Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies

TL;DR

This work develops $k$-explicit local error bounds for Helmholtz FEM solutions, showing that local errors are controlled by a combination of the local best-approximation error and low-frequency content, with constants independent of the wavenumber $k$ when using $k$-weighted Sobolev norms. Two main results are presented: a $k$-explicit local bound in $H^1_k$ with an $L^2$-type term on a slightly larger region, and a variant where the $L^2$ term is replaced by a negative-norm term under additional regularity and quasi-uniformity assumptions at the wavelength scale. These bounds extend classical local error results to the Helmholtz setting, including transmission problems and PML/ DtN truncations, and imply that Helmholtz FE solutions are locally quasi-optimal modulo low frequencies. Numerical experiments corroborate the theory, illustrating how high-frequency components are bounded by local approximation while low-frequency content propagates and dominates in appropriate subregions. Collectively, the results offer a precise framework for understanding localized error propagation in Helmholtz FEM and provide practical guidance for mesh design and frequency-aware error control.

Abstract

For $h$-FEM discretisations of the Helmholtz equation with wavenumber $k$, we obtain $k$-explicit analogues of the classic local FEM error bounds of [Nitsche, Schatz 1974], [Wahlbin 1991], [Demlow, Guzmán, Schatz 2011], showing that these bounds hold with constants independent of $k$, provided one works in Sobolev norms weighted with $k$ in the natural way. We prove two main results: (i) a bound on the local $H^1$ error by the best approximation error plus the $L^2$ error, both on a slightly larger set, and (ii) the bound in (i) but now with the $L^2$ error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the $k$-explicit analogue of the main result of [Demlow, Guzmán, Schatz, 2011]. The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of $k^{-1}$) and is the $k$-explicit analogue of the results of [Nitsche, Schatz 1974], [Wahlbin 1991]. Since our Sobolev spaces are weighted with $k$ in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies $\lesssim k$). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error.

Figures (12)

• Figure 1.1: Illustration of the distance $\partial_<(\Omega_0,\Omega_1)$ defined by \ref{['eq:defDistInf']}, with $\Omega_0$ hatched and $\Omega_1$ shaded.
• Figure 2.1: A mesh with two square regions $S_1$ and $S_2$ each with a uniform mesh size (different to each other).
• Figure 2.2: For Experiment 1 in §\ref{['sec:num:mesh']}, plot of the quantity $\log(10^{-12} + {\left\lvert{\Re(u - u_h)}\right\rvert})/\log(10)$, for $k = 50$, $p = 4$ and $\theta = 0$. Top left: globally uniform mesh with $h = 1.78k^{-1}$. Top right: globally uniform mesh with $h = 0.38k^{-1}$. Bottom: non-uniform mesh, with a coarse region $h_1= 1.78k^{-1}$ and a fine region with $h_2 = 0.38k^{-1}$.
• Figure 2.3: Same as Figure \ref{['fig:LFpollutionHorizontal']} (including all parameter values) but with $\theta = \pi/2$.
• Figure 2.4: For Experiment 2 in §\ref{['sec:num:mesh']}, plots of the $H^1_k$ error and its low- and high-frequency components in the left square (i.e., the square with the coarser mesh).

Theorems & Definitions (26)

• Theorem 1.1: Local quasioptimality in $H^1_k$ up to an $L^2$ error term
• Theorem 1.2: Local quasioptimality in $H^1_k$ up to an error term in a negative norm
• Lemma 1.3
• Example 3.1
• Lemma 3.2
• Lemma 3.4
• Corollary 3.8: Mapping properties of the adjoint solution operator on ${\cal O}(k^{-1})$ balls
• Theorem 4.1: General version of Theorem \ref{['thm:intro1']}
• Theorem 4.2: General version of Theorem \ref{['thm:intro2']}
• Remark 4.3: Galerkin orthogonality