A Lowest-Order Free-Stabilization Virtual Element Method for the Laplacian Eigenvalue Problem

In this paper, we propose a Virtual Element Method (VEM) for the Laplacian eigenvalue problem, which is designed to avoid the requirement of the stabilization terms in standard VEM bilinear forms. In the present method, the constructions of the bilinear forms depend on higher order polynomial projection. To exactly compute the bilinear forms, we need to modify the virtual element space associated to the higher order polynomial projection. Meanwhile, the continuity and coercivity of the discrete VEM bilinear forms depend on the number of vertices of the polygon. By the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove correct spectral approximation and error estimates for the VEM discrete scheme. Finally, we show numerical examples to verify the theoretical results, including the Laplace eigenvalue problem and the Steklov eigenvalue problem.