Abstract
In this paper, stable mixed formulations are designed and analyzed for the quad div problems under two frameworks presented in [23] and [22], respectively. Analogue discretizations are given with respect to the mixed formulation, and optimal convergence rates are observed, which confirm the theoretical analysis.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 91430215
Award Identifier / Grant number: 91530323
Award Identifier / Grant number: 11471026
Funding statement: The author Ronghong Fan is supported by the National Key Research and Development Program of China (Grant No. 2016YFB0201304) and the Major Research Plan of National Natural Science Foundation of China (Grant Nos. 91430215, 91530323). The authors Yanru Liu and Shuo Zhang are supported partially by the National Natural Science Foundation of China with Grant No. 11471026. The author Shuo Zhang is supported by National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.
Acknowledgements
The authors Yanru Liu and Shuo Zhang are grateful to Professor Ming Wang of Peking University with deep memory for his supervision.
References
[1] S. B. Altan and E. C. Aifantis, On the structure of the mode III crack-tip in gradient elasticity, Scripta Metall. Mater. 26 (1992), no. 2, 319–324. 10.1016/0956-716X(92)90194-JSearch in Google Scholar
[2] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. 10.1007/978-3-642-36519-5Search in Google Scholar
[3] S. C. Brenner, J. Sun and L.-Y. Sung, Hodge decomposition methods for a quad-curl problem on planar domains, J. Sci. Comput. 73 (2017), no. 2–3, 495–513. 10.1007/s10915-017-0449-0Search in Google Scholar
[4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland Publishing, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar
[5] P. G. Ciarlet and P. Raviart, A mixed finite element method for the biharmonic equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York (1974), 125–145. 10.1016/B978-0-12-208350-1.50009-1Search in Google Scholar
[6] G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 34, 3669–3750. 10.1016/S0045-7825(02)00286-4Search in Google Scholar
[7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 1977. 10.1007/978-3-642-96379-7Search in Google Scholar
[8] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar
[9] Z. Li and S. Zhang, A stable mixed element method for the biharmonic equation with first-order function spaces, Comput. Methods Appl. Math. 17 (2017), no. 4, 601–616. 10.1515/cmam-2017-0002Search in Google Scholar
[10] R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal. 16 (1964), 51–78. 10.1007/BF00248490Search in Google Scholar
[11] R. D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Internat. J. Solids Struct. 1 (1965), no. 4, 417–438. 10.1016/0020-7683(65)90006-5Search in Google Scholar
[12] M. Neilan, Discrete and conforming smooth de Rham complexes in three dimensions, Math. Comp. 84 (2015), no. 295, 2059–2081. 10.1090/S0025-5718-2015-02958-5Search in Google Scholar
[13] X.-C. Tai and R. Winther, A discrete de Rham complex with enhanced smoothness, Calcolo 43 (2006), no. 4, 287–306. 10.1007/s10092-006-0124-6Search in Google Scholar
[14] M. Wang, Z.-C. Shi and J. Xu, A new class of Zienkiewicz-type non-conforming element in any dimensions, Numer. Math. 106 (2007), no. 2, 335–347. 10.1007/s00211-007-0063-4Search in Google Scholar
[15] M. Wang, Z.-C. Shi and J. Xu, Some n-rectangle nonconforming elements for fourth order elliptic equations, J. Comput. Math. 25 (2007), no. 4, 408–420. Search in Google Scholar
[16] M. Wang and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numer. Math. 103 (2006), no. 1, 155–169. 10.1007/s00211-005-0662-xSearch in Google Scholar
[17] M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations, Math. Comp. 76 (2007), no. 257, 1–18. 10.1090/S0025-5718-06-01889-8Search in Google Scholar
[18]
M. Wang and J. Xu,
Minimal finite element spaces for
[19] J. Xu, private communication, 2016. Search in Google Scholar
[20] A. Ženíšek, Polynomial approximation on tetrahedrons in the finite element method, J. Approx. Theory 7 (1973), 334–351. 10.1016/0021-9045(73)90036-1Search in Google Scholar
[21] S. Zhang, A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math. 59 (2009), no. 1, 219–233. 10.1016/j.apnum.2008.02.002Search in Google Scholar
[22] S. Zhang, Decoupled mixed element schemes for fourth order problems, preprint (2016), https://arxiv.org/abs/1611.10029. Search in Google Scholar
[23] S. Zhang, Regular decomposition and a framework of order reduced methods for fourth order problems, Numer. Math. 138 (2018), no. 1, 241–271. 10.1007/s00211-017-0902-xSearch in Google Scholar
[24] S. Zhang and Z. Zhang, Invalidity of decoupling a biharmonic equation to two Poisson equations on non-convex polygons, Int. J. Numer. Anal. Model. 5 (2008), no. 1, 73–76. Search in Google Scholar
[25]
B. Zheng, Q. Hu and J. Xu,
A nonconforming finite element method for fourth order curl equations in
© 2019 Walter de Gruyter GmbH, Berlin/Boston
