Asymptotic expansions of eigenvalues by both the Crouzeix-Raviart and enriched Crouzeix-Raviart elements

TL;DR

This paper derives optimal asymptotic expansions for eigenvalues obtained from Crouzeix-Raviart and enriched Crouzeix-Raviart elements, enabling fourth-order convergence of extrapolated eigenvalues under smooth eigenfunction conditions.

Contribution

The paper introduces a novel approach using the relation to Raviart-Thomas elements and their superclose properties to derive eigenvalue expansions for nonconforming elements.

Findings

Derived asymptotic expansions for eigenvalues

Achieved fourth-order convergence with extrapolation

Utilized the relation to Raviart-Thomas elements for analysis

Abstract

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix--Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart--Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart--Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.