Locking-Free and Gradient-Robust \({\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})\)-Conforming HDG Methods for Linear Elasticity


Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new: it assures that dominant gradient fields in the momentum balance do not lead to spurious displacements. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    Ahmed, N., Linke, A., Merdon, C.: On really locking-free mixed finite element methods for the transient incompressible Stokes equations. SIAM J. Numer. Anal., pp. 185–209 (2018)

  2. 2.

    Akbas, M., Gallouet, T., Gassmann, A., Linke, A., Merdon, C.: A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem. arXiv:1911.01295, (2019)

  3. 3.

    Akbas, M., Linke, A., Rebholz, L.G., Schroeder, P.W.: The analogue of grad-div stabilization in DG methods for incompressible flows: limiting behavior and extension to tensor-product meshes. Comput. Methods Appl. Mech. Eng. 341, 917–938 (2018)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Arnold, D.N., Brezzi, F., Douglas Jr., J.: PEERS: a new mixed finite element for plane elasticity. Jpn J. Appl. Math. 1, 347–367 (1984)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Arnold, D.N., Douglas Jr., J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45, 1–22 (1984)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76, 1699–1723 (2007)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Arnold, D. N., Qin, J.: Quadratic velocity/linear pressure stokes elements, in Advances in Computer Methods for Partial Differential Equations VII, IMACS, pp. 28–34 (1992)

  8. 8.

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

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Arnold, D.N., Winther, R.: Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci. 13, 295–307 (2003)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Auricchio, F., Beirão da Veiga, L., Lovadina, C., Reali, A.: A stability study of some mixed finite elements for large deformation elasticity problems. Comput. Methods Appl. Mech. Eng. 194, 1075–1092 (2005)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Auricchio, F., Beirão da Veiga, L., Lovadina, C., Reali, A.: The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations. Comput. Methods Appl. Mech. Eng. 199, 314–323 (2010)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Auricchio, F., Beirão da Veiga, L., Lovadina, C., Reali, A., Taylor, R.L., Wriggers, P.: Approximation of incompressible large deformation elastic problems: some unresolved issues. Comput. Mech. 52, 1153–1167 (2013)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Auricchio, F., da Veiga, L.B.a, Buffa, A., Lovadina, C., Reali, A., Sangalli, G.: A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation. Comput. Methods Appl. Mech. Eng. 197, 160–172 (2007)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51, 794–812 (2013)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)

  16. 16.

    Braess, D.: Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer, Berlin (2013)

    Google Scholar 

  17. 17.

    Brenner, S.C.: Korn’s inequalities for piecewise \(H^1\) vector fields. Math. Comput. 73, 1067–1087 (2004)

    MATH  Google Scholar 

  18. 18.

    Brenner, S.C., Sung, L.-Y.: Linear finite element methods for planar linear elasticity. Math. Comput. 59, 321–338 (1992)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Chertock, A., Dudzinski, M., Kurganov, A., Lukáčová-Medviďová, M.: Well-balanced schemes for the shallow water equations with Coriolis forces. Numer. Math. 138, 939–973 (2018)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Cockburn, B., Fu, G.: Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by \(M\)-decompositions. IMA J. Numer. Anal. 38, 566–604 (2018)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Cockburn, B., Kanschat, G., Schotzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74, 1067–1095 (2005)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Cockburn, B., Schötzau, D., Wang, J.: Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Mech. Eng. 195, 3184–3204 (2006)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Cockburn, B., Shi, K.: Superconvergent HDG methods for linear elasticity with weakly symmetric stresses. IMA J. Numer. Anal. 33, 747–770 (2013)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Cotter, C.J., Thuburn, J.: A finite element exterior calculus framework for the rotating shallow-water equations. J. Comput. Phys. 257, 1506–1526 (2014)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Di Pietro, D.A., Ern, A., Linke, A., Schieweck, F.: A discontinuous skeletal method for the viscosity-dependent Stokes problem. Comput. Methods Appl. Mech. Eng. 306, 175–195 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Elguedj, T., Bazilevs, Y., Calo, V., Hughes, T.: \(\overline{B}\) and \(\overline{F}\) projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order nurbs elements. Comput. Methods Appl. Mech. Eng. 197, 2732–2762 (2008)

    MATH  Google Scholar 

  28. 28.

    Falk, R.S.: Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57, 529–550 (1991)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Frerichs, D., Merdon, C.: Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the stokes problem, (2020)

  30. 30.

    Fu, G.: A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77, 237–252 (2019)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Fu, G., Lehrenfeld, C.: A strongly conservative hybrid DG/mixed FEM for the coupling of Stokes and Darcy flow. J. Sci. Comput. 77, pp. 1605–1620

  32. 32.

    Fu, G., Lehrenfeld, C., Linke, A., Streckenbach, T.: Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity. arXiv:2001.08610, (2020)

  33. 33.

    Gauger, N.R., Linke, A., Schroeder, P.W.: On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond. SMAI J. Comput. Math. 5, 89–129 (2019)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Gerbeau, J.-F., Le Bris, C., Bercovier, M.: Spurious velocities in the steady flow of an incompressible fluid subjected to external forces. Internat. J. Numer. Methods Fluids 25, 679–695 (1997)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Gopalakrishnan, J., Guzmán, J.: Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal. 49, 1504–1520 (2011)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Gosse, L., Leroux, A.-Y.: Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes, C. R. Acad. Sci. Paris Sér. I Math. 323, 543–546 (1996)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Greenberg, J.M., Leroux, A.Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16 (1996)

    MathSciNet  Google Scholar 

  38. 38.

    Guzmán, J., Neilan, M.: Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions. Numer. Math. 126, 153–171 (2014)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Guzmán, J., Shu, C.-W., Sequeira, F.A.: \({{\rm H(div)}}\) conforming and DG methods for incompressible Euler’s equations. IMA J. Numer. Anal. 37, 1733–1771 (2017)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Hansbo, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191, 1895–1908 (2002)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Hansbo, P., Larson, M.G.: Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, M2AN Math. Model. Numer. Anal. 37, 63–72 (2003)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)

    Google Scholar 

  43. 43.

    Hong, Q., Kraus, J., Xu, J., Zikatanov, L.: A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations. Numer. Math. 132, 23–49 (2016)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Hughes, T.J., Cohen, M., Haroun, M.: Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Eng. Des. 46, 203–222 (1978)

    Google Scholar 

  45. 45.

    Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    John, V.: Finite Element Methods for Incompressible Flow Problems. Springer, Berlin (2016)

    Google Scholar 

  47. 47.

    John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59, 492–544 (2017)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Kanschat, G., Riviere, B.: A finite element method with strong mass conservation for Biot’s linear consolidation model. J. Sci. Comput. 77, 1762–1779 (2018)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Kreuzer, C., Verfürth, R., Zanotti, P.: Quasi-optimal and pressure robust discretizations of the Stokes equations by moment- and divergence-preserving operators. arXiv:2002.11454, (2020)

  50. 50.

    Lederer, P.L., Lehrenfeld, C., Schöberl, J.: Hybrid discontinuous Galerkin methods with relaxed \(H({\rm div})\)-conformity for incompressible flows. Part I. SIAM J. Numer. Anal. 56, 2070–2094 (2018)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Lederer, P.L., Lehrenfeld, C., Schöberl, J.: Hybrid discontinuous Galerkin methods with relaxed \(H({\rm div})\)-conformity for incompressible flows. Part II. ESAIM Math. Model. Numer. Anal. 53, 503–522 (2019)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Lederer, P.L., Linke, A., Merdon, C., Schöberl, J.: Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements. SIAM J. Numer. Anal. 55, 1291–1314 (2017)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Lederer, P.L., Schöberl, J.: Polynomial robust stability analysis for \(H({\rm div})\)-conforming finite elements for the Stokes equations. IMA J. Numer. Anal. 38, 1832–1860 (2018)

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Lehrenfeld, C., Galerkin, Hybrid Discontinuous, methods for solving incompressible flow problems, : Diploma Thesis. MathCCES/IGPM, RWTH Aachen (2010)

  55. 55.

    Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016)

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Linke, A.: A divergence-free velocity reconstruction for incompressible flows. C. R. Math. Acad. Sci. Paris 350, 837–840 (2012)

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268, 782–800 (2014)

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Linke, A., Matthies, G., Tobiska, L.: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM Math. Model. Numer. Anal. 50, 289–309 (2016)

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Linke, A., Merdon, C.: On velocity errors due to irrotational forces in the Navier-Stokes momentum balance. J. Comput. Phys. 313, 654–661 (2016)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 311, 304–326 (2016)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Malkus, D.S., Hughes, T.J.: Mixed finite element methods – reduced and selective integration techniques: A unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81 (1978)

    MATH  Google Scholar 

  62. 62.

    Marquardt, O., Boeck, S., Freysoldt, C., Hickel, T., Schulz, S., Neugebauer, J., O’Reilly, E.P.: A generalized plane-wave formulation of k.p formalism and continuum-elasticity approach to elastic and electronic properties of semiconductor nanostructures. Comput. Mater. Sci. 95, 280–287 (2014)

    Google Scholar 

  63. 63.

    Natale, A., Shipton, J., Cotter, C.J.: Compatible finite element spaces for geophysical fluid dynamics. Dyn. Stat. Climate Syst. 1, (2016)

  64. 64.

    Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)

    MathSciNet  MATH  Google Scholar 

  65. 65.

    Olshanskii, M.A., Rebholz, L.G.: Application of barycenter refined meshes in linear elasticity and incompressible fluid dynamics. Electron. Trans. Numer. Anal. 38, 258–274 (2011)

    MathSciNet  MATH  Google Scholar 

  66. 66.

    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)

    MathSciNet  MATH  Google Scholar 

  67. 67.

    Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87, 69–93 (2018)

    MathSciNet  MATH  Google Scholar 

  68. 68.

    Schöberl, J., C++11 Implementation of Finite Elements in NGSolve, : ASC Report 30/2014. Vienna University of Technology, Institute for Analysis and Scientific Computing (2014)

  69. 69.

    Schroeder, P.W., Lehrenfeld, C., Linke, A., Lube, G.: Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations. SeMA J. 75, 629–653 (2018)

    MathSciNet  MATH  Google Scholar 

  70. 70.

    Scott, L.R., Vogelius, M., Conforming finite element methods for incompressible, and nearly incompressible continua, in Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., : vol. 22 of Lectures in Appl. Math., Amer. Math. Soc. Providence, RI 1985, 221–244 (1983)

  71. 71.

    Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19, 111–143 (1985)

    MathSciNet  MATH  Google Scholar 

  72. 72.

    Soon, S.-C., Cockburn, B., Stolarski, H.: A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Eng. 80, 1058–1092 (2009)

    MathSciNet  MATH  Google Scholar 

  73. 73.

    Verfürth, R., Zanotti, P.: A quasi-optimal Crouzeix-Raviart discretization of the Stokes equations. SIAM J. Numer. Anal. 57, 1082–1099 (2019)

    MathSciNet  MATH  Google Scholar 

  74. 74.

    Vogelius, M.: An analysis of the \(p\)-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates. Numer. Math. 41, 39–53 (1983)

    MathSciNet  MATH  Google Scholar 

  75. 75.

    Wihler, T.: Locking-free adaptive discontinuous galerkin fem for linear elasticity problems. Math. Comput. 75, 1087–1102 (2006)

    MathSciNet  MATH  Google Scholar 

  76. 76.

    Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74, 543–554 (2005)

    MathSciNet  MATH  Google Scholar 

  77. 77.

    Zhang, S.: On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26, 456–470 (2008)

    MathSciNet  MATH  Google Scholar 

  78. 78.

    Zhang, S.: Divergence-free finite elements on tetrahedral grids for \(k\ge 6\). Math. Comput. 80, 669–695 (2011)

    MATH  Google Scholar 

  79. 79.

    Zhang, S.: Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids. Calcolo 48, 211–244 (2011)

    MathSciNet  MATH  Google Scholar 

  80. 80.

    Zienkiewics, O.C., Taylor, R.L., Too, J.M.: Reduced integration techniques in general analysis of plates and shells. Int. J. Numer. Meth. Eng. 5, 275–290 (1971)

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Guosheng Fu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

G. Fu gratefully acknowledge the partial support of this work from U.S. National Science Foundation through Grant DMS-2012031.

Appendix. The BDM Interpolator for Discontinuous Functions

Appendix. The BDM Interpolator for Discontinuous Functions

The BDM interpolator for discontinuous functions is defined element-by-element for \(\varvec{v}_T \in {\varvec{H}}^1(T)\) through

$$\begin{aligned} (\Pi _V \varvec{v}_T \! \cdot \! \varvec{n}, \varphi )_{F}&= ( \{\!\!\{ \varvec{v}_T \! \cdot \! \varvec{n} \}\!\!\}_*, \varphi )_{F}&\forall ~ \varphi \!\in \! {\mathbb {P}}^{k}\!(F), F \!\in \! \partial T, \end{aligned}$$
$$\begin{aligned} (\Pi _V \varvec{v}_T, \varphi )_{T}&= (\varvec{v}_T, \varphi )_{T}&\forall ~ \varphi \in \varvec{{\mathcal {N}}}^{k-2}(T), \end{aligned}$$

with \(\varvec{{\mathcal {N}}}^{k-2}:=[{\mathbb {P}}^{k-2}(T)]^d + [{\mathbb {P}}^{k-2}(T)]^d \times x\) and \(\{\!\!\{ \cdot \}\!\!\}_*\) the usual DG average operator, cf. [21, 39].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fu, G., Lehrenfeld, C., Linke, A. et al. Locking-Free and Gradient-Robust \({\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})\)-Conforming HDG Methods for Linear Elasticity. J Sci Comput 86, 39 (2021). https://doi.org/10.1007/s10915-020-01396-6

Download citation


  • Linear elasticity
  • Nearly incompressible
  • Locking phenomenon
  • Volume-locking
  • Gradient-robustness
  • Discontinuous Galerkin
  • \({\varvec{H}}\) (div)-conforming HDG

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 76S05
  • 76D07