# Virtual Element for the Buckling Problem of Kirchhoff-Love plates

## TL;DR

This paper introduces a high-order virtual element method for accurately solving the complex fourth-order buckling eigenvalue problem in Kirchhoff-Love plates on polygonal meshes, with proven spectral approximation and optimal error estimates.

## Contribution

It develops a $C^1$ conforming virtual element discretization of arbitrary order for the buckling problem, with theoretical spectral approximation and error analysis.

## Key findings

- The scheme accurately approximates the spectrum of the buckling problem.
- Optimal order error estimates are established for eigenfunctions.
- Numerical experiments confirm theoretical results on various meshes.

## Abstract

In this paper, we develop a virtual element method (VEM) of high order to solve the fourth order plate buckling eigenvalue problem on polygonal meshes. We write a variational formulation based on the Kirchhoff-Love model depending on the transverse displacement of the plate. We propose a $C^1$ conforming virtual element discretization of arbitrary order $k\ge2$ and we use the so-called Babuska--Osborn abstract spectral approximation theory to show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the buckling modes (eigenfunctions) and a double order for the buckling coefficients (eigenvalues). Finally, we report some numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes.