Guaranteed Lower and upper bounds for eigenvalues of second order elliptic operators in any dimension

TL;DR

This paper introduces a novel method for obtaining guaranteed lower and upper bounds for eigenvalues of second order elliptic operators in any dimension, using a generalized Crouzeix-Raviart element and a simple post-processing approach.

Contribution

It presents a new, efficient method that requires only solving one eigenvalue problem and no prior eigenvalue information, with proven asymptotic bounds and practical guarantees.

Findings

The method provides asymptotic lower bounds for eigenvalues.

A simple, inexpensive approach for guaranteed upper bounds is also developed.

Numerical results confirm the theoretical guarantees.

Abstract

In this paper, a new method is proposed to produce guaranteed lower bounds for eigenvalues of general second order elliptic operators in any dimension. Unlike most methods in the literature, the proposed method only needs to solve one discrete eigenvalue problem but not involves any base or intermediate eigenvalue problems, and does not need any a priori information concerning exact eigenvalues either. Moreover, it just assumes basic regularity of exact eigenfunctions. This method is defined by a novel generalized Crouzeix-Raviart element which is proved to yield asymptotic lower bounds for eigenvalues of general second order elliptic operators, and a simple post-processing method. As a byproduct, a simple and cheap method is also proposed to obtain guaranteed upper bounds for eigenvalues, which is based on generalized Crouzeix-Raviart element approximate eigenfunctions, an averaging interpolation from the the generalized Crouzeix-Raviart element space to the conforming linear element space, and an usual Rayleigh-Ritz procedure. The ingredients for the analysis consist of a crucial projection property of the canonical interpolation operator of the generalized Crouzeix-Raviart element, explicitly computable constants for two interpolation operators. Numerics are provided to demonstrate the theoretical results.