A new nonconforming mixed finite element scheme for second order eigenvalue problem

Abstract

A new nonconforming mixed finite element method (MFEM for short) is established for the Laplace eigenvalue problem. Firstly, the optimal order error estimates for both the eigenvalue and eigenpair (the original variable u and the auxiliary variable p→=∇u) are deduced, the lower bound of eigenvalue is estimated simultaneously. Then, by use of the special property of the nonconforming EQ1rot element (the consistency error is of order O(h2) in broken H1-norm, which is one order higher than its interpolation error), the techniques of integral identity and interpolation postprocessing, we derive the superclose and superconvergence results of order O(h2) for u in broken H1-norm and p→ in L2-norm. Furthermore, with the help of asymptotic expansions, the extrapolation solution of order O(h3) for eigenvalue is obtained. Finally, some numerical results are presented to validate our theoretical analysis.