A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number

TL;DR

This paper provides error estimates and stability analysis for a hybridizable discontinuous Galerkin method applied to the Helmholtz equation with high wave numbers, demonstrating stability without mesh constraints and verifying results numerically.

Contribution

It introduces a stability proof for the HDG method for high wave number Helmholtz problems without mesh restrictions and analyzes convergence dependence on key parameters.

Findings

HDG method is stable for high wave numbers without mesh constraints

Convergence depends on wave number, mesh size, and polynomial degree

Numerical experiments confirm theoretical error estimates

Abstract

This paper analyzes the error estimates of the hybridizable discontinuous Galerkin (HDG) method for the Helmholtz equation with high wave number in two and three dimensions. The approximation piecewise polynomial spaces we deal with are of order $p\geq 1$. Through choosing a specific parameter and using the duality argument, it is proved that the HDG method is stable without any mesh constraint for any wave number $\kappa$. By exploiting the stability estimates, the dependence of convergence of the HDG method on $\kappa,h$ and $p$ is obtained. Numerical experiments are given to verify the theoretical results.