Abstract
This paper studies the mixed element method for the boundary value problem of the biharmonic equation
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 91430215
Award Identifier / Grant number: 11101415
Award Identifier / Grant number: 11471026
Funding statement: The author Z. Li is thankful to the support from LSEC, AMSS during his visit and has been supported partially by NSFC grant no. 91430215. The author S. Zhang is partially supported by NSFC grant nos. 11101415 and 11471026.
Acknowledgements
The authors would like to thank the anonymous referees for their comments which helped us to improve the work.
References
[1]
M. Amara and F. Dabaghi,
An optimal
[2] D. N. Arnold and R. S. Falk, A uniformly accurate finite element method for the Reissner–Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. 10.1137/0726074Search in Google Scholar
[3] I. Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1973), no. 3, 179–192. 10.1007/BF01436561Search in Google Scholar
[4] E. M. Behrens and J. Guzmán, A mixed method for the biharmonic problem based on a system of first-order equations, SIAM J. Numer. Anal. 49 (2011), no. 2, 789–817. 10.1137/090775245Search in Google Scholar
[5] C. Bernardi, V. Girault and Y. Maday, Mixed spectral element approximation of the Navier–Stokes equations in the stream-function and vorticity formulation, IMA J. Numer. Anal. 12 (1992), no. 4, 565–608. 10.1093/imanum/12.4.565Search in Google Scholar
[6] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Berlin, 2013. 10.1007/978-3-642-36519-5Search in Google Scholar
[7] J. H. Bramble and R. S. Falk, A mixed-Lagrange multiplier finite element method for the polyharmonic equation, RAIRO Modélisation Math. Anal. Numér. 19 (1985), no. 4, 519–557. 10.1051/m2an/1985190405191Search in Google Scholar
[8] J. H. Bramble and J. Xu, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal. 26 (1989), no. 6, 1267–1275. 10.1137/0726073Search in Google Scholar
[9]
S. C. Brenner and L.-Y. Sung,
[10] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Franc. Automat. Inform. Rech. Operat. R 8 (1974), no. 2, 129–151. 10.1051/m2an/197408R201291Search in Google Scholar
[11] F. Brezzi and M. Fortin, Numerical approximation of Mindlin–Reissner plates, Math. Comp. 47 (1986), no. 175, 151–158. 10.1090/S0025-5718-1986-0842127-7Search in Google Scholar
[12] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Ser. Comput. Math. 15, Springer, New York, 2012. Search in Google Scholar
[13] X.-L. Cheng, W. Han and H.-C. Huang, Some mixed finite element methods for biharmonic equation, J. Comput. Appl. Math. 126 (2000), no. 1, 91–109. 10.1016/S0377-0427(99)00342-8Search in Google Scholar
[14] P. G. Ciarlet and P.-A. Raviart, A mixed finite element method for the biharmonic equation, Mathematical Aspects of Finite Elements in Partial Differential Equations (Madison 1974), Academic Press, New York (1974), 125–145. 10.1016/B978-0-12-208350-1.50009-1Search in Google Scholar
[15] M. Comodi, The Hellan–Herrmann–Johnson method: Some new error estimates and postprocessing, Math. Comp. 52 (1989), no. 185, 17–29. 10.1090/S0025-5718-1989-0946601-7Search in Google Scholar
[16] R. S. Falk, Approximation of the biharmonic equation by a mixed finite element method, SIAM J. Numer. Anal. 15 (1978), no. 3, 556–567. 10.1137/0715036Search in Google Scholar
[17] R. S. Falk, Finite elements for the Reissner–Mindlin plate, Mixed Finite Elements, Compatibility Conditions, and Applications (Cetraro 2006), Lecture Notes in Math. 1939, Springer, Berlin (2008), 195–232. 10.1007/978-3-540-78319-0_5Search in Google Scholar
[18] C. Feng and S. Zhang, Optimal solver for Morley element discretization of biharmonic equation on shape-regular grids, J. Comput. Math. 34 (2016), no. 2, 159–173. 10.4208/jcm.1510-m2014-0085Search in Google Scholar
[19] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Spring, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar
[20] K. Hellan, Analysis of elastic plates in flexure by a simplified finite element method, Acta Polytechn. Scand. Civil Eng. Building Construct. Ser. (46):1–29, 1967. Search in Google Scholar
[21] L. R. Herrmann, Finite-element bending analysis for plates, Amer. Soc. Civil Eng. Eng. Mech. Divis. J. 93 (1967), 13–26. 10.1061/JMCEA3.0000891Search in Google Scholar
[22]
J. Hu and Z.-C. Shi,
The best
[23] C. Johnson, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math. 21 (1973), no. 1, 43–62. 10.1007/BF01436186Search in Google Scholar
[24] W. Krendl and W. Zulehner, A decomposition result for biharmonic problems and the Hellan–Herrmann–Johnson method, Electron. Trans. Numer. Anal. 45 (2016), 257–282. Search in Google Scholar
[25] T. Miyoshi, A finite element method for the solutions of fourth order partial differential equations, Kumamoto J. Sci. Math. 9 (1973), no. 2, 87–116. Search in Google Scholar
[26] P. Monk, A mixed finite element method for the biharmonic equation, SIAM J. Numer. Anal. 24 (1987), no. 4, 737–749. 10.1137/0724048Search in Google Scholar
[27] A. Oukit and R. Pierre, Mixed finite element for the linear plate problem: The Hermann–Miyoshi model revisited, Numer. Math. 74 (1996), no. 4, 453–477. 10.1007/s002110050225Search in Google Scholar
[28] J. Pitkäranta, Analysis of some low-order finite element schemes for Mindlin–Reissner and Kirchhoff plates, Numer. Math. 53 (1988), no. 1–2, 237–254. 10.1007/BF01395887Search in Google Scholar
[29] R. Rannacher, On nonconforming an mixed finite element methods for plate bending problems. The linear case, RAIRO Anal. Numer. 13 (1979), no. 4, 369–387. 10.1051/m2an/1979130403691Search in Google Scholar
[30] Z.-C. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013. Search in Google Scholar
[31] S. Zhang, Stable mixed element schemes for plate models on multiply-connected domains, preprint (2017), https://arxiv.org/abs/1702.06401. Search in Google Scholar
[32] S. Zhang, Y. Xi and X. Ji, A multi-level mixed element method for the eigenvalue problem of biharmonic equation, preprint (2016), https://arxiv.org/abs/1606.05419. 10.1007/s10915-017-0592-7Search in Google Scholar
[33] S. Zhang and J. Xu, Optimal solvers for fourth-order PDEs discretized on unstructured grids, SIAM J. Numer. Anal. 52 (2014), no. 1, 282–307. 10.1137/120878148Search in Google Scholar
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