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A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

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Abstract

This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress σ = −∇2u is sought in the Sobolev space H(divdiv, Ω; \(\mathbb{S}\)) simultaneously with the displacement u in L2(Ω). By stemming from the structure of H(div, Ω; \(\mathbb{S}\)) conforming elements for the linear elasticity problems proposed by Hu and Zhang (2014), the H(divdiv, Ω; \(\mathbb{S}\)) conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H (div, Ω; \(\mathbb{S}\)) conforming spaces of Pk symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for u are composed of the piecewise Pk−2 polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k ⩾ 3, and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis.

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References

  1. Adini A, Clough R. Analysis of Plate Bending by the Finite Element Method. https://books.google.com.sg/books/about/Analysi_of_Plate_Bending_by_the_Finite.html?id=0qrsHAAACAAJ&redir_esc=y, 1960

  2. Argyris J, Fried I, Scharpf D. The TUBA family of elements for the matrix displacement method. J Roy Aero Soc, 1968, 72: 514–517

    Google Scholar 

  3. Arnold D N, Awanou G, Winther R. Finite elements for symmetric tensors in three dimensions. Math Comp, 2008, 77: 1229–1251

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold D N, Falk R S, Winther R. Differential complexes and stability of finite element methods II: The elasticity complex. In: Compatible Spatial Discretizations. The IMA Volumes in Mathematics and Its Applications, vol. 142. New York: Springer, 2006, 47–67

    Chapter  Google Scholar 

  5. Arnold D N, Falk R S, Winther R. Finite element exterior calculus, homological techniques, and applications. Acta Numer, 2006, 15: 1–155

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold D N, Falk R S, Winther R. Finite element exterior calculus: From Hodge theory to numerical stability. Bull Amer Math Soc NS, 2010, 47: 281–354

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnold D N, Hu K. Complexes from complexes. Found Comput Math, 2021, in press

  8. Arnold D N, Winther R. Mixed finite elements for elasticity. Numer Math, 2002, 92: 401–419

    Article  MathSciNet  MATH  Google Scholar 

  9. Behrens E M, Guzmán J. A mixed method for the biharmonic problem based on a system of first-order equations. SIAM J Numer Anal, 2011, 49: 789–817

    Article  MathSciNet  MATH  Google Scholar 

  10. Bernšteǐn I N, Gelfand I M, Gelfand S I. Differential operators on the base affine space and a study of g-modules. In: Proceedings of Summer School of Bolyai János Mathematical Society. Lie Groups and Their Representations. New York: Wiley, 1975, 21–64

    Google Scholar 

  11. Boffi D, Brezzi F, Fortin M. Mixed Finite Element Methods and Applications. Heidelberg: Springer, 2013

    Book  MATH  Google Scholar 

  12. Brenner S C, Gudi T, Sung L Y. A weakly over-penalized symmetric interior penalty method for the biharmonic problem. Electron Trans Numer Anal, 2010, 37: 214–238

    MathSciNet  MATH  Google Scholar 

  13. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. New York: Springer-Verlag, 1991

    MATH  Google Scholar 

  14. Chen H, Chen S, Qiao Z. C0-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem. Numer Math, 2013, 124: 99–119

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen L, Huang X. Finite elements for divdiv-conforming symmetric tensors. arXiv:2005.01271, 2020

  16. Chen L, Huang X. Finite elements for divdiv-conforming symmetric tensors in three dimensions. arxiv:2007.12399, 2020

  17. Christiansen S H, Hu J, Hu K. Nodal finite element de Rham complexes. Numer Math, 2018, 139: 411–446

    Article  MathSciNet  MATH  Google Scholar 

  18. Christiansen S H, Hu K. Finite element systems for vector bundles: Elasticity and curvature. arXiv:1906.09128, 2020

  19. Christiansen S H, Hu K, Sande E. Poincaré path integrals for elasticity. J Math Pures Appl (9), 2020, 135: 83–102

    Article  MathSciNet  MATH  Google Scholar 

  20. Ciarlet P G. The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland, 1978

    MATH  Google Scholar 

  21. Ciarlet P G, Raviart P A. A mixed finite element method for the biharmonic equation. In: Proceedings of a Symposium Conducted by the Mathematics Research Center of the University of Wisconsin-Madison. Mathematical Aspects of Finite Elements In Partial Differential Equations. Amsterdam: Academic Press, 1974, 125–145

    Google Scholar 

  22. Douglas J Jr, Dupont T, Percell P, et al. A family of C1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal Numér, 1979, 13: 227–255

    Article  MathSciNet  MATH  Google Scholar 

  23. Fraeijs De Veubeke B. Variational principles and the patch test. Internat J Numer Methods Engrg, 1974, 8: 783–801

    Article  MathSciNet  MATH  Google Scholar 

  24. Führer T, Heuer N. Fully discrete DPG methods for the Kirchhoff-Love plate bending model. Comput Methods Appl Mech Engrg, 2019, 343: 550–571

    Article  MathSciNet  MATH  Google Scholar 

  25. Führer T, Heuer N, Niemi A H. An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation. Math Comp, 2019, 88: 1587–1619

    Article  MathSciNet  MATH  Google Scholar 

  26. Gao B, Zhang S, Wang M. A note on the nonconforming finite elements for elliptic problems. J Comput Math, 2011, 29: 215–226

    Article  MathSciNet  MATH  Google Scholar 

  27. Gerasimov T, Stylianou A, Sweers G. Corners give problems when decoupling fourth order equations into second order systems. SIAM J Numer Anal, 2012, 50: 1604–1623

    Article  MathSciNet  MATH  Google Scholar 

  28. Girault V, Raviart P A. Finite Element Methods for Navier-Stokes Equations. Berlin: Springer-Verlag, 1986

    Book  MATH  Google Scholar 

  29. Guzmán J, Leykekhman D, Neilan M. A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem. Calcolo, 2012, 49: 95–125

    Article  MathSciNet  MATH  Google Scholar 

  30. Herrmann L. Finite element bending analysis for plates. J Engrg Mech Div, 1967, 93: 13–26

    Article  Google Scholar 

  31. Hu J. Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case. J Comput Math, 2015, 33: 283–296

    Article  MathSciNet  MATH  Google Scholar 

  32. Hu J, Huang Y, Zhang S. The lowest order differentiable finite element on rectangular grids. SIAM J Numer Anal, 2011, 49: 1350–1368

    Article  MathSciNet  MATH  Google Scholar 

  33. Hu J, Tian S, Zhang S. A family of 3D H2-nonconforming tetrahedral finite elements for the biharmonic equation. Sci China Math, 2020, 63: 1505–1522

    Article  MathSciNet  MATH  Google Scholar 

  34. Hu J, Zhang S. A family of conforming mixed finite elements for linear elasticity on triangular grids. arXiv:1406.7457, 2014

  35. Hu J, Zhang S. A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci China Math, 2015, 58: 297–307

    Article  MathSciNet  MATH  Google Scholar 

  36. Hu J, Zhang S. The minimal conforming Hk finite element spaces on Rn rectangular grids. Math Comp, 2015, 84: 563–579

    Article  MathSciNet  MATH  Google Scholar 

  37. Hu J, Zhang S. An error analysis method SPP-BEAM and a construction guideline of nonconforming finite elements for fourth order elliptic problems. J Comput Math, 2020, 38: 195–222

    Article  MathSciNet  MATH  Google Scholar 

  38. Johnson C. On the convergence of a mixed finite-element method for plate bending problems. Numer Math, 1973, 21: 43–62

    Article  MathSciNet  MATH  Google Scholar 

  39. Lascaux P, Lesaint P. Some nonconforming finite elements for the plate bending problem. Rev Française Autom Inform Rech Opér Sér Rouge Anal Numér, 1975, 9: 9–53

    MathSciNet  MATH  Google Scholar 

  40. Miyoshi T. A finite element method for the solutions of fourth order partial differential equations. Kumamoto J Sci Math, 1972, 9: 87–116

    MathSciNet  MATH  Google Scholar 

  41. Morley L S D. A triangular equilibrium element with linearly varying bending moments for plate bending problems. Aeronautical J, 1967, 71: 715–719

    Article  Google Scholar 

  42. Powell M J D, Sabin M A. Piecewise quadratic approximations on triangles. ACM Trans Math Software, 1977, 3: 316–325

    Article  MathSciNet  MATH  Google Scholar 

  43. Shi Z, Wang M. Finite Element Methods. Beijing: Science Press, 2013

    Google Scholar 

  44. Veubeke B F D. A conforming finite element for plate bending. Internat J Solids Structures, 1968, 4: 95–108

    Article  MATH  Google Scholar 

  45. Wang M, Shi Z, Xu J. A new class of Zienkiewicz-type non-conforming element in any dimensions. Numer Math, 2007, 106: 335–347

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang M, Zu P, Zhang S. High accuracy nonconforming finite elements for fourth order problems. Sci China Math, 2012, 55: 2183–2192

    Article  MathSciNet  MATH  Google Scholar 

  47. Yang X. Non-standard finite element methods for the thin structure. PhD Thesis. Beijing: Peking University, 2017

    Google Scholar 

  48. Ženíšek A. Polynomial approximation on tetrahedrons in the finite element method. J Approx Theory, 1973, 7: 334–351

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhang S. A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl Numer Math, 2009, 59: 219–233

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang S, Zhang Z. Invalidity of decoupling a biharmonic equation to two Poisson equations on non-convex polygons. Int J Numer Anal Model, 2008, 5: 73–76

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 11625101 and 11421101).

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jun Hu & Min Zhang

  2. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, 45127, Germany

    Rui Ma

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Min Zhang

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  1. Jun Hu
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  2. Rui Ma
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  3. Min Zhang
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Correspondence to Min Zhang.

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Hu, J., Ma, R. & Zhang, M. A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids. Sci. China Math. 64, 2793–2816 (2021). https://doi.org/10.1007/s11425-020-1883-9

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  • DOI: https://doi.org/10.1007/s11425-020-1883-9

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