A locking-free mixed virtual element discretization for the elasticity eigenvalue problem

Abstract: In this paper, we introduce a mixed virtual element method to approximate the eigenvalues and eigenfunctions of the two-dimensional elasticity eigenvalue problem. Under standard assumptions on the meshes, we prove the convergence of the discrete solution operator to the continuous one as the mesh size tends to zero. Using the theory of compact operators, we analyze the convergence of the method and derive error estimates for both the eigenvalues and eigenfunctions. We validate our theoretical results with a series of numerical tests, in which we compute convergence orders and show that the method is locking-free and capable of accurately approximating the spectrum independently of the shape of the polygons on the meshes.