Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation (Q2862520)

The class of hybridizable discontinuous Galerkin (HDG) methods was first introduced in the context of diffusion problems as an alternative to other existing discontinuous Galerkin methods, displaying a significantly smaller number of coupled unknowns and a better accuracy. These methods were then generalized to convection-diffusion equations, the heat equation and the incompressible and compressible Navier-Stokes equations, among others.NEWLINENEWLINEIn [J. Comput. Phys. 230, No. 10, 3695--3718 (2011; Zbl 1364.76093)], \textit{N. C. Nguyen} et al. extended the family of HDG methods to wave equations in acoustics and elastodynamics, reporting numerical experiments that showed that the velocity and the gradient converge with the optimal order of \(k+1\) in the \(L^2\)-norm when polynomials of degree \(k \geq 0\) are used. In addition, a local post processing allowed to get superconvergence in the \(L^2\)-norm of order \(k+2\) for the original scalar unknown with polynomials of degree \(k \geq 1\). It is the purpose of the paper under review to provide a sound mathematical basis for these observed behavior of the methods. Specifically, an a priori error analysis of the method for the acoustic wave equation in the time continuous case is carried out, rigorously proving the empirical results under suitable hypothesis on the functions involved in the equation. The proof of the convergence properties relies on a projection-based technique previously introduced for diffusion problems. It turns out that, although the problem is purely hyperbolic, it is still possible to use elliptic regularity results in this setting.