A virtual element method for the Laplacian eigenvalue problem in mixed form

Abstract

In this paper, the virtual element method for the approximation of Laplacian eigenvalue problem in mixed form is studied. We show that the discrete form satisfies the hypotheses required by the Brezzi-Babǔska theory. Under some assumptions on polygonal meshes, we employ the spectral theory of compact operators to prove the spectral approximation and the optimal order for the eigenvalues. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.