A mixed virtual element method for the vibration problem of clamped Kirchhoff plate
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A mixed virtual element method for the vibration problem of clamped Kirchhoff plate Jian Meng1 · Liquan Mei1 Received: 16 February 2020 / Accepted: 21 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we give a presentation of virtual element method for the approximation of the vibration problem of clamped Kirchhoff plate, which involves the biharmonic eigenvalue problem. Following the theory of Babˇuska and Osborn, the error estimates of the discrete scheme for the degree k ≥ 2 of polynomials are standard results. However, when considering the case k = 1, we can not apply the technical framework of classical eigenvalue problem directly. Based on the spectral approximation theory, the theory of mixed virtual element method and mixed finite element method for the Stokes problem, the convergence analysis for eigenvalues and eigenfunctions is analyzed and proved. Finally, some numerical experiments are reported to show that the proposed numerical scheme can achieve the optimal convergence order. Keywords Virtual element method · Polygonal meshes · Biharmonic eigenvalue problem · Spectral approximation · Error estimates Mathematics Subject Classification (2010) 65N30 · 65N25 · 74K10
1 Introduction In modern scientific and engineering applications, the numerical solution of eigenvalue problem is of fundamental importance in many fields [16, 18]. Different numerical methods have been proposed to solve eigenvalue problems, such as finite difference method [40], finite element method [5, 17, 53], spectral method [26], Communicated by: Lourenco Beirao da Veiga Liquan Mei
[email protected] Jian Meng [email protected] 1
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China
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Adv Comput Math
(2020) 46:68
hybrid high-order method [23], discontinuous Galerkin method [29, 42], and weak Galerkin method [56]. The virtual element method (VEM) was introduced in [6], which was born as an extension of the finite element method (FEM) to polygons or polyhedra, and it is also a variational formulation of the mimetic finite difference (MFD) method [24]. The main idea of VEM is that its discrete space consists of the polynomials and additional non-polynomials defined by local PDE problems implicitly. In particular, one has to compute suitable projections on polynomial spaces for the computation of non-polynomial functions, typically unknown, virtual functions. Consequently, the VEM has the capability of dealing with very general polygonal meshes and constructing highly regular discrete spaces [13]. So far, the VEM has been applied for solving many problems; for example, see [3, 10, 12, 15, 21]. Moreover, the VEM has also been applied to a variety of eigenvalue problems, such as the Steklov eigenvalue problem [45, 47, 48], the Laplacian eigenvalue problem [27, 28, 34, 35, 44], the acoustic vibration problem [14], the vibration problem and buckling problem of Kirchhoff plate [43, 49, 51], the tran
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