Stabilization in relation to wavenumber in HDG methods

Jay Gopalakrishnan,Stéphane Lanteri,Ronan Perrussel,Nicole Olivares

doi:10.1186/s40323-015-0032-x

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.

Highlights

Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex
Results on unisolvent stabilization We turn to the question of how we can choose a value for the stabilization parameter τ that will guarantee that the local matrices are not singular
In the p = 0 case, after statically condensing the element matrices and following the procedure leading to (25), we find that the discrete wavenumber kh for the hybrid Raviart–Thomas (HRT) method satisfies the two dimension(al) (2D) dispersion relation c12 + c22 2(hk)2 − 12 + c12c22 4(hk)2 + 48 +2 − 24 = 0, (36)

Summary

Introduction

Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex. Wave propagation through complex structures, composed of both propagating and absorbing media, are routinely simulated using numerical methods. Among the various numerical methods used, the Hybrid Discontinuous Galerkin (HDG) method has emerged as an attractive choice for such simulations. The easy passage to high order using interface unknowns, condensation of all interior variables, availability of error estimators and adaptive algorithms, are some of the reasons for the adoption of HDG methods. It is important to design numerical methods that remain stable as the wavenumber varies in the complex plane. In applications like computational lithography, one finds absorbing materials with complex refractive index in parts of the domain of simulation.

Methods

Results

Conclusion