Mesh-Uniform Stability and Error Estimates for HDG Discretizations of the Helmholtz Equation

Abstract

We revisit the hybridizable discontinuous Galerkin method of Cui and Zhang for the Helmholtz equation with first-order absorbing boundary condition. Their analysis proves stability without imposing a mesh constraint coupling $h$ and $κ$, but the explicit stability bound still contains negative powers of the mesh size. We prove that this mesh-dependent blow-up is not intrinsic to the HDG discretization. Under the same geometric and mesh framework as in the reference analysis, for fixed polynomial degree and for stabilization parameters uniformly bounded above and below, we establish a generalized stability estimate whose constant is independent of the mesh size. For every prescribed compact interval $0<κ_0\leκ\leκ_1<\infty$, the stability constant can be chosen locally uniformly with respect to $κ$. The proof replaces the Rellich-identity and inverse-estimate argument by a compactness argument based on the HDG discrete distributional gradient and the Crouzeix--Raviart lifting mechanism. As consequences, we obtain well-posedness, convergence for minimal $L^2$ data, and projection-based error estimates in which the negative powers of the mesh size appearing in the previous theory are removed. Numerical experiments confirm the predicted mesh-uniform behavior and show a clear contrast with the mesh-dependent growth suggested by the previous