Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element

TL;DR

This paper develops asymptotically exact a posteriori error estimates for eigenvalues using nonconforming Crouzeix-Raviart elements, enabling high-accuracy eigenvalue approximations with efficient post-processing.

Contribution

It introduces two novel methods to accurately estimate errors for eigenvalues with nonconforming elements, overcoming previous challenges related to nonconformity and consistency errors.

Findings

Achieves asymptotically exact error estimates for eigenvalues

Enables high-accuracy eigenvalue approximations with a single eigenproblem solve

Demonstrates effectiveness through theoretical analysis and numerical tests

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 one is the construction of high accuracy gradient recovery algorithms, the second one 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 allows the use of high accuracy gradient recovery techniques. Further, a post-processing 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 to solve only one eigenvalue problem and admits high accuracy eigenvalue approximations both theoretically and numerically.