# A virtual element method for the vibration problem of Kirchhoff plates

## TL;DR

This paper introduces a virtual element method for accurately approximating the vibration spectra of thin Kirchhoff plates on polygonal meshes, with proven optimal error estimates and confirmed through numerical experiments.

## Contribution

It develops a simple, conforming VEM scheme for Kirchhoff plate vibrations that achieves optimal spectral approximation and error estimates, extending to non-convex polygons.

## Key findings

- The scheme accurately approximates eigenvalues and eigenfunctions.
- Numerical experiments confirm theoretical error estimates.
- The method works on various mesh types, including non-convex polygons.

## 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 $H^2(\Omega)$ 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 schemeprovides 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.