Skip to main content
SpringerLink
Log in

Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, S., Cockburn, B.: A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. 25, 515–521 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adini, A., Clough, R.: Analysis of plate bending by the finite element method, technical report, NSF Report G. 7337 (1961)

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput. 35, 1039–1062 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bazeley, G., Cheung, Y., Irons, B., Zienkiewicz, O.: Triangular elements in plate bending—conforming and nonconforming solutions. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base: Dayton, Ohio, pp. 547–576 (1965)

  9. Beirão da Veiga, L., Niiranen, J., Stenberg, R.: A posteriori error estimates for the Morley plate bending element. Numer. Math. 106, 165–179 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  11. Bramble, J.H., Pasciak, J.E.: New convergence estimates for multigrid algorithms. Math. Comput. 49, 311–329 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bramble, J.H., Zhang, X.: Multigrid methods for the biharmonic problem discretized by conforming \(C^1\) finite elements on nonnested meshes. Numer. Funct. Anal. Optim. 16, 835–846 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brenner, S.C.: An optimal-order nonconforming multigrid method for the biharmonic equation. SIAM J. Numer. Anal. 26, 1124–1138 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brenner, S.C.: A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity, Math. Comp. 63, 435–460, S1–S5 (1994)

  15. Brenner, S.C.: Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comput. 68, 25–53 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brenner, S.C., Sung, L.-Y.: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22, 83–118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brenner, S.C., Sung, L.-Y.: Multigrid algorithms for \(C^0\) interior penalty methods. SIAM J. Numer. Anal. 44, 199–223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  19. Carstensen, C., Gallistl, D., Hu, J.: A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes. Comput. Math. Appl. 68, 2167–2181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, L.: iFEM: An integrated finite element methods package in MATLAB, technical report, University of California at Irvine (2008)

  21. Chen, L.: Multigrid methods for constrained minimization problems and application to saddle point problems. arXiv:1601.04091 (2016)

  22. Chen, L., Hu, J., Huang, X.: Fast auxiliary space preconditioner for linear elasticity in mixed form. Math. Comp. (2017). https://doi.org/10.1090/mcom/3285

  23. Chen, L., Hu, J., Huang, X.: Stabilized mixed finite element methods for linear elasticity on simplicial grids in \(\mathbb{R}^n\). Comput. Methods Appl. Math. 17, 17–31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam (1978)

    MATH  Google Scholar 

  25. Ciarlet, P.G.: On Korn’s inequality. Chin. Ann. Math. Ser. B 31, 607–618 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Comodi, M.I.: The Hellan–Herrmann–Johnson method: some new error estimates and postprocessing. Math. Comput. 52, 17–29 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: 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. Eng. 191, 3669–3750 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. RAIRO Anal. Numér. 14, 249–277 (1980)

    MathSciNet  MATH  Google Scholar 

  29. Feng, K., Shi, Z.-C.: Mathematical Theory of Elastic Structures. Springer, Berlin (1996)

    Google Scholar 

  30. de Veubeke, D.F.: Displacement and equilibrium models in the finite element method, ch. 9. In: Zienkiewicz, O., Holister, G.S. (eds.) Stress Analysis, pp. 145–197. Wiley, New York (1965)

    Google Scholar 

  31. Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)

    MATH  Google Scholar 

  32. Hellan, K.: Analysis of elastic plates in flexure by a simplified finite element method. Acta Polytechnica Scandinavia, Civ. Eng. Ser. 46, Trondheim (1967)

  33. Herrmann, L.R.: Finite element bending analysis for plates. J. Eng. Mech. Div. 93, 49–83 (1967)

    Google Scholar 

  34. Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the higher order case. J. Comput. Math. 33, 283–296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Hu, J., Zhang, S.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the lower order case. Math. Models Methods Appl. Sci. 26, 1649–1669 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Huang, J., Huang, X., Xu, Y.: Convergence of an adaptive mixed finite element method for Kirchhoff plate bending problems. SIAM J. Numer. Anal. 49, 574–607 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Krendl, W., Rafetseder, K., Zulehner, W.: A decomposition result for biharmonic problems and the Hellan–Herrmann–Johnson method. Electron. Trans. Numer. Anal. 45, 257–282 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Analyse Numérique 9, 9–53 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lee, Y.-J., Wu, J., Xu, J., Zikatanov, L.: A sharp convergence estimate for the method of subspace corrections for singular systems of equations. Math. Comput. 77, 831–850 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Q. 19, 149–169 (1968)

    Google Scholar 

  44. Pechstein, A., Schöberl, J.: Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 21, 1761–1782 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Peisker, P., Rust, W., Stein, E.: Iterative solution methods for plate bending problems: multigrid and preconditioned cg algorithm. SIAM J. Numer. Anal. 27, 1450–1465 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  46. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells, 2nd edn. CRC Press, New York (2006)

    Google Scholar 

  47. Shi, Z.-C., Xu, X.: A \(V\)-cycle multigrid method for TRUNC plate element. Comput. Methods Appl. Mech. Eng. 188, 483–493 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér. 25, 151–167 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  49. Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996). International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, M.: The \(W\)-cycle multigrid method for finite elements with nonnested spaces. Adv. Math. (China) 23, 238–250 (1994)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, M., Shi, Z.-C., Xu, J.: Some \(n\)-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math. 25, 408–420 (2007)

    MathSciNet  MATH  Google Scholar 

  53. Wang, M., Xu, J.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103, 155–169 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, M., Xu, J.: Minimal finite element spaces for \(2m\)-th-order partial differential equations in \(R^n\). Math. Comput. 82, 25–43 (2013)

    Article  MATH  Google Scholar 

  55. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581–613 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xu, X., Li, L.: A \(V\)-cycle multigrid method and additive multilevel preconditioners for the plate bending problem discretized by conforming finite elements. Appl. Math. Comput. 93, 233–258 (1998)

    MathSciNet  MATH  Google Scholar 

  57. Xu, X.-J., Li, L.-K.: A \(V\)-cycle multigrid method for the plate bending problem discretized by nonconforming finite elements. J. Comput. Math. 17, 533–544 (1999)

    MathSciNet  MATH  Google Scholar 

  58. Zhang, S.: An optimal order multigrid method for biharmonic, \(C^1\) finite element equations. Numer. Math. 56, 613–624 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhao, J.: Convergence of V-cycle and F-cycle multigrid methods for the biharmonic problem using the Morley element. Electron. Trans. Numer. Anal. 17, 112–132 (2004)

    MathSciNet  MATH  Google Scholar 

  60. Zhao, J.: Convergence of V- and F-cycle multigrid methods for the biharmonic problem using the Hsieh–Clough–Tocher element. Numer. Methods Partial Differ. Equ. 21, 451–471 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhou, S.Z., Feng, G.: A multigrid method for the Zienkiewicz element approximation of biharmonic equations. Hunan Daxue Xuebao 20, 1–6 (1993)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Long Chen

  2. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, China

    Long Chen

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

    Jun Hu

  4. Department of Mathematics, Wenzhou University, Wenzhou, 325035, China

    Xuehai Huang

Authors
  1. Long Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuehai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuehai Huang.

Additional information

L. Chen: The work of this author was supported by the National Science Foundation (NSF) DMS-1418934, and in part by the Sea Poly Project of Beijing Overseas Talents and the National Natural Science Foundation of China Project 11671159. This work was finished when the first author visited Peking University in the fall of 2015. He would like to thank Peking University for the support and hospitality, as well as for their exciting research atmosphere.

J. Hu: The work of this author was supported by the National Natural Science Foundation of China Projects 11625101, 91430213 and 11421101.

X. Huang: The work of this author was supported by the National Natural Science Foundation of China Projects 11771338 and 11671304, Zhejiang Provincial Natural Science Foundation of China Projects LY17A010010, LY15A010015 and LY15A010016, and Wenzhou Science and Technology Plan Project G20160019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, L., Hu, J. & Huang, X. Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems. J Sci Comput 76, 673–696 (2018). https://doi.org/10.1007/s10915-017-0636-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0636-z

Keywords

Search

Navigation