The main goal of this paper is to present a new strategy of increasing the convergence rate for the numerical solution of the linear finite element eigenvalue problems. This is done by introducing a postprocessing technique for eigenvalues. The postprocessing technique deals with solving a corresponding linear elliptic problem. We prove that the proposed algorithm has the superconvergence property of the eigenvalues and this improvement is attained at a small computational cost. Thus, good finite element approximations for eigenvalues are obtained on the coarse mesh. The numerical examples presented and discussed here show that the resulting postprocessing method is computationally more efficient than the method to which it is applied.

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In this paper we establish the convergence and the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a lumped mass finite-element approximation. Various aspects of lumped mass techniques have been discussed for such eigenvalue problems by Fix (1972), Ishihara (1977), Strang and Fix (1973) and Tong et al. (1971), among others. In our approach the lumping of the mass matrix results from the use of a Lobatto quadrature formula for the integrals over rectangular Lagrange finite elements of degree k