A finite element method for the buckling problem of simply supported Kirchhoff plates

The aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported   Kirchhoff plates subjected to general plane stress tensor. We introduce an auxiliary variable w:=Δuw:=Δu (with uu representing the displacement of the plate) to write a variational formulation of the spectral problem. We propose a conforming discretization of the problem, where the unknowns are approximated by piecewise linear and continuous finite elements. We show 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 present some numerical experiments supporting our theoretical results.