On singular Galerkin discretizations for three models in high-frequency scattering

TL;DR

The paper investigates the potential for discrete ill-posedness in conforming $hp$-finite element discretizations of time-harmonic high-frequency scattering problems, showing that singular system matrices can arise at specific wavenumbers even when the continuous models are well-posed. By constructing Bernkopf-type meshes in 2D and 3D and analyzing the determinant polynomials of the discrete systems, the authors identify explicit spurious frequencies for Helmholtz and Maxwell models across low-order elements, illustrating that discrete unique continuation fails in these cases. The results demonstrate that a discrete resolution condition is not merely a theoretical artefact but a necessary ingredient for discrete stability, and that the presence of a Bernkopf submesh in a mesh can propagate these singularities under scaling. The findings have implications for preconditioning, adaptivity, and the design of stable discretizations, highlighting the nuanced interplay between mesh geometry, polynomial degree, and wavenumber in high-frequency scattering problems.

Abstract

We consider three common mathematical models for time-harmonic high frequency scattering: the Helmholtz equation in two and three spatial dimensions, a transverse magnetic problem in two dimensions, and Maxwell's equation in three dimensions with dissipative boundary conditions such that the continuous problem is well posed. In this paper, we construct meshes for popular (low order) Galerkin finite element discretizations such that the discrete system matrix becomes singular and the discrete problem is not well posed. This implies that a condition "the finite element space has to be sufficiently rich" in the form of a resolution condition - typically imposed for discrete well-posedness - is not an artifact from the proof by a compact perturbation argument but necessary for discrete stability of the Galerkin discretization.

Figures (6)

• Figure 1: Bernkopf mesh in 2D for $\alpha=1/2$.
• Figure 2: Bernkopf mesh in 3D for the choice $\alpha=1/2$.
• Figure 3: Spurious solutions for the $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ Galerkin discretization of the two-dimensional Helmholtz equation for $\alpha=1/2$. Left: $p=1$ and $k\left( \alpha\right) =\pm6$, right: $p=2$ and $k\left( \alpha\right) = \pm 10$.
• Figure 4: Spurious solution for the $\mathbb{P}_{1}$ Galerkin discretization of the 3D Helmholtz problem for the Bernkopf mesh with $\alpha=1/2$ and critical wavenumber $k_{1}\left( 1/2\right) =\pm\sqrt{80}$.
• Figure 5: Spurious solutions for the $\mathbb{P}_{2}$ Galerkin discretization for the Helmholtz problem in 3D for $\alpha=1/2$. Left: $k_{2}\left( 1/2\right) ^{2}=168$, right: $k_{2}\left( 1/2\right) ^{2}=189$.

Theorems & Definitions (16)

• Remark 4
• Proposition 5: MelenkDiss
• Proposition 6: chaumontfrelet_2019a
• Proposition 7
• Remark 8
• Definition 9: Conforming hp finite element space
• Definition 10: Raviart-Thomas elements
• Definition 11: Nédélec elements
• Proposition 12: mm_stas_helm2
• Proposition 13