Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods

TL;DR

This paper reveals that HDG methods can simultaneously produce upper and lower eigenvalue bounds by tuning stabilization parameters, leading to a high-accuracy, cost-effective eigenvalue computation algorithm.

Contribution

It introduces a novel approach to achieve both bounds with HDG methods and designs a high-accuracy eigenvalue algorithm based on this insight.

Findings

HDG methods can produce both upper and lower eigenvalue bounds.

Tuning stabilization parameters controls the bounds.

Numerical results confirm the theoretical properties.

Abstract

In this paper, we observe an interesting phenomenon for a hybridizable discontinuous Galerkin (HDG) method for eigenvalue problems. Specifically, using the same finite element method, we may achieve both upper and lower eigenvalue bounds simultaneously, simply by the fine tuning of the stabilization parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence rate at a lower computational cost. Meanwhile, we demonstrate that certain type of HDG methods can only provide upper bounds. As a by-product, the asymptotic upper bound property of the Brezzi-Douglas-Marini mixed finite element is also established. Numerical results supporting our theory are given.