Stabilization in relation to wavenumber in HDG methods
This work provides theoretical guidance for choosing stabilization parameters in HDG methods to avoid failure and improve accuracy, which is important for researchers simulating wave propagation in complex media.
The paper identifies that HDG methods can produce singular systems for complex wavenumbers and provides a sufficient condition on the stabilization parameter to guarantee unique solvability. For real wavenumbers, an asymptotic dispersion analysis shows that optimal stabilization parameters lie on the imaginary axis, while the mixed hybrid Raviart-Thomas method achieves an order smaller wavenumber errors than HDG.
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.