TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS

This paper extends the two-grid discretization scheme of the conforming finite elementsproposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconformingfinite elements for eigenvalue problems. In particular, two two-grid discretization schemesbased on Rayleigh quotient technique are proposed. By using these new schemes, thesolution of an eigenvalue problem on a fine mesh is reduced to that on a much coarsermesh together with the solution of a linear algebraic system on the fine mesh. The resultingsolution still maintains an asymptotically optimal accuracy. Comparing with the two-griddiscretization scheme of the conforming finite elements, the main advantages of our newschemes are twofold when the mesh size is small enough. First, the lower bounds of theexact eigenvalues in our two-grid discretization schemes can be obtained. Second, the firsteigenvalue given by the new schemes has much better accuracy than that obtained bysolving the eigenvalue problems on the fine mesh directly.