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

摘要

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 πh is reduced to the solution of the eigenvalue problem on a much coarser grid πH and the solution of a linear algebraic system on the fine grid πh. By using spectral approximation theory and Nitsche–Lascaux–Lesaint technique in space H-12(∂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 λk,h of nonconforming Crouzeix–Raviart element approximate the exact eigenvalues from below, the approximate eigenvalues λk,h∗ obtained by the two-grid discretization scheme also approximate the exact ones from below, and the accuracy of λk,h∗ is higher than that of λk,h.