A Hybrid High-Order method for Kirchhoff–Love plate bending problems

Bonaldi, Francesco; Di Pietro, Daniele A.; Geymonat, Giuseppe; Krasucki, Françoise

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 393-421.

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k⩾1 are used as unknowns, we prove convergence in h^{k+1} (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in h^{k+3} is also derived for the L^2-norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

Classification : 65N30, 65N12, 74K20
Mots clés : Hybrid High-Order methods, Kirchhoff–Love plates, biharmonic problems, energy projector

[1] J. Aghili, S. Boyaval and D.A. Di Pietro, Hybridization of mixed high-order methods on general meshes and application to the Stokes equations. Comput. Methods Appl. Math. 15 (2015) 111–134. | DOI | MR | Zbl

[2] M. Amara, D. Capatina-Papaghiuc and A. Chatti, Bending moment mixed method for the Kirchhoff–Love plate model. SIAM J. Numer. Anal. 40 (2002) 1632–1649. | DOI | MR | Zbl

[3] P.F. Antonietti, L. Beirão Da Veiga, S. Scacchi and M. Verani, A C¹ virtual element method for the Cahn–Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54 (2016) 34–56. | DOI | MR | Zbl

[4] P.F. Antonietti, G. Manzini, M. Verani, The fully nonconforming virtual element method for biharmonic problems. Preprint (2016). | arXiv | MR

[5] D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19 (1985) 7–32. | DOI | Numdam | MR | Zbl

[6] K.J. Bathe, Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ (1996). | Zbl

[7] M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22 (2003) 751–756. | DOI | MR | Zbl

[8] E.M. Behrens and J. Guzmán, A mixed method for the biharmonic problem based on a system of first-order equations. SIAM J. Numer. Anal. 49 (2011) 789–817. | DOI | MR | Zbl

[9] L. Beirão Da Veiga and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781. | DOI | MR | Zbl

[10] H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556–581. | DOI | MR | Zbl

[11] D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer-Verlag (2013). | DOI | MR | Zbl

[12] D. Boffi and D.A. Di Pietro, Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes. ESAIM: M2AN 52 (2018) 1–28. | DOI | Numdam | MR | Zbl

[13] S.C. Brenner, C⁰ interior penalty methods, in Frontiers in Numerical Analysis – Durham 2010 Series. Lecture Notes in Computational Science and Engineering. Springer (2010) 79–147. | MR | Zbl

[14] S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Springer (2008). | DOI | MR | Zbl

[15] F. Brezzi, The great beauty of VEMs, in Proc. of the ICM 2014. Vol I of Plenary Lectures (2015) 217–235. | MR | Zbl

[16] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | DOI | MR | Zbl

[17] F. Brezzi, L.D. Marini, Virtual Element Methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253 (2013) 455–462. | DOI | MR | Zbl

[18] F. Chave,D.A. Di Pietro, F. Marche, and F. Pigeonneau, A hybrid high-order method for the Cahn–Hilliard problem in mixed form. SIAM J. Numer. Anal. 54 (2016) 1873–1898. | DOI | MR | Zbl

[19] C. Chinosi and L.D. Marini, Virtual Element Method for fourth order problems: L²-estimates. Comput. Math. Appl. 72 (2016) 1959–1967. | DOI | MR | Zbl

[20] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). Revised reprint SIAM (2002). | MR | Zbl

[21] B. Cockburn, B. Dong and J. Guzmán, A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40 (2009) 141–187. | DOI | MR | Zbl

[22] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. | DOI | MR | Zbl

[23] M.I. Comodi, The Hellan–Herrmann–Johnson method: some new error estimates and post-processing. Math. Comput. 52 (1989) 17–29. | DOI | MR | Zbl

[24] M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics. Springer-Verlag (1980). | Zbl

[25] D.A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes. Math. Comput. 86 (2017) 2159–2191. | DOI | MR | Zbl

[26] D.A. Di Pietro and J. Droniou, W^s,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems. Math. Models Methods Appl. Sci. 27 (2017) 879–908. | DOI | MR | Zbl

[27] D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Mathématiques & Applications. Springer-Verlag, Berlin (2012). | DOI | MR | Zbl

[28] D.A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283 (2015) 1–21. | DOI | MR | Zbl

[29] D.A. Di Pietro and R. Tittarelli, An introduction to Hybrid High-Order methods, in Numerical methods for PDEs. Lectures from the Fall 2016 Thematic Quarter at Institut Henri Poincaré. SEMA-SIMAI Series. Springer (2017). Preprint (2017). | arXiv | MR

[30] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992). | MR | Zbl

[31] P. Grisvard, Elliptic Problems in Nonsmooth Domains. SIAM (2011). | DOI | MR | Zbl

[32] R.A. Horn and F. Zhang, Basic properties of the Schur complement, in The Schur Complement and Its Applications. Vol. 4 of Numerical Methods and Algorithms. Springer, Boston, MA (2005). | DOI | MR

[33] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, 2nd edition. Prentice-Hall (1996). | Zbl

[34] C. Johnson, On the convergence of a mixed finite-element method for plate bending problems. Numer. Math. 21 (1973) 43–62. | DOI | MR | Zbl

[35] P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem. Rev. Française Automat. Informat. Recherche Operationnelle Sér. Rouge Anal. Numer. 9 (1975) 9–53. | Numdam | MR | Zbl

[36] V. Maz’Ya and J. Rossmann, Elliptic Equations in Polyhedral Domains. Vol. 162 of Mathematical Surveys and Monographs. American Mathematical Society (2010). | DOI | MR | Zbl

[37] L. Mu, J.Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods Partial Differ. Equ. 30 (2014) 1003–1029. | DOI | MR | Zbl

[38] J. Nečas, Direct Methods in the Theory of Elliptic Equations. English translation of Les méthodes directes en théorie des équations elliptiques published in 1967 simultaneously by Academia, the Publishing House of the Czechoslovak Academy of Sciences in Prague, and by Masson in Paris. Springer (2012). | DOI | MR | Zbl

[39] O.C. Zienkiewicz, in The Finite Element Method in Engineering Science, 1st edition. McGraw-Hill, New York (1971). | MR | Zbl

[40] O.C. Zienkiewicz, Origins, milestones and directions of the finite element method – a personal view, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.–L. Lions. Vol. IV of Finite Element Methods (Part 2) – Numerical Methods for Solids (Part 2) (1996) 5–67. | MR | Zbl

[41] J. Zhao, S. Chen, and B. Zhang, The nonconforming virtual element method for plate bending problems. Math. Models Methods Appl. Sci. 26 (2016) 1671–1687. | DOI | MR | Zbl