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Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor

    https://doi.org/10.1142/S0218202517500567Cited by:17 (Source: Crossref)

    We propose two classes of mixed finite elements for linear elasticity of any order, with interior penalty for nonconforming symmetric stress approximation. One key point of our method is to introduce some appropriate nonconforming face-bubble spaces based on the local decomposition of discrete symmetric tensors, with which the stability can be easily established. We prove the optimal L2-error estimate for displacement and optimal Hh(div) error estimate for stress by adding an interior penalty term. The elements are easy to be implemented thanks to the explicit formulations of its basis functions. Moreover, the method can be applied to arbitrary simplicial grids for any spatial dimension in a unified fashion. Numerical tests for both 2D and 3D are provided to validate our theoretical results.

    Communicated by F. Brezzi

    AMSC: 65N30, 65N15, 74B05

    References

    • 1. S. Adams and B. Cockburn, A mixed finite element method for elasticity in three dimensions, J. Sci. Comput. 25 (2005) 515–521. Crossref, Web of ScienceGoogle Scholar
    • 2. M. Amara and J.-M. Thomas, Equilibrium finite elements for the linear elastic problem, Numer. Math. 33 (1979) 367–383. Crossref, Web of ScienceGoogle Scholar
    • 3. D. Arnold and G. Awanou, Rectangular mixed finite elements for elasticity, Math. Models Methods Appl. Sci. 15 (2005) 1417–1429. Link, Web of ScienceGoogle Scholar
    • 4. D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251. Crossref, Web of ScienceGoogle Scholar
    • 5. D. Arnold, G. Awanou and R. Winther, Nonconforming tetrahedral mixed finite elements for elasticity, Math. Models Methods Appl. Sci. 24 (2014) 783–796. Link, Web of ScienceGoogle Scholar
    • 6. D. Arnold, J. Douglas, Jr. and C. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984) 1–22. Crossref, Web of ScienceGoogle Scholar
    • 7. D. Arnold, R. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comput. 76 (2007) 1699–1723. Crossref, Web of ScienceGoogle Scholar
    • 8. D. Arnold and R. Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002) 401–419. Crossref, Web of ScienceGoogle Scholar
    • 9. D. Arnold and R. Winther, Nonconforming mixed elements for elasticity, Math. Models Methods Appl. Sci. 13 (2003) 295–307. Link, Web of ScienceGoogle Scholar
    • 10. G. Awanou, A rotated nonconforming rectangular mixed element for elasticity, Calcolo 46 (2009) 49–60. Crossref, Web of ScienceGoogle Scholar
    • 11. I. Babuška, Error-bounds for finite element method, Numer. Math. 16 (1971) 322–333. Crossref, Web of ScienceGoogle Scholar
    • 12. D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity, Commun. Pure Appl. Anal. 8 (2009) 95–121. Crossref, Web of ScienceGoogle Scholar
    • 13. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Anal. Num. 8 (1974) 129–151. Google Scholar
    • 14. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol. 15 (Springer, 1991). CrossrefGoogle Scholar
    • 15. Z. Cai and X. Ye, A mixed nonconforming finite element for linear elasticity, Numer. Methods Partial Differential Equations 21 (2005) 1043–1051. Crossref, Web of ScienceGoogle Scholar
    • 16. L. Chen, iFEM: An integrated finite element methods package in MATLAB, Technical report, University of California Irvine (2008). Google Scholar
    • 17. S.-C. Chen and Y.-N. Wang, Conforming rectangular mixed finite elements for elasticity, J. Sci. Comput. 47 (2011) 93–108. Crossref, Web of ScienceGoogle Scholar
    • 18. B. Cockburn, J. Gopalakrishnan and J. Guzmán, A new elasticity element made for enforcing weak stress symmetry, Math. Comput. 79 (2010) 1331–1349. Crossref, Web of ScienceGoogle Scholar
    • 19. B. Cockburn and K. Shi, Superconvergent HDG methods for linear elasticity with weakly symmetric stresses, IMA J. Numer. Anal. 33 (2012) 747–770. Crossref, Web of ScienceGoogle Scholar
    • 20. M. Farhloul and M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: A unified approach, Numer. Math. 76 (1997) 419–440. Crossref, Web of ScienceGoogle Scholar
    • 21. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (Elsevier, 2000). Google Scholar
    • 22. S. Gong, S. Wu and J. Xu, New hybridized mixed methods for linear elasticity and optimal multilevel solvers, arXiv:1704.07540. Google Scholar
    • 23. J. Gopalakrishnan and J. Guzmán, Symmetric nonconforming mixed finite elements for linear elasticity, SIAM J. Numer. Anal. 49 (2011) 1504–1520. Crossref, Web of ScienceGoogle Scholar
    • 24. J. Gopalakrishnan and J. Guzmán, A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012) 352–372. Crossref, Web of ScienceGoogle Scholar
    • 25. J. Hu, Finite element approximations of symmetric tensors on simplicial grids in n: The higher order case, J. Comput. Math. 33 (2015) 1–14. Crossref, Web of ScienceGoogle Scholar
    • 26. J. Hu, A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation, SIAM J. Numer. Anal. 53 (2015) 1438–1463. Crossref, Web of ScienceGoogle Scholar
    • 27. J. Hu, H. Man and S. Zhang, A simple conforming mixed finite element for linear elasticity on rectangular grids in any space dimension, J. Sci. Comput. 58 (2014) 367–379. Crossref, Web of ScienceGoogle Scholar
    • 28. J. Hu and Z.-C. Shi, Lower order rectangular nonconforming mixed finite elements for plane elasticity, SIAM J. Numer. Anal. 46 (2007) 88–102. Crossref, Web of ScienceGoogle Scholar
    • 29. J. Hu and S. Zhang, A family of conforming mixed finite elements for linear elasticity on triangular grids, arXiv:1406.7457. Google Scholar
    • 30. J. Hu and S. Zhang, A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids, Sci. China Math. 58 (2015) 297–307. Crossref, Web of ScienceGoogle Scholar
    • 31. J. Hu and S. Zhang, Finite element approximations of symmetric tensors on simplicial grids in n: The lower order case, Math. Models Methods Appl. Sci. 26 (2016) 1649–1669. Link, Web of ScienceGoogle Scholar
    • 32. C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978) 103–116. Crossref, Web of ScienceGoogle Scholar
    • 33. Y.-J. Lee, J. Wu, J. Xu and L. Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634–641. Crossref, Web of ScienceGoogle Scholar
    • 34. H.-Y. Man, J. Hu and Z.-C. Shi, Lower order rectangular nonconforming mixed finite element for the three-dimensional elasticity problem, Math. Models Methods Appl. Sci. 19 (2009) 51–65. Link, Web of ScienceGoogle Scholar
    • 35. W. Qiu and L. Demkowicz, Mixed hp-finite element method for linear elasticity with weakly imposed symmetry, Comput. Methods Appl. Mech. Eng. 198 (2009) 3682–3701. Crossref, Web of ScienceGoogle Scholar
    • 36. W. Qiu, J. Shen and K. Shi, An HDG method for linear elasticity with strong symmetric stresses, Math. Comput. (2017). Crossref, Web of ScienceGoogle Scholar
    • 37. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput. 54 (1990) 483–493. Crossref, Web of ScienceGoogle Scholar
    • 38. S.-Y. Yi, Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions, Calcolo 42 (2005) 115–133. Crossref, Web of ScienceGoogle Scholar
    • 39. S.-Y. Yi, A new nonconforming mixed finite element method for linear elasticity, Math. Models Methods Appl. Sci. 16 (2006) 979–999. Link, Web of ScienceGoogle Scholar
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