no. 3
The nonconforming Virtual Element Method for eigenvalue problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 749-774

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

DOI : 10.1051/m2an/2018074
Classification : 65N30, 65N25
Keywords: Nonconforming virtual element, eigenvalue problem, polygonal meshes

Gardini, Francesca 1 ; Manzini, Gianmarco 1 ; Vacca, Giuseppe 1

1 
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     title = {The nonconforming {Virtual} {Element} {Method} for eigenvalue problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
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Gardini, Francesca; Manzini, Gianmarco; Vacca, Giuseppe. The nonconforming Virtual Element Method for eigenvalue problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 749-774. doi: 10.1051/m2an/2018074

[1] R.A. Adams, Sobolev spaces. In Vol. 65 of Pure and Applied Mathematics. Academic Press, New York-London (1975). | MR | Zbl

[2] S. Agmon, Lectures on elliptic boundary value problems. In Vol. 2 of Van Nostrand Mathematical Studies. D. Van Nostrand Co., Inc., Princeton, NJ-Toronto-London (1965). | MR | Zbl

[3] 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. | MR | Zbl | DOI

[4] 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. | MR | Zbl | DOI

[5] P.F. Antonietti, G. Manzini and M. Verani, The fully nonconforming virtual element method for biharmonic problems. Math. Models Methods Appl. Sci. 28 (2018) 387–407. | MR | Zbl | DOI

[6] E. Artioli, S. De Miranda, C. Lovadina and L. Patruno, A stress/displacement virtual element method for plane elasticity problems. Comput. Meth. Appl. Mech. Eng. 325 (2017) 155–174. | MR | Zbl | DOI

[7] B. Ayuso De Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. ESAIM: M2AN 50 (2016) 879–904. | MR | Zbl | Numdam | DOI

[8] I. Babuška and J. Osborn, Eigenvalue problems. In: Handbook of Numerical Analysis. Handb. Numer. Anal. II. North-Holland, Amsterdam (1991) 641–787. | Zbl | MR

[9] 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. | Zbl | MR | DOI

[10] 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. | Zbl | MR | DOI

[11] L. Beirão Da Veiga and G. Manzini, Residual a posteriori error estimation for the virtual element method for elliptic problems. ESAIM: M2AN 49 (2015) 577–599. | Zbl | Numdam | MR | DOI

[12] L. Beirão Da Veiga, D. Mora, G. Rivera and R. Rodriguez, A virtual element method for the acoustic vibration problem, Numer. Math. 136 (2017) 725–763. | Zbl | MR | DOI

[13] L. Beirão Da Veiga, F. Brezzi, F. Dassi, L.D. Marini and A. Russo, Serendipity virtual elements for general elliptic equations in three dimensions. Chin. Ann. Math. Ser. B 39 (2018) 315–334. | Zbl | MR | DOI

[14] L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Serendipity nodal vem spaces. Comput. Fluids 141 (2016) 2–12. | Zbl | MR | DOI

[15] L. Beirão Da Veiga, F. Dassi and A. Russo, High-order virtual element method on polyhedral meshes. Comput. Math. Appl. 74 (2017) 1110–1122. | Zbl | MR | DOI

[16] L. Beirão Da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method. Math. Models Methods Appl. Sci. 27 (2017) 2557–2594. | Zbl | MR | DOI

[17] L. Beirão Da Veiga, C. Lovadina and G. Vacca, Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM: M2AN 51 (2017) 509–535. | MR | Zbl | Numdam | DOI

[18] L. Beirão Da Veiga and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2013) 759–781. | Zbl | MR | DOI

[19] M.F. Benedetto, S. Berrone, A. Borio, S. Pieraccini and S. Scialò, A hybrid mortar virtual element method for discrete fracture network simulations. J. Comput. Phys. 306 (2016) 148–166. | Zbl | MR | DOI

[20] D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. | Zbl | MR | DOI

[21] S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H 1 1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. | Zbl | MR | DOI

[22] S.C. Brenner, Q. Guan and L. Sung, Some estimates for virtual element methods. Comput. Methods Appl. Math. 17 (2017) 553–574. | Zbl | MR | DOI

[23] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. In Vol. 15 of Texts in Applied Mathematics. Springer, New York, NY (2008). | Zbl | MR

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

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

[26] V. Calo, M. Cicuttin, Q. Deng and A. Ern, Spectral approximation of elliptic operators by the Hybrid High-Order Method. J. Math. Comp. 88 (2019) 1559–1586. | Zbl | MR | DOI

[27] A. Cangiani, F. Gardini and G. Manzini, Convergence of the mimetic finite difference method for eigenvalue problems in mixed form. Comput. Methods Appl. Mech. Eng. 200 (2011) 1150–1160. | Zbl | MR | DOI

[28] A. Cangiani, E.H. Georgoulis, T. Pryer and O.J. Sutton, A posteriori error estimates for the virtual element method. Numer. Math. 137 (2017) 857–893. | Zbl | MR | DOI

[29] A. Cangiani, V. Gyrya and G. Manzini, The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54 (2016) 3411–3435. | Zbl | MR | DOI

[30] A. Cangiani, G. Manzini, A. Russo and N. Sukumar, Hourglass stabilization and the virtual element method. Int. J. Numer. Meth. Eng. 102 (2015) 404–436. | Zbl | MR | DOI

[31] A. Cangiani, G. Manzini and O.J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37 (2017) 1317–1354. | Zbl | MR

[32] A. Chernov, L. Beirão Da Veiga, L. Mascotto and A. Russo, Basic principles of h p virtual elements on quasiuniform meshes. Math. Models Methods Appl. Sci. 26 (2016) 1567–1598. | Zbl | MR | DOI

[33] H. Chi, L. Beirão Da Veiga and G.H. Paulino, Some basic formulations of the virtual element method (VEM) for finite deformations. Comput. Methods Appl. Mech. Eng. 318 (2017) 148–192. | Zbl | MR | DOI

[34] P. Ciarlet, Basic error estimates for elliptic problems. In: Finite Element Methods (Part 1). In Vol. 2 of Handbook of Numerical Analysis. Elsevier (1991) 17–351. | Zbl | MR | DOI

[35] B. Cockburn, D.A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: M2AN 50 (2016) 635–650. | Zbl | Numdam | MR | DOI

[36] M. Dauge, Benchmark computations for Maxwell equations for the approximation of highly singular solutions. Available at: http://perso.univ-rennes1.fr/monique.dauge/benchmax.html (2004).

[37] D. 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. | Zbl | MR | DOI

[38] D.A. Di Pietro, J. Droniou and G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes. J. Comput. Phys. 355 (2018) 397–425. | Zbl | MR | DOI

[39] D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. In Vol. 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012). | Zbl | MR | DOI

[40] D.A. Di Pietro and A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Math. Acad. Sci. Paris 353 (2015) 31–34. | Zbl | MR

[41] A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation. ESAIM: M2AN 51 (2017) 1367–1385. | Zbl | Numdam | MR | DOI

[42] F. Gardini and G. Vacca, Virtual element method for second order elliptic eigenvalue problems. IMA J. Numer. Anal. 38 (2018) 2026–2054. | Zbl | MR | DOI

[43] P. Grisvard, Singularities in boundary value problems and exact controllability of hyperbolic systems. In: Optimization, Optimal Control and Partial Differential Equations (Iaşi, 1992). In Vol. 107 of Internat. Ser. Numer. Math. Birkhäuser, Basel (1992) 77–84. | Zbl | MR | DOI

[44] T. Kato, Perturbation Theory for Linear Operators. 2nd edition. Springer-Verlag, Berlin (1976). | Zbl | MR

[45] K. Lipnikov and G. Manzini, High-order mimetic method for unstructured polyhedral meshes. J. Comput. Phys. 272 (2014) 360–385. | Zbl | MR | DOI

[46] L. Mascotto, I. Perugia and A. Pichler, Non-conforming harmonic virtual element method: h -and p -versions. J. Sci. Comput. 77 (2018) 1874–1908. | Zbl | MR | DOI

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

[48] D. Mora, G. Rivera and R. Rodrguez, A posteriori error estimates for a virtual element method for the Steklov eigenvalue problem. Comput. Math. Appl. 74 (2017) 2172–2190. | Zbl | MR | DOI

[49] D. Mora, G. Rivera and I. Velásquez, A virtual element method for the vibration problem of kirchhoff plates. ESAIM: M2AN 52 (2018) 1437–1456. | Zbl | Numdam | MR | DOI

[50] C. Talischi, G.H. Paulino, A. Pereira and I.F.M. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim. 45 (2012) 309–328. | Zbl | MR | DOI

[51] G. Vacca, Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74 (2017) 882–898. | Zbl | MR | DOI

[52] G. Vacca, An H1-conforming virtual element for darcy and brinkman equations. Math. Models Methods Appl. Sci. 28 (2018) 159–194. | Zbl | MR | DOI

[53] P. Wriggers, W. Rust and B. Reddy, A virtual element method for contact. Comput. Mech. 58 (2016) 1039–1050. | Zbl | MR | DOI

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