Two-grid Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems

Abstract

This paper extends the two-grid discretization scheme of the conforming flnite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming flnite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a flne mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the flne mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming flnite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the flrst eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the flne mesh directly.