Error Analysis of an HDG Method with Impedance Traces for the Helmholtz Equation

TL;DR

This paper presents a new analysis of a hybrid discontinuous Galerkin method for the Helmholtz equation, demonstrating stability and optimal convergence without resolution conditions, suitable for iterative solvers.

Contribution

It introduces a novel HDG method with impedance traces that is independent of wavenumber, mesh size, and polynomial degree, with proven stability and convergence.

Findings

Unique discrete solvability without resolution condition

Optimal convergence rates with respect to mesh size

Method tailored for static condensation and iterative solvers

Abstract

In this work, a novel analysis of a hybrid discontinuous Galerkin method for the Helmholtz equation is presented. It uses wavenumber, mesh size and polynomial degree independent stabilisation parameters leading to impedance traces between elements. With analysis techniques based on projection operators unique discrete solvability without a resolution condition and optimal convergence rates with respect to the mesh size are proven. The considered method is tailored towards enabling static condensation and the usage of iterative solvers.