Geodesic
A virtual element method for the vibration problem of Kirchhoff plates
Mora, David1  ; Rivera, Gonzalo2 ; Velásquez, Iván3
1 GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Concepción, Chile
2 Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, Osorno, Chile
3 Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Concepción, Chile
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1437-1456

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The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an H2(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

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Accepté le :
DOI : 10.1051/m2an/2017041
Classification : 65N25, 65N30, 74K20
Keywords: Virtual element method, Kirchhoff plates, spectral problem, error estimates

Mora, David 1 ; Rivera, Gonzalo 2 ; Velásquez, Iván 3

1 GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Concepción, Chile
2 Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, Osorno, Chile
3 Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Concepción, Chile 
@article{M2AN_2018__52_4_1437_0,
     author = {Mora, David and Rivera, Gonzalo and Vel\'asquez, Iv\'an},
     title = {A virtual element method for the vibration problem of {Kirchhoff} plates},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1437--1456},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {4},
     year = {2018},
     doi = {10.1051/m2an/2017041},
     mrnumber = {3875292},
     zbl = {1407.65274},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017041/}
}
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Mora, David; Rivera, Gonzalo; Velásquez, Iván. A virtual element method for the vibration problem of Kirchhoff plates. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1437-1456. doi: 10.1051/m2an/2017041

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