Spectral approximations by the HDG method
Provides rigorous convergence rates for spectral approximations using HDG methods, benefiting numerical analysts and engineers solving eigenvalue problems.
The paper proves that the HDG method approximates eigenvalues of second-order elliptic problems at rate 2k+1 and eigenfunctions at rate k+1, and shows that a postprocessed Rayleigh quotient yields faster convergence at rate 2k+2 for both HDG and BDM methods.
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.