A priori and a posteriori error estimates for a virtual element method for the non-self-adjoint Steklov eigenvalue problem

In this paper we analyze a virtual element method (VEM) for the non-self-adjoint Steklov eigenvalue problem. The conforming VEM on polytopal meshes is used for discretization. We analyze the correct spectral approximation of the discrete scheme and prove an a priori error estimate for the discrete eigenvalues and eigenfunctions. The convergence order of a discrete eigenvalue may decrease if the corresponding eigenfunction has a singularity and it can be improved on a locally refined mesh. The VEM has great flexibility in handling computational meshes. These facts motivate us to construct a computable a posteriori error estimator for the VEM and prove its reliability and efficiency. This estimator can be applied to very general polytopal meshes with hanging nodes. Finally, we show numerical examples to verify the theoretical results, including optimal convergence of discrete eigenvalues on uniformly refined meshes of a square domain and a cube domain, and we demonstrate the efficiency of the estimator on adaptively refined meshes on an L-shaped domain and also discuss the influence of stabilization parameters on the virtual element approximation.