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

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 ≥ 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. Mathematics Subjects Classification: 65N15, 65N30, 65N35, 35J05

Higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1D

“…Another interesting work is the hybrid high-order spectral approximation in [14], which results in two extra orders of superconvergence for the eigenvalues and one extra order of superconvergence for the eigenfunctions. However, all these methods were dealing with elliptic differential operators where the diffusion coefficients are continuous.…”

The p- and hp-versions of the virtual element method for elliptic eigenvalue problems

“…being α * (p) and α * (p) defined in (21), β * being defined in (26), k * and k * being defined in (1), and ν * being defined in (2), and having set Π 0…”

“…We begin with the first one. Applying the definitions of c and c n in (5) and (24), respectively, using (26) and the properties of the L 2 projector, and applying a Poincaré inequality, we deduce…”

“…Despite the novelty of this method, virtual elements for the approximation of eigenvalues have been applied to a plethora of different problems, such as the Poisson problem [37,38], the Poisson problem with a potential term [30], the Steklov eigenvalue problem [48,49], transmission problems [51], the vibration problem of Kirchhoff plates [50], and the acoustic vibration problem [21]. We also highlight that the approximation of eigenvalues with polygonal methods has been targeted in the context of the hybrid-high order method [26] and of the mimetic finite differences [27].…”

A hybrid high‐order formulation for a Neumann problem on polytopal meshes

“…We have proved the convergence of the method, with the optimal rates of convergence when the exact solution is smooth enough: order k + 1 for the flux error, and k + 2 for potential error, when piecewise polynomial of degree at most k are considered for the corresponding approximations. This technique can deal with hanging nodes, as in the Refined mesh (Figure 1c), and also with triangular, quadrilateral and hexagonal meshes (Figure 1a, b, This library has been used to solve many problems as those described in [21][22][23][24][25][26][27][28][29]. On the other hand, HArDCore (Hybrid Arbitrary Degree::Core, https://github.com/jdroniou/HArDCore) is a C++ code focused on HHO methods, but it can be useful for a wide range of hybrid methods.…”