J. Galkowski

activity

20112026

most citedRestriction Bounds for the Free Resolvent and Resonances in Lossy Scattering

34 citations · 193 across the 55 of their papers we have counts for

collaborators

69 papers

math.AP2026

Cartesian PML truncation for the Helmholtz equation is exponentially accurate at high frequency

Martin Averseng, Jeffrey Galkowski, Euan A. Spence

For a wide class of Helmholtz scattering problems (namely, those fitting in the black-box framework of Sjöstrand--Zworski) we prove that the error incurred by approximating the rad…

math.NA2026

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

Jeffrey Galkowski, Mostafa Meliani, Euan A. Spence

We consider the $h p$ -FEM applied to the Helmholtz scattering problem with wavenumber $k$ , truncated with a perfectly-matched layer. The scatterer consists of a combination of Diric…

math-ph2026

Localised Davies generators for (pseudo)differential operators

Jeffrey Galkowski, Maciej Zworski

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state…

math.AP2026

1D Scattering through time dependent media with memory

Jeffrey Galkowski, Zhen Huang, Maciej Zworski

We construct a scattering matrix with operator valued entries describing solutions to the 1+1 wave equation where permittivities has memory and depends on time and space. It is the…

math.NA2025

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

Jeffrey Galkowski, Euan A. Spence

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…

math.NA2025

Helmholtz boundary integral methods and the pollution effect

Jeffrey Galkowski, Manas Rachh, Euan A. Spence

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 in…

math.NA2025

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

Martin Averseng, Jeffrey Galkowski, Euan A. Spence

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 ri…

math.SP2025

Surface plasmons in metamaterial cavities

Yan-Long Fang, Jeffrey Galkowski

We study scattering by metamaterials with negative indices of refraction, which are known to support \emph{surface plasmons} -- long-lived states that are highly localized at the b…

math.NA2025

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

Jeffrey Galkowski, Euan A. Spence

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…

math.NA2024

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

Jeffrey Galkowski, Shihua Gong, Ivan G. Graham +2

We discuss parallel (additive) and sequential (multiplicative) variants of overlapping Schwarz methods for the Helmholtz equation in $R^{d}$ , with large real wavenumber and…

math.NA2024

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

Théophile Chaumont-Frelet, Jeffrey Galkowski, Euan A. Spence

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 var…

math.NA2024

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

Jeffrey Galkowski, Shihua Gong, Ivan G. Graham +2

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 approx…

math-ph2024★ 1 cited

Classical-Quantum correspondence in Lindblad evolution

Jeffrey Galkowski, Zhen Huang, Maciej Zworski

We show that for the Lindblad evolution defined using (at most) quadratically growing classical Hamiltonians and (at most) linearly growing classical jump functions (quantized into…

math.AP2023★ 2 cited

An abstract formulation of the flat band condition

Jeffrey Galkowski, Maciej Zworski

Motivated by the study of flat bands in models of twisted bilayer graphene (TBG), we give abstract conditions which guarantee the existence of a discrete set of parameters for whic…

math.NA2023

Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies

Martin Averseng, Euan A. Spence, Jeffrey Galkowski

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], [Wahlbi…

math.AP2023

Propagation for Schrödinger operators with potentials singular along a hypersurface

Jeffrey Galkowski, Jared Wunsch

In this article, we study propagation of defect measures for Schrödinger operators, $- h^{2} Δ_{g} + V$ , on a Riemannian manifold $(M, g)$ of dimension $n$ with $V$ having conormal singular…

math.NA2023★ 1 cited

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

Jeffrey Galkowski, Euan A. Spence

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 $k ≫ 1$ , the $\textit{as…

math.AP2022

Asymptotics for the spectral function on Zoll manifolds

Yaiza Canzani, Jeffrey Galkowski, Blake Keeler

Let $(M, g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$ . The positive defin…

math.NA2022

Lower bounds for piecewise polynomial approximations of oscillatory functions

Jeffrey Galkowski

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and h…

math.SP2022

The scattering phase: seen at last

Jeffrey Galkowski, Pierre Marchand, Jian Wang +1

The scattering phase, defined as $lo g det S (λ) /2 π i$ where $S (λ)$ is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when deali…

math.SP2022

Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case

Jeffrey Galkowski, Leonid Parnovski, Roman Shterenberg

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let $H : L^{2} (R) \to L^{2} (R)$ have the form…

math.AP2022

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

Jeffrey Galkowski, David Lafontaine, Euan A. Spence +1

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 th…

math.DS2022

Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant

Yaiza Canzani, Jeffrey Galkowski

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$ , a notion analogous to full Lebesgue…

math.NA2022

Does the Helmholtz boundary element method suffer from the pollution effect?

Jeffrey Galkowski, Euan A. Spence

In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $≲ k$ requires $\sim k^{d}$ degrees of freedom. A numerical method for solving the…

Cartesian PML truncation for the Helmholtz equation is exponentially accurate at high frequency

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

Localised Davies generators for (pseudo)differential operators

1D Scattering through time dependent media with memory

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

Helmholtz boundary integral methods and the pollution effect

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

Surface plasmons in metamaterial cavities

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

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

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

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

Classical-Quantum correspondence in Lindblad evolution

An abstract formulation of the flat band condition

Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies

Propagation for Schrödinger operators with potentials singular along a hypersurface

Sharp preasymptotic error bounds for the Helmholtz h-FEM

Asymptotics for the spectral function on Zoll manifolds

Lower bounds for piecewise polynomial approximations of oscillatory functions

The scattering phase: seen at last

Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case

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

Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics are predominant

Does the Helmholtz boundary element method suffer from the pollution effect?

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

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

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