Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods

We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.Comment: Submitted as a conference proceeding; minor revisio

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor $L^2$ approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

We introduce a space–time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one-dimensional scheme of Kretzschmar et al. (IMA J Numer Anal 36:1599–1635, 2016). Test and trial discrete functions are space–time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space–time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on “tent-pitched” meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds

Modèle SIBC pour la simulation FDTD d'une cible recouverte de métamatériaux

International audiencePour contrôler la diffraction d'une cible éclairée par une onde radar, une technique est d'utiliser des métamatériaux avec des propriétés d'absorption ou de redirection du champ. Leur prise en compte dans les outils de simulation peut s'avérer complexe surtout lorsqu'ils sont appliqués sous forme de couches minces hétérogènes à la surface de la cible. Ceci peut être réalisé en déterminant d'abord un modèle d'impédance de surface dépendant de l'angle d'incidence, qui est ensuite implémenté dans un solveur 3D FDTD

Modèle SIBC pour la simulation FDTD d'une cible recouverte de métamatériaux

International audiencePour contrôler la diffraction d'une cible éclairée par une onde radar, une technique est d'utiliser des métamatériaux avec des propriétés d'absorption ou de redirection du champ. Leur prise en compte dans les outils de simulation peut s'avérer complexe surtout lorsqu'ils sont appliqués sous forme de couches minces hétérogènes à la surface de la cible. Ceci peut être réalisé en déterminant d'abord un modèle d'impédance de surface dépendant de l'angle d'incidence, qui est ensuite implémenté dans un solveur 3D FDTD