Stabilization in relation to wavenumber in HDG methods

TL;DR

This paper analyzes the stability and dispersion properties of HDG methods for wave equations with complex and real wavenumbers, providing conditions for unique solvability and optimal stabilization parameters.

Contribution

It establishes a sufficient condition on the HDG stabilization parameter for complex wavenumbers and compares dispersion errors with the hybrid Raviart-Thomas method.

Findings

Certain stabilization parameters cause HDG failure for complex wavenumbers.

Opposite sign of real part of stabilization parameter and imaginary part of wavenumber ensures stability.

Hybrid Raviart-Thomas method exhibits smaller dispersion errors than HDG.

Abstract

Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex. Numerical methods of the Hybrid Discontinuous Galerkin (HDG) type are considered for simulating waves that satisfy the Helmholtz and Maxwell equations. It is shown that these methods, when wrongly used, give rise to singular systems for complex wavenumbers. A sufficient condition on the HDG stabilization parameter for guaranteeing unique solvability of the numerical HDG system, both for Helmholtz and Maxwell systems, is obtained for complex wavenumbers. For real wavenumbers, results from a dispersion analysis are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal that some choices of the stabilization parameter are better than others. To summarize the findings, there are values of the HDG stabilization parameter that will cause the HDG method to fail for complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. When the wavenumber is real, values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors are found on the imaginary axis. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method showed that its wavenumber errors are an order smaller than those of the HDG method.