Trefftz Discontinuous Galerkin methods for scattering by periodic structures

TL;DR

This work develops a TDG framework for plane-wave scattering by periodic gratings governed by the 2D Helmholtz equation $\Delta u + k^2 \varepsilon u = 0$, using plane-wave Trefftz spaces and a truncated DtN map on the cell to enforce radiation. The method is proven coercive and quasi-optimal, with an explicit Rellich-identity–based stability bound that remains robust away from Rayleigh–Wood anomalies, and it supports exact analytic assembly for polygonal meshes. Numerical experiments across multi-material layers, corners, and impenetrable obstacles confirm exponential convergence in the plane-wave count and demonstrate the method’s ability to handle complex geometries and guided modes, including non-unique configurations. The approach provides an efficient, accurate, and scalable tool for simulating periodic diffraction gratings, with clear guidance on DtN truncation effects and stability. Public MATLAB code accompanies the study for reproducibility and further exploration.

Abstract

We propose a Trefftz discontinuous Galerkin (TDG) method for the approximation of plane wave scattering by periodic diffraction gratings, modelled by the two-dimensional Helmholtz equation. The periodic obstacle may include penetrable and impenetrable regions. The TDG method requires the approximation of the Dirichlet-to-Neumann (DtN) operator on the periodic cell faces, and relies on plane wave discrete spaces. For polygonal meshes, all linear-system entries can be computed analytically. Using a Rellich identity, we prove a new explicit stability estimate for the Helmholtz solution, which is robust in the small material jump limit.

Figures (16)

• Figure 1: Geometry of the periodic scattering region $\Omega_0$ and the Dirichlet obstacle $D$. Continuous lines separate regions of $\Omega$ with constant permittivity $\varepsilon$.
• Figure 2: The truncated domain $\Omega=(0,L)\times(-H,H)\setminus \overline D$.
• Figure 3: Periodicity of the mesh.
• Figure 4: Flat interface example of §\ref{['s:2regions']}, lossless case (i) ($\varepsilon^-=1.5$, $\theta=-\pi/3$). Left to right: real part and absolute value of the numerical solution; absolute value of the pointwise error (in logarithmic color scale) for $h=1.5$ and $p=30$; convergence of the $L^2(\Omega)$ and $H^1(\Omega)$ relative error norms for $p \in \{3,\ldots,30\}$ on the same mesh.
• Figure 5: Same plots as in Figure \ref{['fig:no_abs']} for the lossy case (ii) of §\ref{['s:2regions']} ($\varepsilon^-=(1.25 + 0.1{\mathrm i})^2$, $\theta=-\pi/4$).

Theorems & Definitions (25)

• Definition 2.1: Quasi-periodic function
• Proposition 2.2: Fourier expansion
• Lemma 2.3: Pinto2020
• Definition 2.4: Truncated DtN operator
• Lemma 2.5
• proof
• Remark 2.6
• Proposition 2.7
• proof
• Lemma 3.1: Rellich identity