A virtual element method for the vibration problem of Kirchhoff plates

Abstract

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 H 2 (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. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. 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.

The p- and hp-versions of the virtual element method for elliptic eigenvalue problems

“…The proof follows the line e.g. of that of [50,Theorem 4.3]. Let w n be a discrete eigenfunction associated with the discrete eigenvalue λ n , and let w and λ be the corresponding exact eigenfunction and eigenvalue, respectively.…”

The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows

“…with α * and α * positive constants independent of the element E. For instance, a standard choice (26) and properties (39) imply that the discrete form a E h (·, ·) satisfies the consistency and stability properties. The global approximated bilinear form a h (·, ·) : V h × V h → R is defined by simply summing the local contributions:…”

A C1 Virtual Element Method on polyhedral meshes

“…Although the main motivation of VEM is the use of general polytopal partitions, its flexibility can lead also to different advantages. One, initiated in [28,18] and further investigated in [3,11,46,47], is the possibility to develop C 1 conforming spaces, still keeping the accuracy order and the number of degrees of freedom at a reasonable level. More specifically, the lowest degree requires only three degrees of freedom for each vertex independently for the shape of the elements.…”