Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods

TL;DR

The paper addresses computing Laplacian eigenvalues using HDG methods and shows that a gradient-based HDG discretization can yield both lower and upper eigenvalue bounds by tuning the stabilization parameter $\gamma$, while a divergence-based HDG discretization yields only upper bounds. It introduces a high-accuracy post-processing algorithm that combines bound sequences to achieve superior convergence, and it proves a by-product upper bound for Brezzi-Douglas-Marini elements. Theoretical results are complemented by extensive numerical experiments on squares with standard and high-order elements and with discontinuous coefficients, confirming quadratic convergence of discrete eigenvalues and quartic convergence after post-processing. The work provides a cost-efficient, reliable HDG-based framework for eigenvalue computations with tunable bounds and accelerated accuracy.

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.

Figures (4)

• Figure 1: The errors of the first eigenvalue $\lambda_h^L, \lambda_h^U$ and $\widehat{\lambda}_h$
• Figure 2: The errors of the first eigenvalue $\lambda_h^L, \lambda_h^U$ and $\widehat{\lambda}_h$, $k = 2$
• Figure 3: The partition of domain $\Omega$
• Figure :

Theorems & Definitions (5)

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