Abstract
The \(H^m\)-nonconforming virtual elements of any order k on any shape of polytope in \({\mathbb {R}}^n\) with constraints \(m> n\) and \(k\ge m\) are constructed in a universal way. A generalized Green’s identity for \(H^m\) inner product with \(m>n\) is derived, which is essential to devise the \(H^m\)-nonconforming virtual elements. By means of the local \(H^m\) projection and a stabilization term using only the boundary degrees of freedom, the \(H^m\)-nonconforming virtual element methods are proposed to approximate solutions of the m-harmonic equation. The norm equivalence of the stabilization on the kernel of the local \(H^m\) projection is proved by using the bubble function technique, the Poincaré inquality and the trace inequality, which implies the well-posedness of the virtual element methods. The optimal error estimates for the \(H^m\)-nonconforming virtual element methods are achieved from an estimate of the weak continuity and the error estimate of the canonical interpolation. Finally, the implementation of the nonconforming virtual element method is discussed.
Similar content being viewed by others
References
Antonietti, P.F., Manzini, G., Verani, M.: The fully nonconforming virtual element method for biharmonic problems. Math. Models Methods Appl. Sci. 28(2), 387–407 (2018)
Antonietti, P.F., Manzini, G., Verani, M.: The conforming virtual element method for polyharmonic problems. Comput. Math. Appl. 79(7), 2021–2034 (2020)
Argyris, J., Fried, I., Scharpf, D.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 701–709 (1968)
Ayuso de Dios, B., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50(3), 879–904 (2016)
Beirão Da Veiga, L., Brezzi, F., Dassi, F., Marini, L.D., Russo, A.: Serendipity virtual elements for general elliptic equations in three dimensions. Chin. Ann. Math. Ser. B 39(2), 315–334 (2018)
Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Serendipity nodal VEM spaces. Comput. Fluids 141, 2–12 (2016)
Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Virtual element implementation for general elliptic equations. In: Barrenechea, G.R., Brezzi, F., Cangiani, A., Georgoulis, E.H. (eds.) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, pp. 39–71. Springer, Cham (2016)
Beirão da Veiga, L., Dassi, F., Russo, A.: A \(C^1\) virtual element method on polyhedral meshes. Comput. Math. Appl. 79(7), 1936–1955 (2020)
Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)
Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The Hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541–1573 (2014)
Beirão da Veiga, L., Manzini, G.: A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34(2), 759–781 (2014)
Bramble, J.H., Zlámal, M.S.: Triangular elements in the finite element method. Math. Comput. 24, 809–820 (1970)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, third edition edn. Springer, New York (2008)
Brenner, S.C., Sung, L.-Y.: Virtual element methods on meshes with small edges or faces. Math. Models Methods Appl. Sci. 28(7), 1291–1336 (2018)
Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)
Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 55:5 (2018)
Chen, L., Huang, X.: Nonconforming virtual element method for \(2m\)th order partial differential equations in \(\mathbb{R}^n\). Math. Comput. 89(324), 1711–1744 (2020)
Droniou, J., Ilyas, M., Lamichhane, B.P., Wheeler, G.E.: A mixed finite element method for a sixth-order elliptic problem. IMA J. Numer. Anal. 39(1), 374–397 (2019)
Gallistl, D.: Stable splitting of polyharmonic operators by generalized Stokes systems. Math. Comput. 86(308), 2555–2577 (2017)
Gudi, T., Neilan, M.: An interior penalty method for a sixth-order elliptic equation. IMA J. Numer. Anal. 31(4), 1734–1753 (2011)
Hu, J., Huang, Y., Zhang, S.: The lowest order differentiable finite element on rectangular grids. SIAM J. Numer. Anal. 49(4), 1350–1368 (2011)
Hu, J., Zhang, S.: The minimal conforming \(H^k\) finite element spaces on \(R^n\) rectangular grids. Math. Comput. 84(292), 563–579 (2015)
Hu, J., Zhang, S.: A canonical construction of \(H^m\)-nonconforming triangular finite elements. Ann. Appl. Math. 33(3), 266–288 (2017)
Hu, J., Zhang, S.: A cubic \(H^3\)-nonconforming finite element. Commun. Appl. Math. Comput. 1(1), 81–100 (2019)
Russo, A.: On the choice of the internal degrees of freedom for the nodal virtual element method in two dimensions. Comput. Math. Appl. 72(8), 1968–1976 (2016)
Schedensack, M.: A new discretization for \(m\)th-Laplace equations with arbitrary polynomial degrees. SIAM J. Numer. Anal. 54(4), 2138–2162 (2016)
Wang, M., Xu, J.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103(1), 155–169 (2006)
Wang, M., Xu, J.: Minimal finite element spaces for \(2m\)-th-order partial differential equations in \(R^n\). Math. Comput. 82(281), 25–43 (2013)
Wang, Y.: A nonconforming Crouzeix–Raviart type finite element on polygonal meshes. Math. Comput. 88(315), 237–271 (2019)
Wu, S., Xu, J.: \(\cal{P}_m\) interior penalty nonconforming finite element methods for \(2m\)-th order PDEs in \(R^{n}\). arXiv:1710.07678 (2017)
Wu, S., Xu, J.: Nonconforming finite element spaces for \(2m\)th order partial differential equations on \(\mathbb{R}^n\) simplicial grids when \(m=n+1\). Math. Comput. 88(316), 531–551 (2019)
Ženíšek, A.: Interpolation polynomials on the triangle. Numer. Math. 15, 283–296 (1970)
Ženíšek, A.: Tetrahedral finite \(C^{(m)}\)-elements. Acta Univ. Carolinae-Math. Phys. 15(1–2), 189–193 (1974)
Zhang, S.: A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59(1), 219–233 (2009)
Zhang, S.: On the full \(C_1\)-\(Q_k\) finite element spaces on rectangles and cuboids. Adv. Appl. Math. Mech. 2(6), 701–721 (2010)
Zhao, J., Chen, S., Zhang, B.: The nonconforming virtual element method for plate bending problems. Math. Models Methods Appl. Sci. 26(9), 1671–1687 (2016)
Zhao, J., Zhang, B., Chen, S., Mao, S.: The Morley-type virtual element for plate bending problems. J. Sci. Comput. 76(1), 610–629 (2018)
Acknowledgements
The author was supported by the National Natural Science Foundation of China Project 11771338, and the Fundamental Research Funds for the Central Universities 2019110066.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Huang, X. Nonconforming virtual element method for 2mth order partial differential equations in \({\mathbb {R}}^n\) with \(m>n\). Calcolo 57, 42 (2020). https://doi.org/10.1007/s10092-020-00381-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10092-020-00381-7
Keywords
- \(H^m\)-nonconforming virtual element
- Generalized Green’s identity
- Polyharmonic equation
- Norm equivalence
- Error analysis