A mixed virtual element method for the vibration problem of clamped Kirchhoff plate

Abstract

In this paper, we give a presentation of virtual element method for the approximation of the vibration problem of clamped Kirchhoff plate, which involves the biharmonic eigenvalue problem. Following the theory of Babǔska and Osborn, the error estimates of the discrete scheme for the degree k ≥ 2 of polynomials are standard results. However, when considering the case k = 1, we can not apply the technical framework of classical eigenvalue problem directly. Based on the spectral approximation theory, the theory of mixed virtual element method and mixed finite element method for the Stokes problem, the convergence analysis for eigenvalues and eigenfunctions is analyzed and proved. Finally, some numerical experiments are reported to show that the proposed numerical scheme can achieve the optimal convergence order.