Spectral approximation of elliptic operators by the Hybrid High-Order method
For researchers in numerical analysis and computational PDEs, this provides rigorous convergence rates for eigenvalue approximation using HHO methods, improving upon existing HDG error estimates.
The paper proves that the Hybrid High-Order method approximates eigenvalues of elliptic operators with convergence rate \(h^{2k+2}\) and eigenfunctions with \(h^{k+1}\) for smooth eigenfunctions, improving on HDG methods. Numerical examples confirm the rates and show superconvergence of eigenvalues as \(h^{2k+4}\) for a specific stabilization parameter in 1D.
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.