The nonconforming Virtual Element Method for eigenvalue problems

Gardini, Francesca; Manzini, Gianmarco; Vacca, Giuseppe

doi:10.1051/m2an/2018074

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

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 749-774.

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Résumé

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.

MR Zbl

DOI : 10.1051/m2an/2018074

Classification : 65N30, 65N25
Mots-clés : Nonconforming virtual element, eigenvalue problem, polygonal meshes

Affiliations des auteurs :

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

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     author = {Gardini, Francesca and Manzini, Gianmarco and Vacca, Giuseppe},
     title = {The nonconforming {Virtual} {Element} {Method} for eigenvalue problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {749--774},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {3},
     year = {2019},
     doi = {10.1051/m2an/2018074},
     mrnumber = {3959470},
     zbl = {1431.65214},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018074/}
}

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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. http://www.numdam.org/articles/10.1051/m2an/2018074/

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