The $hp$-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect

TL;DR

The work establishes that the hp-FEM, when applied to the Helmholtz exterior Dirichlet problem truncated by a radial PML, achieves quasioptimality with a $k$-independent constant under explicit mesh-degree conditions: $hk/p\le C_1$ and $p\ge C_2\log k$, provided the solution operator is polynomially bounded and the domain/coefficient data satisfy analytic or smoothing assumptions. A key novelty is extending the high/low-frequency decomposition from previous works on outward Helmholtz solutions to PML-truncated problems, using semiclassical analysis, functional calculus on a torus, and PML ellipticity. The decomposition yields a regular, analytic (or highly regular) low-frequency part and a high-frequency part controlled by semiclassical ellipticity, with a negligible residual; this separation is what eliminates pollution for hp-FEM in the PML setting. The results rely on a black-box scattering framework and robust PML estimates, showing that the PML-truncated problem inherits the favorable $k$-scaling of the non-PML analysis and allowing quasioptimal hp-FEM error bounds that scale like the best hp-approximation. Overall, the paper provides rigorous, $k$-explicit guarantees for using hp-FEM with PML truncation in high-frequency Helmholtz scattering, with clear implications for efficient computational wave propagation in complex media.

Abstract

We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber $k$ in [Galkowski, Lafontaine, Spence, 2021]. We show that the $hp$-FEM applied to this problem does not suffer from the pollution effect, in that there exist $C_1,C_2>0$ such that if $hk/p\leq C_1$ and $p \geq C_2 \log k$ then the Galerkin solutions are quasioptimal (with constant independent of $k$), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in $k$ (which occurs for "most" $k$ by [Lafontaine, Spence, Wunsch, 2021]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This $hp$-FEM result is obtained via a decomposition of the PML solution into "high-" and "low-frequency" components, analogous to the decomposition for the original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence, Wunsch, 2022]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large $k$).

Figures (6)

• Figure 1.1: The regions where $v_{{{\cal A}}, {\rm near}}$ and $v_{{{\cal A}}, {\rm far}}$ appearing in Theorem \ref{['thm:LSW4']} are analytic, entire, or $O(k^{-\infty})$.
• Figure 3.1: The black-box setting. The symbol $\simeq$ is used to denote equality in the sense of \ref{['eq:bbreq1']} and \ref{['eq:defref']}.
• Figure 5.1: Decomposition of the PML solution described in §\ref{['subsec:abdec']} (when $\rho\neq 1$ in \ref{['eq:lowenest']})
• Figure 5.2: The cut-off functions $\varphi_0, \widetilde{\varphi}_0, \varphi_1, \widetilde{\varphi}_1, \varphi_{\rm tr}$, and $\widetilde{\varphi}_{\rm tr}$ described at the start of §\ref{['subsec:high']}.
• Figure 5.3: The cut-off functions $\rho_1, \rho_2, \gamma_1, \gamma_2$ defined at the start of §\ref{['subsec:low']}.

Theorems & Definitions (49)

• Definition 1.1: Helmholtz Exterior Dirichlet problem
• Definition 1.2: Polynomial-boundedness of the solution operator
• Theorem 1.3
• Remark 1.4: Link with other notation used in the literature
• Remark 1.5: Smoothness of the PML scaling function $f_\theta$
• Theorem 1.6: Radial PMLs are exponentially accurate for $k$ large
• Lemma 1.7: Variational formulation of the PML problem \ref{['eq:PML']}
• Remark 1.8: Plane-wave scattering
• Theorem 1.12
• Remark 1.13: Non-conforming error