boundary integral conditions.Acta Math. Sci. 17 (1997), 405-412. Zbl 0907.65096, MR 1613231, 10.1016/S0252-9602(17)30859-7 |
| Reference:
|
[19] Lin, Q., Zhu, Q.: Preprocessing and Postprocessing for the Finite Element Method.Shanghai Sci. Tech. Publishers Shanghai (1994), Chinese. |
| Reference:
|
[20] Mercier, B., Osborn, J., Rappaz, J., Raviat, P.-A.: Eigenvalue approximation by mixed and hybrid method.Math. Comput. 36 (1981), 427-453. MR 0606505, 10.1090/S0025-5718-1981-0606505-9 |
| Reference:
|
[21] Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element.Numer. Methods Partial Differ. Equations 8 (1992), 97-111. Zbl 0742.76051, MR 1148797, 10.1002/num.1690080202 |
| Reference:
|
[22] Shaidurov, V.: Multigrid Methods for Finite Elements.Kluwer Academic Publishers Dordrecht (1995). Zbl 0837.65118, MR 1335921 |
| Reference:
|
[23] Wang, J., Ye, X.: Superconvergence of finite element approximations for the Stokes problem by projection methods.SIAM J. Numer. Anal. 39 (2001), 1001-1013. Zbl 1002.65118, MR 1860454, 10.1137/S003614290037589X |
| Reference:
|
[24] Yang, Y.: An Analysis of the Finite Element Method for Eigenvalue Problems.Guizhou People Public Press Guizhou (2004), Chinese. |
| Reference:
|
[25] Ye, X.: Superconvergence of nonconforming finite element method for the Stokes equations.Numer. Methods Partial Differ. Equations 18 (2002), 143-154. Zbl 1003.65121, MR 1902289, 10.1002/num.1036 |
| Reference:
|
[26] Zhou, A., Li, J.: The full approximation accuracy for the stream function-vorticity-pressure method.Numer. Math. 68 (1994), 427-435. Zbl 0823.65110, MR 1313153, 10.1007/s002110050070 |
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