A Stabilized Virtual Element Method for the Stokes Eigenvalue Problem

In this paper, we propose and analyze a stabilized lower-order and equal-order virtual element method for the Stokes eigenvalue problem. The proposed method employs the same lowest-order conforming virtual element space and the same degrees of freedoms for both velocity and pressure. By adopting a mesh-dependent stabilization term associated with the pressure to the local bilinear form, we show the inf-sup condition and derive the stability for the source problem. We moreover employ the discrete spectral theory of compact operators to study the spectral approximation and obtain the optimal order for the eigenvalues. The proposed stabilized conforming lowest-order VEM of the Stokes eigenvalue problem is new, to the authors’ knowledge. Numerical results on polygonal and polyhedral meshes are presented to illustrate the performance of the proposed method and the theoretical results obtained.