| Title:
|
Superconvergence of a stabilized approximation for the Stokes eigenvalue problem by projection method (English) |
| Author:
|
Huang, Pengzhan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
361-370 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper presents a superconvergence result based on projection method for stabilized finite element approximation of the Stokes eigenvalue problem. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The paper complements the work of Li et al. (2012), which establishes the superconvergence result of the Stokes equations by the stabilized finite element method. Moreover, numerical tests confirm the theoretical analysis. (English) |
| Keyword:
|
Stokes eigenvalue problem |
| Keyword:
|
stabilized method |
| Keyword:
|
lowest equal-order pair |
| Keyword:
|
projection method |
| Keyword:
|
superconvergence |
| MSC:
|
65B99 |
| MSC:
|
65N25 |
| MSC:
|
65N30 |
| MSC:
|
76D07 |
| idZBL:
|
Zbl 06362233 |
| idMR:
|
MR3233549 |
| DOI:
|
10.1007/s10492-014-0061-7 |
| . |
| Date available:
|
2014-07-14T08:53:52Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143868 |
| . |
| Reference:
|
[1] Bochev, P., 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:
|
[2] 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:
|
[3] Chen, W., Lin, Q.: Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method.Appl. Math., Praha 51 (2006), 73-88. Zbl 1164.65489, MR 2197324, 10.1007/s10492-006-0006-x |
| Reference:
|
[4] Chen, H., Wang, J.: An interior estimate of superconvergence for finite element solutions for second-order elliptic problems on quasi-uniform meshes by local projections.SIAM J. Numer. Anal. 41 (2003), 1318-1338 (electronic). Zbl 1058.65118, MR 2034883, 10.1137/S0036142902410039 |
| Reference:
|
[5] Chou, S. H., Ye, X.: Superconvergence of finite volume methods for the second-order elliptic problem.Comput. Methods Appl. Mech. Eng. 196 (2007), 3706-3712. Zbl 1173.65354, MR 2339996, 10.1016/j.cma.2006.10.025 |
| Reference:
|
[6] Cui, M., Ye, X.: Superconvergence of finite volume methods for the Stokes equations.Numer. Methods Partial Differ. Equations 25 (2009), 1212-1230. Zbl 1170.76037, MR 2541808, 10.1002/num.20399 |
| Reference:
|
[7] Hecht, F., Pironneau, O., Hyaric, A. Le, Ohtsuka, K.: FREEFEM++, version 2.3-3, 2008. Software avaible at http://www.freefem.org.. |
| Reference:
|
[8] Heimsund, B. O., Tai, X. C., Wang, J. P.: Superconvergence for the gradient of finite element approximations by $L^2$ projections.SIAM J. Numer. Anal. 40 (2002), 1263-1280. Zbl 1047.65095, MR 1951894, 10.1137/S003614290037410X |
| Reference:
|
[9] Huang, P. Z., He, Y. N., Feng, X. L.: Numerical investigations on several stabilized finite element methods for the Stokes eigenvalue problem.Math. Probl. Eng. 2011 (2011), Article ID 745908, pp. 14. Zbl 1235.74286, MR 2826898 |
| Reference:
|
[10] Huang, P. Z., He, Y. N., Feng, X. L.: Two-level stabilized finite element method for Stokes eigenvalue problem.Appl. Math. Mech., Engl. Ed. 33 (2012), 621-630. Zbl 1266.65192, MR 2978223, 10.1007/s10483-012-1575-7 |
| Reference:
|
[11] Huang, P. Z., Zhang, T., Ma, X. L.: Superconvergence by $L^2$-projection for a stabilized finite volume method for the stationary Navier-Stokes equations.Comput. Math. Appl. 62 (2011), 4249-4257. Zbl 1236.76017, MR 2859980, 10.1016/j.camwa.2011.10.012 |
| Reference:
|
[12] Jia, S., Xie, H., Yin, X., Gao, S.: Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods.Appl. Math., Praha 54 (2009), 1-15. Zbl 1212.65434, MR 2476018, 10.1007/s10492-009-0001-0 |
| Reference:
|
[13] Li, J.: Penalty finite element approximations for the Stokes equations by $L^2$ projection.Math. Methods Appl. Sci. 32 (2009), 470-479. MR 2493591, 10.1002/mma.1051 |
| Reference:
|
[14] Li, J., He, Y. N.: Superconvergence of discontinuous Galerkin finite element method for the stationary Navier-Stokes equations.Numer. Methods Partial Differ. Equations 23 (2007), 421-436. Zbl 1107.76046, MR 2289460, 10.1002/num.20188 |
| Reference:
|
[15] Li, J., He, Y. N.: 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:
|
[16] Li, J., He, Y. N., Wu, J. H.: A local superconvergence analysis of the finite element method for the Stokes equations by local projections.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 6499-6511. Zbl 1227.65115, MR 2834057, 10.1016/j.na.2011.06.033 |
| Reference:
|
[17] Li, J., Mei, L. Q., Chen, Z. X.: Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh $L^2$ projection.Numer. Methods Partial Differ. Equations 28 (2012), 115-126. Zbl 1234.65038, MR 2864661, 10.1002/num.20610 |
| Reference:
|
[18] Li, J., Wang, J., Ye, X.: Superconvergence by $L^2$-projections for stabilized finite element methods for the Stokes equations.Int. J. Numer. Anal. Model. 6 (2009), 711-723. MR 2574761 |
| Reference:
|
[19] Liu, H. P., Yan, N. N.: Enhancing finite element approximation for eigenvalue problems by projection method.Comput. Methods Appl. Mech. Eng. 233/236 (2012), 81-91. Zbl 1253.74107, MR 2924022, 10.1016/j.cma.2012.04.009 |
| 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] Wang, J.: Superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems.J. Math. Study 33 (2000), 229-243. Zbl 0987.65108, MR 1868268 |
| Reference:
|
[22] Wang, J., Ye, X.: Superconvergence of finite element approximations for the Stokes problem by projection methods.SIAM J. Numer. Anal. 39 (2001), 1001-1013 (electronic). Zbl 1002.65118, MR 1860454, 10.1137/S003614290037589X |
| Reference:
|
[23] 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:
|
[24] 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 |
| . |
