Asymptotically Exact A Posteriori Error Estimates of Eigenvalues by the Crouzeix--Raviart Element and Enriched Crouzeix--Raviart Element

Abstract

Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix--Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties: the first is the construction of high accuracy gradient recovery algorithms, and the second is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are proposed to solve the second difficulty in the present paper. In particular, this solution allows the use of high accuracy gradient recovery techniques. Further, a postprocessing algorithm is designed by utilizing asymptotically exact a posteriori error estimators to construct the weights of a combination of two approximate eigenvalues. This algorithm requires solving only one eigenvalue problem and admits high accuracy eigenvalue approximations both theoretically and numerically.