Finite element approximations for the eigenvalue problem of the Laplace operator is discussed. A gradient recovery scheme is proposed to enhance the accuray of the numerical eigenvalues. By reconstructing the numerical solution and its gradient, it is possible to produce more accurate numerical eigenvalues. Furthermore, the recovered gradient can be used to form an a posteriori error estimator to guide an adaptive mesh refinement. Therefore, this method works not only for structured meshes, but also for unstructured and adaptive meshes.

Additional computational cost for this post-processing technique is only $O(N)$ ($N$ is the total degrees of freedom), comparing with $O(N^2)$ cost for the original problem.