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New low-order mixed finite element methods for linear elasticity (2310.13920v3)

Published 21 Oct 2023 in math.NA and cs.NA

Abstract: New low-order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order normal-normal face bubble space. The reduced counterpart has only $d(d+1)2$ degrees of freedom. Basis functions are explicitly given in terms of barycentric coordinates. Low-order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient $\lambda$, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.

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References (56)
  1. S. Adams and B. Cockburn. A mixed finite element method for elasticity in three dimensions. J. Sci. Comput., 25(3):515–521, 2005.
  2. M. Amara and J.-M. Thomas. Equilibrium finite elements for the linear elastic problem. Numer. Math., 33(4):367–383, 1979.
  3. A family of higher order mixed finite element methods for plane elasticity. Numer. Math., 45:1–22, 1984.
  4. D. Arnold and R. Winther. Mixed finite elements for elasticity. Numer. Math., (92):401–419, 2002.
  5. D. N. Arnold and G. Awanou. Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 15(9):1417–1429, 2005.
  6. Finite elements for symmetric tensors in three dimensions. Math. Comp., 77(263):1229–1251, 2008.
  7. Nonconforming tetrahedral mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 24(4):783–796, 2014.
  8. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp., 76(260):1699–1723, 2007.
  9. D. N. Arnold and K. Hu. Complexes from complexes. Found. Comput. Math., 21(6):1739–1774, 2021.
  10. D. N. Arnold and R. Winther. Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci., 13(3):295–307, 2003.
  11. G. Awanou. A rotated nonconforming rectangular mixed element for elasticity. Calcolo, 46:49–60, 2009.
  12. G. Awanou. Two remarks on rectangular mixed finite elements for elasticity. J. Sci. Comput., 50(1):91–102, 2012.
  13. K. Bell. A refined triangular plate bending finite element. Internat. J. Numer. Methods Engrg., 1(1):101–122, 1969.
  14. Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal., 8(1):95–121, 2009.
  15. Mixed finite element methods and applications. Springer, Heidelberg, 2013.
  16. S. C. Brenner. Korn’s inequalities for piecewise H1superscript𝐻1H^{1}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT vector fields. Math. Comp., 73(247):1067–1087, 2004.
  17. Linear finite element methods for planar linear elasticity. Math. Comp., 59(200):321–338, 1992.
  18. Z. Cai and X. Ye. A mixed nonconforming finite element for linear elasticity. Numer. Methods Partial Differential Equations, 21(6):1043–1051, 2005.
  19. Stabilized mixed finite element methods for linear elasticity on simplicial grids in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Comput. Methods Appl. Math., 17(1):17–31, 2017.
  20. Fast auxiliary space preconditioners for linear elasticity in mixed form. Math. Comp., 87(312):1601–1633, 2018.
  21. L. Chen and X. Huang. Finite elements for divdiv-conforming symmetric tensors. arXiv preprint arXiv:2005.01271, 2020.
  22. L. Chen and X. Huang. Finite elements for div- and divdiv-conforming symmetric tensors in arbitrary dimension. SIAM J. Numer. Anal., 60(4):1932–1961, 2022.
  23. L. Chen and X. Huang. Geometric decompositions of H⁢(div)𝐻divH({\rm div})italic_H ( roman_div )-conforming finite element tensors, part I: Vector and matrix functions. arXiv preprint arXiv:2112.14351, 2022.
  24. L. Chen and X. Huang. A new div-div-conforming symmetric tensor finite element space with applications to the biharmonic equation. Math. Comp., arXiv preprint arXiv:2305.11356, 2023.
  25. S. Chen and Y. Wang. Conforming rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 47:93–108, 2011.
  26. A new elasticity element made for enforcing weak stress symmetry. Math. Comp., 79(271):1331–1349, 2010.
  27. Global estimates for mixed methods for second order elliptic equations. Math. Comp., 44(169):39–52, 1985.
  28. M. Eastwood. A complex from linear elasticity. In The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999), number 63, pages 23–29, 2000.
  29. M. Farhloul and M. Fortin. Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math., 76(4):419–440, 1997.
  30. New hybridized mixed methods for linear elasticity and optimal multilevel solvers. Numer. Math., 141(2):569–604, 2019.
  31. J. Gopalakrishnan and J. Guzmán. Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal., 49(4):1504–1520, 2011.
  32. J. Gopalakrishnan and J. Guzmán. A second elasticity element using the matrix bubble. IMA J. Numer. Anal., 32(1):352–372, 2012.
  33. K. Hellan. Analysis of elastic plates in flexure by a simplified finite element method, volume 46 of Acta polytechnica Scandinavica. Civil engineering and building construction series. Norges tekniske vitenskapsakademi, Trondheim, 1967.
  34. L. R. Herrmann. Finite element bending analysis for plates. Journal of the Engineering Mechanics Division, 93(EM5):49–83, 1967.
  35. J. Hu. Finite element approximations of symmetric tensors on simplicial grids in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT: the higher order case. J. Comput. Math., 33(3):283–296, 2015.
  36. 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(3):1438–1463, 2015.
  37. A new mixed finite element for the linear elasticity problem in 3D. arXiv preprint arXiv:2303.05805, 2023.
  38. A simple conforming mixed finite element for linear elasticity on rectangular grids in any space dimension. J. Sci. Comput., 58(2):367–379, 2014.
  39. J. Hu and Z.-C. Shi. Lower order rectangular nonconforming mixed finite elements for plane elasticity. SIAM J. Numer. Anal., 46(1):88–102, 2007.
  40. J. Hu and S. Zhang. A family of conforming mixed finite elements for linear elasticity on triangular grids. arXiv preprint arXiv:1406.7457, 2014.
  41. J. Hu and S. Zhang. A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math., 58(2):297–307, 2015.
  42. J. Hu and S. Zhang. Finite element approximations of symmetric tensors on simplicial grids in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT: the lower order case. Math. Models Methods Appl. Sci., 26(9):1649–1669, 2016.
  43. J. Huang and X. Huang. Local and parallel algorithms for fourth order problems discretized by the Morley-Wang-Xu element method. Numer. Math., 119(4):667–697, 2011.
  44. C. Johnson. On the convergence of a mixed finite-element method for plate bending problems. Numer. Math., 21:43–62, 1973.
  45. C. Johnson and B. Mercier. Some equilibrium finite element methods for two-dimensional problems in continuum mechanics. Numer. Math., 30:103–116, 1978.
  46. Lower order rectangular nonconforming mixed finite element for the three-dimensional elasticity problem. Math. Models Methods Appl. Sci., 19(1):51–65, 2009.
  47. J.-C. Nédélec. A new family of mixed finite elements in 𝐑3superscript𝐑3{\bf R}^{3}bold_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Numer. Math., 50(1):57–81, 1986.
  48. M. Okabe. Explicit interpolation formulas for the Bell triangle. Comput. Methods Appl. Mech. Engrg., 117(3-4):411–421, 1994.
  49. A. Pechstein and J. Schöberl. Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 21(8):1761–1782, 2011.
  50. W. Qiu and L. Demkowicz. Mixed h⁢pℎ𝑝hpitalic_h italic_p-finite element method for linear elasticity with weakly imposed symmetry. Comput. Methods Appl. Mech. Engrg., 198(47-48):3682–3701, 2009.
  51. A. Sinwel. A New Family of Mixed Finite Elements for Elasticity. PhD thesis, Johannes Kepler University Linz, 2009.
  52. R. Stenberg. Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér., 25(1):151–167, 1991.
  53. Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor. Math. Models Methods Appl. Sci., 27(14):2711–2743, 2017.
  54. X. Xie and J. Xu. New mixed finite elements for plane elasticity and Stokes equations. Sci. China Math., 54(7):1499–1519, 2011.
  55. S.-Y. Yi. Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions. Calcolo, 42(2):115–133, 2005.
  56. S.-Y. Yi. A new nonconforming mixed finite element method for linear elasticity. Math. Models Methods Appl. Sci., 16(7):979–999, 2006.
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