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

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 pi(h) is reduced to the solution of the eigenvalue problem on a much coarser grid pi(H) and the solution of a linear algebraic system on the fine grid pi(h). By using spectral approximation theory and Nitsche-Lascaux-Lesaint technique in space H(-1/2)(partial derivative Omega), we prove that the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking H = root h p. And the numerical experiments indicate that when the eigenvalues lambda(k,h) of nonconforming Crouzeix-Raviart element approximate the exact eigenvalues from below, the approximate eigenvalues lambda*(k,h) obtained by the two-grid discretization scheme also approximate the exact ones from below, and the accuracy of lambda*(k,h) is higher than that of lambda(k,h). (C) 2011 Elsevier Inc. All rights reserved.