Spectral approximations by the HDG method

TL;DR

This paper analyzes the spectral approximation capabilities of the HDG method for elliptic eigenvalue problems, demonstrating convergence rates and proposing a postprocessing technique for improved eigenvalue accuracy.

Contribution

It establishes convergence rates for eigenvalues and eigenfunctions in the HDG method and introduces a postprocessing approach for faster eigenvalue convergence.

Findings

Eigenvalues converge at rate 2k+1 for smooth eigenfunctions.

Eigenfunctions converge at rate k+1.

Postprocessed eigenvalues can converge at rate 2k+2.

Abstract

We consider the numerical approximation of the spectrum of a second-order elliptic eigenvalue problem by the hybridizable discontinuous Galerkin (HDG) method. We show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate 2k+1 and k+1, respectively. Here k is the degree of the polynomials used to approximate the solution, its flux, and the numerical traces. Our numerical studies show that a Rayleigh quotient-like formula applied to certain locally postprocessed approximations can yield eigenvalues that converge faster at the rate 2k + 2 for the HDG method as well as for the Brezzi-Douglas-Marini (BDM) method. We also derive and study a condensed nonlinear eigenproblem for the numerical traces obtained by eliminating all the other variables.