A Penalized Crouzeix-Raviart Element Method for Second Order Elliptic Eigenvalue Problems

TL;DR

This paper introduces a penalized Crouzeix-Raviart element method for second order elliptic eigenvalue problems, enabling high-accuracy eigenvalue approximations through adjustable penalty parameters and an algorithm for optimal tuning.

Contribution

It proposes a novel penalized Crouzeix-Raviart method with a penalty tuning algorithm, improving eigenvalue approximation accuracy for elliptic operators.

Findings

Discrete eigenvalues can be made highly accurate with proper penalty tuning

The method effectively balances local approximation and global continuity

Numerical tests confirm the method's high performance

Abstract

In this paper we propose a penalized Crouzeix-Raviart element method for eigenvalue problems of second order elliptic operators. The key idea is to add a penalty term to tune the local approximation property and the global continuity property of the discrete eigenfunctions. The feature of this method is that by adjusting the penalty parameter, the resulted discrete eigenvalues can be in a state of "chaos", and consequently a large portion of them can be reliable and approximate the exact ones with high accuracy. Furthermore, we design an algorithm to select such a quasi-optimal penalty parameter. Finally, we provide numerical tests to demonstrate the performance of the proposed method.