Spectral approximation of elliptic operators by the Hybrid High-Order method

TL;DR

This paper analyzes the spectral approximation of second-order elliptic operators using the Hybrid High-Order method, proving convergence rates for eigenvalues and eigenfunctions and demonstrating superconvergence in certain cases.

Contribution

It provides new theoretical convergence rates for the HHO method's eigenvalue and eigenfunction approximations, improving upon previous error estimates for similar methods.

Findings

Eigenvalues converge as h^{2t} and eigenfunctions as h^{t} in H^1-seminorm.

For smooth eigenfunctions, eigenvalues converge at rate h^{2k+2} and eigenfunctions at h^{k+1}.

Numerical experiments confirm theoretical rates and show superconvergence for eigenvalues in 1D.

Abstract

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $k\geq0$. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. Our theoretical findings, which improve recent error estimates for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension for smooth eigenfunctions that the eigenvalues superconverge as $h^{2k+4}$ for a specific value of the stabilization parameter.