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Euan A. Spence

Publications and source records attributed to Euan A. Spence.

At least 19 recordsLinked to original sources
arXiv2026

Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems

We study overlapping Schwarz methods for the Helmholtz equation posed in any dimension with large, real wavenumber and smooth variable wave speed. The radiation condition is approximated by a Cartesian perfectly-matched layer (PML). The domain-decomposition subdomains are overlapping hyperrectangles with Cartesian PMLs at their boundaries. The overlaps of the subdomains and the widths of the PMLs are all taken to be independent of the wavenumber. For both parallel (i.e., additive) and sequential (i.e., multiplicative) methods, we show that after a specified number of iterations -- depending on the behaviour of the geometric-optic rays -- the error is smooth and smaller than any negative power of the wavenumber. For the parallel method, the specified number of iterations is less than the maximum number of subdomains, counted with their multiplicity, that a geometric-optic ray can intersect. These results, which are illustrated by numerical experiments, are the first wavenumber-explicit results about convergence of overlapping Schwarz methods for the Helmholtz equation, and the first wavenumber-explicit results about convergence of any domain-decomposition method for the Helmholtz equation with a non-trivial scatterer (here a variable wave speed).

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arXiv2026

The $hp$-FEM does not suffer from the pollution effect for piecewise-smooth Helmholtz problems with Gevrey regularity at boundaries

We consider the $hp$-FEM applied to the Helmholtz scattering problem with wavenumber $k$, truncated with a perfectly-matched layer. The scatterer consists of a combination of Dirichlet, Neumann, and penetrable obstacles together with variable coefficients. Provided that the Helmholtz solution operator is polynomially bounded in $k$, all coefficients are piecewise smooth, all boundary surfaces are Gevrey and all coefficients restricted to boundary surfaces are Gevrey together with all their normal derivatives, we show that the $hp$-FEM is quasioptimal when $p\geq 1+\varepsilon \log k$ and $hk/p$ is sufficiently small; i.e., the $hp$-FEM does not suffer from the pollution effect. This result generalises the analogous results in both [Bernkopf, Chaumont-Frelet, Melenk 2025] (proved for piecewise analytic coefficients and analytic boundaries) and [Galkowski, Lafontaine, Spence, Wunsch 2024] (proved for smooth coefficients that are analytic near analytic obstacles) to a much larger class of scatterers.

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arXiv2026

Theory of two-level Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation

We analyse the classic two-level additive Schwarz domain-decomposition GMRES preconditioner for finite-element discretisations of the Helmholtz equation with large wavenumber $k$, where both the fine and coarse spaces consist of piecewise polynomials with polynomial degree increasing like $\log k$. We exhibit choices of these fine and coarse spaces such that -- up to factors of $\log k$ -- both are pollution free (with the ratio of the coarse-space dimension to the fine-space dimension arbitrarily small), the number of degrees of freedom per subdomain is constant, and the number of GMRES iterations is proved to be bounded independently of $k$. These are the first $k$-explicit convergence results about a two-level Schwarz preconditioner for high-frequency Helmholtz with a coarse space that is pollution free and does not consist of problem-adapted basis functions.

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arXiv2026

Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media

This paper analyzes the convergence of parallel overlapping domain-decomposition methods with impedance boundary conditions for the time-harmonic Maxwell equations in heterogeneous media. We prove that the parallel iterative method is well-posed in an appropriate function space, and characterize the error propagation operator through impedance-to-impedance maps that describe interactions between neighboring subdomains. For strip domain decompositions, we derive explicit convergence estimates in terms of the norms of the impedance-to-impedance maps. At the discrete level, we develop the finite-element counterpart of these results based on Nédélec-element discretisations. Under the assumption that the discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined, we show that the discrete method inherits the convergence behavior of the continuous method. We illustrate this theory with numerical experiments for strip domain decompositions, and also present numerical experiments for checkerboard domain decompositions that go beyond our theory.

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arXiv2026

Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems

The $h$-version of the finite-element method ($h$-FEM) applied to the high-frequency Helmholtz equation has been a classic topic in numerical analysis since the 1990s. It is now rigorously understood that (using piecewise polynomials of degree $p$ on a mesh of a maximal width $h$) the conditions "$(hk)^p ρ$ sufficiently small" and "$(hk)^{2p} ρ$ sufficiently small" guarantee, respectively, $k$-uniform quasioptimality (QO) and bounded relative error (BRE), where $ρ$ is the norm of the solution operator with $ρ\sim k$ for non-trapping problems. Empirically, these conditions are observed to be optimal in the context of $h$-FEM with a uniform mesh. This paper demonstrates that QO and BRE can be achieved using certain non-uniform meshes that violate the conditions above on $h$ and involve coarser meshes away from trapping and in the perfectly matched layer (PML). The main theorem details how varying the meshwidth in one region affects errors both in that region and elsewhere. One notable consequence is that, for any scattering problem (trapping or nontrapping), in the PML one only needs $hk$ to be sufficiently small; i.e. there is no pollution in the PML. The motivating idea for the analysis is that the Helmholtz data-to-solution map behaves differently depending on the locations of both the measurement and data, in particular, on the properties of billiards trajectories (i.e. rays) through these sets. Because of this, it is natural that the approximation requirements for finite-element spaces in a subset should depend on the properties of billiard rays through that set. Inserting this behaviour into the latest duality arguments for the FEM applied to the high-frequency Helmholtz equation allows us to retain detailed information about the influence of $\textit{both}$ the mesh structure $\textit{and}$ the behaviour of the true solution on local errors in FEM.

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arXiv2026

Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis

We consider the numerical solution of high-frequency scattering problems modeled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and techniques from $\textit{semiclassical analysis}$ have provided a new perspective and been used to settle several long-standing open problems in this area. Semiclassical analysis works in phase space (i.e., position and frequency) and describes rigorously the extent to which solutions of high-frequency PDEs are dictated by the properties of the corresponding geometric-optic rays. The goals of the article are to (i) give a introduction to semiclassical analysis aimed at non-experts and (ii) showcase some of the numerical-analysis results about finite-element methods, boundary-element methods, and domain-decomposition methods obtained using semiclassical techniques.

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arXiv2026

Helmholtz boundary integral methods and the pollution effect

This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation methods -- with subspaces consisting of $\textit{either}$ piecewise polynomials (in 2-d for collocation, in any dimension for Galerkin) $\textit{or}$ trigonometric polynomials (in 2-d) -- as well as a fully discrete quadrature (Nyström) method based on trigonometric polynomials (in 2-d). For each of these methods, we prove -- in many cases for the first time -- rigorous results about the fundamental question: how quickly must the number of degrees of freedom (the dimension of the approximation space) grow with $k$ to maintain accuracy of the computed solution? Importantly, we determine which of these methods suffer from $\textit{the pollution effect}$. That is, we address the question: must the number of points per wavelength $\to \infty$ to maintain accuracy as $k\to\infty$?

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arXiv2026

Massively parallel Schwarz methods for the high frequency Helmholtz equation

We investigate the parallel one-level overlapping Schwarz method for solving finite element discretization of high-frequency Helmholtz equations. The resulting linear systems are large, indefinite, ill-conditioned, and complex-valued. We present a practical variant of the restricted additive Schwarz method with Perfectly Matched Layer transmission conditions (RAS-PML), which was originally analyzed in a theoretical setting in {\tt arXiv:2404.02156}, with some numerical experiments given in {\tt arXiv:2408.16580}. In our algorithm, the width of the overlap and the additional PML layer on each subdomain is allowed to decrease with $\mathcal{O}(k^{-1} \log(k))$, as the frequency $k \rightarrow \infty$, and this is observed to ensure good convergence while avoiding excessive communication. In experiments, the proposed method achieves $\mathcal{O}(k^d)$ parallel scalability under Cartesian domain decomposition and exhibits $\mathcal{O}(k)$ iteration counts and convergence time for $d$-dimensional Helmholtz problems ($d = 2,3$) as $k$ increases. In this preliminary note we restrict to experiments on 2D problems with constant wave speed. Details, analysis and extensions to variable wavespeed and 3D will be given in future work.

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arXiv2026

Preconditioning FEM discretisations of the high-frequency Helmholtz and Maxwell equations by either perturbing the coefficients or adding absorption

This paper investigates the following question: given a Galerkin matrix corresponding to a finite-element discretisation of either the Helmholtz or time-harmonic Maxwell equations with variable coefficients, suppose that the coefficients of the underlying PDE are perturbed; how good an approximate inverse (i.e., preconditioner) is the resulting Galerkin matrix to the original Galerkin matrix? An important special case is when the perturbation consists of adding absorption (in the spirit of "shifted Laplacian preconditioning"). The results of this paper improve the Helmholtz results in [Gander, Graham, Spence, 2015] and [Graham, Pembery, Spence, 2021], and extend these results to the time-harmonic Maxwell equations, confirming a conjecture in the recent preprint [Li, Hu, arXiv 2501.18305].

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arXiv2026

Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations

We prove sharp wavenumber-explicit error bounds for first- or second-family-Nédélec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the $h$-version of the finite-element method, with fixed polynomial degree $p$, applied to the time-harmonic Maxwell equations, the $\textit{asymptotic regime}$ is when the meshwidth, $h$, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the $\textit{preasymptotic regime}$ is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-family Nédélec elements or higher-than-lowest-order second-family Nédélec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.

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arXiv2025

Quasi-Monte Carlo methods for uncertainty quantification of wave propagation and scattering problems modelled by the Helmholtz equation

We analyse and implement a quasi-Monte Carlo (QMC) finite element method (FEM) for the forward problem of uncertainty quantification (UQ) for the Helmholtz equation with random coefficients, both in the second-order and zero-order terms of the equation, thus modelling wave scattering in random media. The problem is formulated on the infinite propagation domain, after scattering by the heterogeneity, and also (possibly) a bounded impenetrable scatterer. The spatial discretization scheme includes truncation to a bounded domain via a perfectly matched layer (PML) technique and then FEM approximation. A special case is the problem of an incident plane wave being scattered by a bounded sound-soft impenetrable obstacle surrounded by a random heterogeneous medium, or more simply, just scattering by the random medium. The random coefficients are assumed to be affine separable expansions with infinitely many independent uniformly distributed and bounded random parameters. As quantities of interest for the UQ, we consider the expectation of general linear functionals of the solution, with a special case being the far-field pattern of the scattered field. The numerical method consists of (a) dimension truncation in parameter space, (b) application of an adapted QMC method to compute expected values, and (c) computation of samples of the PDE solution via PML truncation and FEM approximation. Our error estimates are explicit in $s$ (the dimension truncation parameter), $N$ (the number of QMC points), $h$ (the FEM grid size) and (most importantly), $k$ (the Helmholtz wavenumber). The method is also exponentially accurate with respect to the PML truncation radius. Illustrative numerical experiments are given.

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arXiv2025

Schwarz methods with PMLs for Helmholtz problems: fast convergence at high frequency

We discuss parallel (additive) and sequential (multiplicative) variants of overlapping Schwarz methods for the Helmholtz equation in $\mathbb{R}^d$, with large real wavenumber and smooth variable wave speed. The radiation condition is approximated by a Cartesian perfectly-matched layer (PML). The domain-decomposition subdomains are overlapping hyperrectangles with Cartesian PMLs at their boundaries. In a recent paper ({\tt arXiv:2404.02156}), the current authors proved (for both variants) that, after a specified number of iterations -- depending on the behaviour of the geometric-optic rays -- the error is smooth and smaller than any negative power of the wavenumber $k$. For the parallel method, the specified number of iterations is less than the maximum number of subdomains, counted with their multiplicity, that a geometric-optic ray can intersect. The theory, which is given at the continuous level and makes essential use of semi-classical analysis, assumes that the overlaps of the subdomains and the widths of the PMLs are all independent of the wavenumber. In this paper we extend the results of {\tt arXiv:2404.02156} by experimentally studying the behaviour of the methods in the practically important case when both the overlap and the PML width decrease as the wavenumber increases. We find that (at least for constant wavespeed), the methods remain robust to increasing $k$, even for miminal overlap, when the PML is one wavelength wide.

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arXiv2025

Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems

We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite-element discretisations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned matrix to be close to the identity, and covers DD subdomains of arbitrary size, arbitrary absorbing layers/boundary conditions on both the global and local Helmholtz problems, and coarse spaces not necessarily related to the subdomains. The assumptions on the coarse space are satisfied by the approximation spaces using problem-adapted basis functions that have been recently analysed as coarse spaces for the Helmholtz equation, as well as all spaces that are known to be quasi-optimal via a Schatz-type argument. As an example, we apply this theory when the coarse space consists of piecewise polynomials; these are then the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.

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arXiv2025

Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering

We consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable obstacle, whose shape is uncertain and modelled via a radial spectral parameterization with random coefficients. Using recent results on the stability of Helmholtz transmission problems with piecewise constant coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Mathematical Models and Methods in Applied Sciences, 29 (2019), pp. 317-354] we obtain frequency-explicit statements on the holomorphic dependence of the scattered field and the far-field pattern on the stochastic shape parameters. This paves the way for applying general results on the efficient construction of high-dimensional surrogate models. We also take into account the effect of domain truncation by means of perfectly matched layers (PML). In addition, spatial regularity estimates which are explicit in terms of the wavenumber $k$ permit us to quantify the impact of finite-element Galerkin discretization using high-order Lagrangian finite-element spaces.

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arXiv2025

The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains

We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?" has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper we prove that for nontrapping problems solved using straight elements the geometric error is order $kh$, which is then less than the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, we prove that, even for problems with strong trapping, using degree four (in 2-d) or degree five (in 3-d) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.

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arXiv2024

Sharp preasymptotic error bounds for the Helmholtz $h$-FEM

In the analysis of the $h$-version of the finite-element method (FEM), with fixed polynomial degree $p$, applied to the Helmholtz equation with wavenumber $k\gg 1$, the $\textit{asymptotic regime}$ is when $(hk)^p C_{\rm sol}$ is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here $C_{\rm sol}$ is the $L^2 \to L^2$ norm of the Helmholtz solution operator, with $C_{\rm sol} \sim k$ for nontrapping problems. In the $\textit{preasymptotic regime}$, one expects that if $(hk)^{2p}C_{\rm sol}$ is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition $\textit{either}$ realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball $\textit{or}$ approximated either by a radial perfectly-matched layer (PML) or an impedance boundary condition. Previously, such bounds for $p>1$ were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalisation of the "elliptic-projection" argument (the argument used to obtain the result for $p=1$) which can be applied to a wide variety of abstract Helmholtz-type problems.

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arXiv2024

Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed

We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of $z$, the norm of the solution operator is bounded by that function. This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows superalgebraically, with these frequencies often called $\textit{quasi-resonances}$. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations.

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