We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.
DOI: 10.1051/m2an/2015090
Keywords: Virtual element method, nonconforming method, Poisson equation, elliptic problems, unstructured meshes
@article{M2AN_2016__50_3_879_0, author = {Ayuso de Dios, Blanca and Lipnikov, Konstantin and Manzini, Gianmarco}, title = {The nonconforming virtual element method}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {879--904}, publisher = {EDP-Sciences}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/m2an/2015090}, mrnumber = {3507277}, zbl = {1343.65140}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015090/} }
TY - JOUR AU - Ayuso de Dios, Blanca AU - Lipnikov, Konstantin AU - Manzini, Gianmarco TI - The nonconforming virtual element method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 879 EP - 904 VL - 50 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015090/ DO - 10.1051/m2an/2015090 LA - en ID - M2AN_2016__50_3_879_0 ER -
%0 Journal Article %A Ayuso de Dios, Blanca %A Lipnikov, Konstantin %A Manzini, Gianmarco %T The nonconforming virtual element method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 879-904 %V 50 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015090/ %R 10.1051/m2an/2015090 %G en %F M2AN_2016__50_3_879_0
Ayuso de Dios, Blanca; Lipnikov, Konstantin; Manzini, Gianmarco. The nonconforming virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 3, pp. 879-904. doi : 10.1051/m2an/2015090. http://www.numdam.org/articles/10.1051/m2an/2015090/
R.A. Adams, Sobolev spaces, Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). | MR | Zbl
Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. | DOI | MR | Zbl
, , , and ,Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO M2AN 19 (1985) 7–32. | DOI | Numdam | MR | Zbl
and ,Gauss-Legendre elements: A stable, higher order non-conforming finite element family. Comput. 79 (2007) 1–21. | DOI | MR | Zbl
and ,Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. | DOI | MR | Zbl
, and ,L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method. Vol. 11 of Modeling, Simulations and Applications, 1st edition. Springer-Verlag, New York (2014). | MR | Zbl
Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl
, , , , and ,Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl
, and ,The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl Sci. 24 (2014) 1541–1573. | DOI | MR | Zbl
, , and ,The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Engrg. 280 (2014) 135–156. | DOI | MR | Zbl
, , and ,Poincaré–Friedrichs inequalities for piecewise functions. SIAM J. Numer. Anal. 41 (2003) 306–324. | DOI | MR | Zbl
,S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts Appl. Math. Springer-Verlag, New York (1994). | MR | Zbl
Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253 (2013) 455–462. | DOI | MR | Zbl
and ,Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl
, and ,Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. | DOI | Numdam | MR | Zbl
, and ,Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | DOI | Numdam | MR | Zbl
, and ,Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47 (2009) 2612–2637. | DOI | MR | Zbl
, and ,P.G. Ciarlet, Basic Error Estimates for Elliptic Problems. In Vol. II of Handb. Numer. Anal. North-Holland, Amsterdam (1991) 17–351. | MR | Zbl
The Hellan–Herrmann–Johnson method: some new error estimates and postprocessing. Math. Comput. 52 (1989) 17–29. | DOI | MR | Zbl
,Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973) 33–75. | Numdam | MR | Zbl
and ,Nonconforming finite elements for the Stokes problem. Math. Comput. 52 (1989) 437–456. | DOI | MR | Zbl
and ,A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Engrg. 283 (2015) 1–21. | DOI | MR | Zbl
and ,An extension of the Crouzeix-Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84 (2015) 1–31. | DOI | MR | Zbl
and ,An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14 (2014) 461–472. | DOI | MR | Zbl
, and ,Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529–550. | DOI | MR | Zbl
,A three-dimensional quadratic nonconforming element. Numer. Math. 46 (1985) 269–279. | DOI | MR | Zbl
,A non-conforming piecewise quadratic finite element on triangles. Int. J. Numer. Methods Eng. 19 (1983) 505–520. | DOI | MR | Zbl
and ,A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79 (2010) 2169–2189. | DOI | MR | Zbl
,Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275–311. | DOI | MR | Zbl
and ,K. Lipnikov and G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. Accepted for publication in J. Comput. Phys. | MR
The mimetic finite difference method for 3D magnetostatics fields problems. J. Comput. Phys. 230 (2011) 305–328. | DOI | MR | Zbl
, , and ,Mimetic finite difference method. J. Comput. Phys. 257 (2014) 1163–1227. | DOI | MR | Zbl
, and ,An inexpensive method for the evaluation of the solution of the lowest order Raviart–Thomas mixed method. SIAM J. Numer. Anal. 22 (1985) 493–496. | DOI | MR | Zbl
,Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. | DOI | MR | Zbl
and ,A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25 (2015) 1421–1445. | DOI | MR | Zbl
, and ,Nonconforming finite-element discretization of convex variational problems. IMA J. Numer. Anal. 31 (2011) 847–864. | DOI | MR | Zbl
,Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys. 257 (2014) 1394–1422. | DOI | MR | Zbl
, , , and ,Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differ. Equ. 8 (1992) 97–111. | DOI | MR | Zbl
and ,Crouzeix-Velte decompositions for higher-order finite elements. Comput. Math. Appl. 51 (2006) 967–986. | DOI | MR | Zbl
and ,G. Strang, Variational Crimes in the Finite Element Method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Proc. of Sympos., Univ. Maryland, Baltimore, Md. Academic Press, New York (1972) 689–710. | MR | Zbl
G. Strang and G.J. Fix, An analysis of the finite element method. Prentice-Hall Series in Automatic Computation. Prentice-Hall Inc., Englewood Cliffs, N. J. (1973). | MR | Zbl
A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715–719. | DOI | Numdam | MR | Zbl
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