Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements

Abstract

An algorithm is proposed to give explicit lower bounds of the Stokes eigenvalues by utilizing two nonconforming finite element methods: Crouzeix–Raviart (CR) element and enriched Crouzeix–Raviart (ECR) element. Compared with the existing literatures which give lower eigenvalue bounds under the asymptotic condition that the mesh size is “small enough”, the proposed algorithm in this paper drops the asymptotic condition and provide explicit lower bounds even for a rough mesh. Numerical experiments are also performed to validate the theoretical results.