On perfectly matched layers for discontinuous Petrov–Galerkin methods

Abstract

In this article, several discontinuous Petrov-Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov-Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two-and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

The DPG-star method

“…Note that abstract ultraweak formulations encompass many physical models besides the Poisson example given above. Other important examples with similar functional settings include convection-dominated diffusion [25,14], Stokes flow [47,13], linear elasticity [7,12], and acoustics [34,43,50].…”

A 3D DPG Maxwell approach to nonlinear Raman gain in fiber laser amplifiers

“…Towards this end, PMLs have been widely used in finite element implementations. Most notably, [8] and the recent work [62] use DPG methods to implement ultraweak formulations for various wave propagation phenomena. In this case, a stretched coordinate PML for the ultraweak formulation is used, with stretching along the z-axis, since outgoing waves need to be attenuated in only the z-direction.…”

“…In this case, a stretched coordinate PML for the ultraweak formulation is used, with stretching along the z-axis, since outgoing waves need to be attenuated in only the z-direction. We refer the interested reader to [62] and Appendix A for implementation details. Figure 2 indicates Figure 2.…”

“…The theoretical foundations of the subject were established in [16,17,10,33,11]. This active area of research has been successfully employed to problems in linear elasticity [38,29], time harmonic wave propagation, including DPG versions of perfectly matched layers (PMLs) [54,62], compressible and incompressible Navier-Stokes [25,23,12,24], fluid flow [39], viscoelasticity [27] and space-time formulations [20,35,25,22]. Moreover, versions of DPG for polygonal meshes have been introduced in [61].…”

“…Thus, we are able to compute the cross sectional power via the time-averaged Poynting vector (irradiance) using the DPG trace variables. In this context, we also note the fact that we utilize a frequency domain PML (see [6,60,13,31,45,46,62]), which is also implemented using the ultraweak DPG formulation. As we shall explain, the use of a PML is critically important, and one cannot adequately observe the gain phenomenon with simpler impedance boundary conditions.…”