| Title:
|
Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem (English) |
| Author:
|
Feng, Xinlong |
| Author:
|
Weng, Zhifeng |
| Author:
|
Xie, Hehu |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
59 |
| Issue:
|
6 |
| Year:
|
2014 |
| Pages:
|
615-630 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper provides an accelerated two-grid stabilized mixed finite element scheme for the Stokes eigenvalue problem based on the pressure projection. With the scheme, the solution of the Stokes eigenvalue problem on a fine grid is reduced to the solution of the Stokes eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. By solving a slightly different linear problem on the fine grid, the new algorithm significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate. Finally, numerical experiments are shown to verify the high efficiency and the theoretical results of the new method. (English) |
| Keyword:
|
accelerated two grid method |
| Keyword:
|
Stokes eigenvalue problem |
| Keyword:
|
stabilized method |
| Keyword:
|
equal-order pair |
| Keyword:
|
error estimate |
| MSC:
|
65N12 |
| MSC:
|
65N25 |
| MSC:
|
65N30 |
| MSC:
|
76D07 |
| idZBL:
|
Zbl 06391453 |
| idMR:
|
MR3277730 |
| DOI:
|
10.1007/s10492-014-0076-0 |
| . |
| Date available:
|
2014-11-10T09:10:00Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143991 |
| . |
| Reference:
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[1] Babuška, I., Osborn, J. E.: 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:
|
[2] Babuška, I., Osborn, J.: Eigenvalue problems.P. G. Ciarlet, et al. Handbook of Numerical Analysis. Volume II: Finite element methods (Part 1) North-Holland Amsterdam (1991), 641-787. MR 1115240 |
| Reference:
|
[3] Becker, R., Hansbo, P.: A simple pressure stabilization method for the Stokes equation.Commun. Numer. Methods Eng. 24 (2008), 1421-1430. Zbl 1153.76036, MR 2474694, 10.1002/cnm.1041 |
| Reference:
|
[4] Bochev, P. B., Dohrmann, C. R., Gunzburger, M. D.: Stabilization of low-order mixed finite elements for the Stokes equations.SIAM J. Numer. Anal. 44 (2006), 82-101 (electronic). Zbl 1145.76015, MR 2217373, 10.1137/S0036142905444482 |
| Reference:
|
[5] Boffi, D.: Finite element approximation of eigenvalue problems.Acta Numerica 19 (2010), 1-120. Zbl 1242.65110, MR 2652780, 10.1017/S0962492910000012 |
| Reference:
|
[6] Chen, H., He, Y., Li, Y., Xie, H.: A multigrid method based on shifted-inverse power technique for eigenvalue problem.http://arxiv.org/pdf/1401.5378v3 (2014). MR 3386235 |
| Reference:
|
[7] Chen, H., Jia, S., Xie, H.: Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems.Appl. Math., Praha 54 (2009), 237-250. Zbl 1212.65431, MR 2530541, 10.1007/s10492-009-0015-7 |
| Reference:
|
[8] Chen, H., Jia, S., Xie, H.: Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem.Appl. Numer. Math. 61 (2011), 615-629. Zbl 1209.65126, MR 2754580, 10.1016/j.apnum.2010.12.007 |
| Reference:
|
[9] Chien, C. S., Jeng, B. W.: A two-grid discretization scheme for semilinear elliptic eigenvalue problems.SIAM J. Sci. Comput. 27 (2006), 1287-1304. Zbl 1095.65100, MR 2199749, 10.1137/030602447 |
| Reference:
|
[10] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems.Studies in Mathematics and Its Applications. Vol. 4 North-Holland Publishing Company, Amsterdam (1978). Zbl 0383.65058, MR 0520174 |
| Reference:
|
[11] Feng, X., Kim, I., Nam, H., Sheen, D.: Locally stabilized $P_1$-nonconforming quadrilateral and hexahedral finite element methods for the Stokes equations.J. Comput. Appl. Math. 236 (2011), 714-727. Zbl 1233.65088, MR 2853496, 10.1016/j.cam.2011.06.009 |
| Reference:
|
[12] Golub, G. H., Loan, C. F. Van: Matrix Computations. (3rd ed.).The Johns Hopkins Univ. Press Baltimore (1996). MR 1417720 |
| Reference:
|
[13] Hackbusch, W.: Multi-Grid Methods and Applications.Springer Series in Computational Mathematics 4 Springer, Berlin (1985). Zbl 0595.65106 |
| Reference:
|
[14] Hu, X., Cheng, X.: Acceleration of a two-grid method for eigenvalue problems.Math. Comput. 80 (2011), 1287-1301. Zbl 1232.65141, MR 2785459, 10.1090/S0025-5718-2011-02458-0 |
| Reference:
|
[15] Huang, P., He, Y., Feng, X.: Numerical investigations on several stabilized finite element methods for the Stokes eigenvalue problem.Math. Probl. Eng. 2011 (2011), Article ID 745908, 14 pages. Zbl 1235.74286, MR 2826898 |
| Reference:
|
[16] Huang, P., He, Y., Feng, X.: Two-level stabilized finite element method for the Stokes eigenvalue problem.Appl. Math. Mech., Engl. Ed. 33 (2012), 621-630. MR 2978223, 10.1007/s10483-012-1575-7 |
| Reference:
|
[17] Kolman, K.: A two-level method for nonsymmetric eigenvalue problems.Acta Math. Appl. Sin., Engl. Ser. 21 (2005), 1-12. Zbl 1084.65109, MR 2123599, 10.1007/s10255-005-0209-z |
| Reference:
|
[18] Li, J., He, Y.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations.J. Comput. Appl. Math. 214 (2008), 58-65. Zbl 1132.35436, MR 2391672, 10.1016/j.cam.2007.02.015 |
| Reference:
|
[19] Li, H., Yang, Y.: The adaptive finite element method based on multi-scale discretizations for eigenvalue problems.Comput. Math. Appl. 65 (2013), 1086-1102. Zbl 1266.65196, MR 3028637, 10.1016/j.camwa.2013.01.043 |
| Reference:
|
[20] Lovadina, C., Lyly, M., Stenberg, R.: A posteriori estimates for the Stokes eigenvalue problem.Numer. Methods Partial Differ. Equations 25 (2009), 244-257. Zbl 1169.65109, MR 2473688, 10.1002/num.20342 |
| Reference:
|
[21] Mercier, B., Osborn, J., Rappaz, J., Raviart, P.-A.: Eigenvalue approximation by mixed and hybrid methods.Math. Comput. 36 (1981), 427-453. Zbl 0472.65080, MR 0606505, 10.1090/S0025-5718-1981-0606505-9 |
| Reference:
|
[22] Peters, G., Wilkinson, J. H.: Inverse iteration, ill-conditioned equations and Newton's method.SIAM Rev. 21 (1979), 339-360. MR 0535118, 10.1137/1021052 |
| Reference:
|
[23] Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems. (2nd ed.).Springer Series in Computational Mathematics 24 Springer, Berlin (2008). Zbl 1155.65087, MR 2454024 |
| Reference:
|
[24] Weng, Z., Feng, X., Zhai, S.: Investigations on two kinds of two-grid mixed finite element methods for the elliptic eigenvalue problem.Comput. Math. Appl. 64 (2012), 2635-2646. Zbl 1268.65157, MR 2970840, 10.1016/j.camwa.2012.07.009 |
| Reference:
|
[25] Xu, J.: A novel two-grid method for semilinear elliptic equations.SIAM J. Sci. Comput. 15 (1994), 231-237. Zbl 0795.65077, MR 1257166, 10.1137/0915016 |
| Reference:
|
[26] Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs.SIAM J. Numer. Anal. 33 (1996), 1759-1777. Zbl 0860.65119, MR 1411848, 10.1137/S0036142992232949 |
| Reference:
|
[27] Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems.Math. Comput. 70 (2001), 17-25. Zbl 0959.65119, MR 1677419, 10.1090/S0025-5718-99-01180-1 |
| Reference:
|
[28] Yang, Y., Bi, H.: Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems.SIAM J. Numer. Anal. 49 (2011), 1602-1624. Zbl 1236.65143, MR 2831063, 10.1137/100810241 |
| Reference:
|
[29] Yang, Y., Fan, X.: Generalized Rayleigh quotient and finite element two-grid discretization schemes.Sci. China, Ser. A 52 (2009), 1955-1972. Zbl 1188.65151, MR 2545001, 10.1007/s11425-009-0016-8 |
| Reference:
|
[30] Yin, X., Xie, H., Jia, S., Gao, S.: Asymptotic expansions and extrapolations of eigenvalues for the Stokes problem by mixed finite element methods.J. Comput. Appl. Math. 215 (2008), 127-141. Zbl 1149.65090, MR 2400623, 10.1016/j.cam.2007.03.028 |
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