The Weak Galerkin and Crouzeix-Raviart element method for elastic eigenvalue problems

TL;DR

This paper develops a weak Galerkin method for elastic eigenvalue problems, proving it to be locking-free and providing lower bounds, with additional analysis of Crouzeix-Raviart elements and supporting numerical experiments.

Contribution

Introduces an abstract framework for weak Galerkin methods applied to elastic eigenvalue problems, establishing their lower bound properties and extending analysis to Crouzeix-Raviart elements.

Findings

WG method is locking-free and provides asymptotic lower bounds.

Crouzeix-Raviart element also exhibits lower bound properties.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we first introduce an abstract framework to solve the eigenvalue problem by weak Galerkin (WG) method. By the application of the framework, WG method is proved to be locking-free and gives asymptotic lower bounds for the elastic eigenvalue problem. Also, we analyze the lower bound property for Crouzeix-Raviart (CR) element as an extensional work. In the end, we present some numerical experiments to support the theoretical results.