The nonconforming virtual element method

Ayuso de Dios, Blanca; Lipnikov, Konstantin; Manzini, Gianmarco

doi:10.1051/m2an/2015090

Ayuso de Dios, Blanca ^1,² ; Lipnikov, Konstantin ³ ; Manzini, Gianmarco ^2,³

¹ Institut für Mathematik, Technische Universität Hamburg-Harburg, Am Schwarzenberg-Campus 3, D-21073 Hamburg, Germany
² Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) – CNR, 27100 Pavia, Italy
³ Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 879-904.

Résumé

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.

Reçu le : 2015-04-26

MR Zbl | 6 citations dans Numdam

DOI : 10.1051/m2an/2015090

Classification : 65N30, 65N12, 65G99, 76R99
Mots clés : Virtual element method, nonconforming method, Poisson equation, elliptic problems, unstructured meshes

Affiliations des auteurs :

Ayuso de Dios, Blanca ^{1, 2} ; Lipnikov, Konstantin ³ ; Manzini, Gianmarco ^{2, 3}

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     title = {The nonconforming virtual element method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {879--904},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {3},
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     doi = {10.1051/m2an/2015090},
     mrnumber = {3507277},
     zbl = {1343.65140},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015090/}
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Ayuso de Dios, Blanca; Lipnikov, Konstantin; Manzini, Gianmarco. The nonconforming virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 879-904. doi : 10.1051/m2an/2015090. http://www.numdam.org/articles/10.1051/m2an/2015090/

Bibliographie
Cité par

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

B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. | DOI | MR | Zbl

D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO M2AN 19 (1985) 7–32. | DOI | Numdam | MR | Zbl

A.E. Baran and G. Stoyan, Gauss-Legendre elements: A stable, higher order non-conforming finite element family. Comput. 79 (2007) 1–21. | DOI | MR | Zbl

L. Beirão Da Veiga, K. Lipnikov and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. | DOI | MR | Zbl

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

L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl Sci. 24 (2014) 1541–1573. | DOI | MR | Zbl

F.M. Benedetto, S. Berrone, S. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Engrg. 280 (2014) 135–156. | DOI | MR | Zbl

S.C. Brenner, Poincaré–Friedrichs inequalities for piecewise $H^{1}$ 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

F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253 (2013) 455–462. | DOI | MR | Zbl

F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl

F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. | DOI | Numdam | MR | Zbl

F. Brezzi, R.S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | DOI | Numdam | MR | Zbl

A. Cangiani, G. Manzini and A. Russo, Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47 (2009) 2612–2637. | DOI | MR | Zbl

P.G. Ciarlet, Basic Error Estimates for Elliptic Problems. In Vol. II of Handb. Numer. Anal. North-Holland, Amsterdam (1991) 17–351. | MR | Zbl

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

M. Crouzeix and P.-A. Raviart, 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

M. Crouzeix and R.S. Falk, Nonconforming finite elements for the Stokes problem. Math. Comput. 52 (1989) 437–456. | DOI | MR | Zbl

D.A. Di Pietro and A. Ern, 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

D.A. Di Pietro and S. Lemaire, 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

D.A. Di Pietro, A. Ern and S. Lemaire, 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

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

M. Fortin, A three-dimensional quadratic nonconforming element. Numer. Math. 46 (1985) 269–279. | DOI | MR | Zbl

M. Fortin and M. Soulie, A non-conforming piecewise quadratic finite element on triangles. Int. J. Numer. Methods Eng. 19 (1983) 505–520. | DOI | MR | Zbl

T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79 (2010) 2169–2189. | DOI | MR | Zbl

J.G. Heywood and R. Rannacher, 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

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

K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa, The mimetic finite difference method for 3D magnetostatics fields problems. J. Comput. Phys. 230 (2011) 305–328. | DOI | MR | Zbl

K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. 257 (2014) 1163–1227. | DOI | MR | Zbl

L.D. Marini, 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

G. Matthies and L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. | DOI | MR | Zbl

D. Mora, G. Rivera and R. Rodriguez, A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25 (2015) 1421–1445. | DOI | MR | Zbl

C. Ortner, Nonconforming finite-element discretization of convex variational problems. IMA J. Numer. Anal. 31 (2011) 847–864. | DOI | MR | Zbl

A. Palha, P.P. Rebelo, R. Hiemstra, J. Kreeft and M. Gerritsma, 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

R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differ. Equ. 8 (1992) 97–111. | DOI | MR | Zbl

G. Stoyan and A.E. Baran, Crouzeix-Velte decompositions for higher-order finite elements. Comput. Math. Appl. 51 (2006) 967–986. | DOI | MR | Zbl

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

R. Verfürth. A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715–719. | DOI | Numdam | MR | Zbl

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