Spectral approximation of elliptic operators by the hybrid high-order method

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 ≥ 0. 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 ∈ [s, k + 1] depends on the smoothness of the eigenfunctions, and s > 1/2 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.