Numerical approximation of a potentials formulation for the elasticity vibration problem

This paper deals with a numerical approximation of the elasticity vibration problem based on a potentials decomposition. Decomposing the displacements field into potentials is a well-known tool in elastodynamics that takes advantage of the decoupling of pressure waves and shear waves inside a homogeneous isotropic media. In the spectral problem on a bounded domain, this decomposition decouples the elasticity equations into two Laplacian-like equations that only interact at the boundary. We show that spurious eigenvalues appear when Lagrangian finite elements are used to discretize the problem. Then, we propose an alternative weak formulation which avoids this drawback. A finite element discretization of this weak formulation based again on Lagrangian finite elements is proposed and tested by means of some numerical experiments, which show convergence and absence of spurious modes.