A two-grid method of the non-conforming Crouzeix–Raviart element for the Steklov eigenvalue problem

Abstract

This paper discusses a high efficient scheme for the Steklov eigenvalue problem. A two-grid discretization scheme of nonconforming Crouzeix–Raviart element is established. With this scheme, the solution of a Steklov eigenvalue problem on a fine grid π h is reduced to the solution of the eigenvalue problem on a much coarser grid π H and the solution of a linear algebraic system on the fine grid π h . By using spectral approximation theory and Nitsche–Lascaux–Lesaint technique in space H - 1 2 ( ∂ Ω ) , we prove that the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking H = h . And the numerical experiments indicate that when the eigenvalues λ k, h of nonconforming Crouzeix–Raviart element approximate the exact eigenvalues from below, the approximate eigenvalues λ k , h ∗ obtained by the two-grid discretization scheme also approximate the exact ones from below, and the accuracy of λ k , h ∗ is higher than that of λ k, h .