Euan Spence: Papers

Below is a list of the majority of my papers and preprints organised by subject area. A complete list of my papers can be found here.

Papers about the Laplace equation

• S.N. Chandler-Wilde, E.A. Spence, Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains, Numer. Math., vol. 156, pages 1325–1384,(2024) arxiv copy

• S.N. Chandler-Wilde, E.A. Spence, Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains, Numer. Math., vol. 150, pages 299-371 (2022) arxiv copy, correction

Papers about the Maxwell equations

• [New]  E.A. Spence, Preconditioning FEM discretisations of the high-frequency Maxwell equations by either perturbing the coefficients or adding absorption

• T. Chaumont-Frelet, J. Galkowski, E.A. Spence, Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations

• T. Chaumont-Frelet, A. Moiola, E.A. Spence, Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media. J. Math. Pure. Appl., vol. 179, pages 183-218 (2023) arxiv copy

• M. Bonazzoli, V. Dolean, I.G. Graham, E.A. Spence, P.-H. Tournier, Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell's equations with absorption. Math. Comp., vol. 88, pages 2559-2604 (2019) arxiv copy

Papers about the Helmholtz equation

Papers about the Helmholtz equation itself

• E.A. Spence, Y. Zou, J. Wunsch, Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed, J. Differ. Equations, vol. 440, part 2, article number 113441 (2025) arxiv copy

• T. Chaumont-Frelet, E.A. Spence, Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization, SIAM J. Math. Anal., Vol. 55, No. 2, pages 1319–1363 (2023) arxiv version

• J. Galkowski, P. Marchand, E.A. Spence, Eigenvalues of the truncated Helmholtz solution operator under strong trapping, SIAM J. Math. Anal., Vol. 53, No. 6, pages 6724-6770 (2021) arxiv copy

• D. Lafontaine, E.A. Spence, J. Wunsch, For most frequencies, strong trapping has a weak effect in frequency-domain scattering, Comm. Pure Appl. Math., volume 74, issue 10, pages 2025-2063 (2021) arxiv copy

• J. Galkowski, E.A. Spence, J. Wunsch, Optimal constants in non-trapping resolvent estimates and applications in numerical analysis, Pure and Applied Analysis, volume 2, number 1, pages 157-202 (2020) arxiv copy

• I.G. Graham, O.R. Pembery, E.A. Spence, The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances, J. Differ. Equations, vol. 266, issue 6, 2869-2923 (2019), arxiv copy

• S.N. Chandler-Wilde, E.A. Spence, A. Gibbs, V. P. Smyshlyaev, High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis, SIAM J. Math. Anal., volume 52, issue 1, pages 845-893 (2020) arxiv copy

• A. Moiola, E.A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Math. Mod. Meth. App. S., vol. 29, no. 2, 317-354 (2019) arxiv copy

• D. Baskin, E.A. Spence, J. Wunsch, Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations, SIAM J. Math. Anal. vol. 48, no. 1, 229-267 (2016) local official copy, arxiv copy

• E.A. Spence, Wavenumber-explicit bounds in time-harmonic acoustic scattering, SIAM J. Math. Anal., vol. 46, no. 4, 2987-3024 (2014) local official copy

Papers about the accuracy of absorbing boundary conditions/PML for the Helmholtz equation

• J. Galkowski, D. Lafontaine, E.A. Spence, Perfectly-matched-layer truncation is exponentially accurate at high frequency, SIAM J. Math. Anal., vol. 55, number 4, pages 3344-3394 (2023) arxiv copy

• J. Galkowski, D. Lafontaine, E.A. Spence, Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves. IMA. J. Numer. Anal., vol. 44, issue 4, pages 1946-2069 (2024) arxiv copy

Papers about the convergence of the finite element method applied to the Helmholtz equation

• [New]  M. Averseng J. Galkowski, E.A. Spence, Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems

• T. Chaumont-Frelet, E.A. Spence, The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains. IMA. J. Numer. Anal., to appear.

• M. Averseng J. Galkowski, E.A. Spence, Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies, Adv. Comp. Math., vol. 50, article number 112 (2024) arxiv copy

• J. Galkowski, E.A. Spence, Sharp preasymptotic error bounds for the Helmholtz h-FEM, SIAM J. Numer. Anal., vol. 63, no. 1, pages 1-22 (2025) arxiv copy

• J. Galkowski, D. Lafontaine, E.A. Spence, J. Wunsch, The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect, Comm. Math. Sci., vol. 22, no. 7, pages 1761-1816 (2024) arxiv copy

• E.A. Spence, A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation, Adv. Comp. Math., vol. 49, article number 27 (2023) arxiv copy

• J. Galkowski, D. Lafontaine, E.A. Spence, J. Wunsch, Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method , SIAM J. Math. Anal., vol. 55, no. 4, pages 3903-3958 (2023), arxiv copy

• D. Lafontaine, E.A. Spence, J. Wunsch, Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients , Comput. Math. Appl., vol. 113, pages 59-69 (2022) arxiv copy

• D. Lafontaine, E.A. Spence, J. Wunsch, A sharp relative-error bound for the Helmholtz h-FEM at high frequency Numer. Math., vol. 150, pages 137-178 (2022) arxiv copy

Papers about uncertainty quantification for the Helmholtz equation

• I.G. Graham, F.Y. Kuo, D. Nuyens, I.H. Sloan, E.A. Spence, Quasi-Monte Carlo methods for uncertainty quantification of wave propagation and scattering problems modelled by the Helmholtz equation, IMA J. Num. Anal., to appear

• R. Hiptmair, C. Schwab, E.A. Spence, Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering, SIAM/ASA J. Uncertainty Quantification, to appear

• E.A. Spence, J. Wunsch, Wavenumber-explicit parametric holomorphy of Helmholtz solutions in the context of uncertainty quantification, SIAM/ASA J. Uncertainty Quantification, Vol. 11, number 2, pages 567-590 (2023) arxiv copy

• I.G. Graham, O.R. Pembery, E.A. Spence, Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification, Adv. Comp. Math., vol. 47, article number 68 (2021) arxiv copy

• O.R. Pembery, E.A. Spence, The Helmholtz equation in random media: well-posedness and a priori bounds, SIAM/ASA J. Uncertainty Quantification, volume 8, number 1, pages 58–87 (2020) arxiv copy

Papers about domain-decomposition and/or preconditioning the Helmholtz equation

• I.G. Graham, E.A. Spence, Two-level hybrid Schwarz preconditioners with piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation

• J. Galkowski, E.A. Spence, Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems, SIAM J. Numer. Anal., to appear

• J. Galkowski, S. Gong, I.G. Graham, D. Lafontaine, E.A. Spence, Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems

• D. Lafontaine, E.A. Spence, Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition, Pure and Applied Analysis, vol. 5, no. 4, pages 927-972, 2023, arxiv copy

• S. Gong, I.G. Graham, E.A. Spence, Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems, Math. Comp., volume 92, pages 175-215 (2023) arxiv copy

• S. Gong, M.J. Gander, I.G. Graham, D. Lafontaine, E.A. Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation, Numer. Math., volume 152, pages 259-306 (2022) arxiv copy

• S. Gong, I.G. Graham, E.A. Spence, Domain decomposition preconditioners for high-order discretisations of the heterogeneous Helmholtz equation IMA J. Num. Anal, vol. 41, no. 3, pages 2139-2185 (2021) arxiv copy

• I.G. Graham, E.A. Spence, J. Zou, Domain Decomposition with local impedance condition for the Helmholtz equation with absorption, SIAM J. Numer. Anal., vol. 58, number 5, pages 2515–2543 (2020) arxiv copy

• I.G. Graham, E.A. Spence, E. Vainikko, Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption, Math. Comp., vol. 86, pages 2089-2127 (2017) arxiv copy

• M.J. Gander, I.G. Graham, E.A. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed? Numer. Math., vol. 131, issue 3, pages 567-614(2015) local unofficial copy (Note that this is a revision of the preprint titled How should one choose the shift for the shifted Laplacian to be a good preconditioner for the Helmholtz equation?)

Papers about boundary integral equations for the Helmholtz equation

• [New]  J. Galkowski, M. Rachh, E.A. Spence, Helmholtz boundary integral methods and the pollution effect

• J. Galkowski, E.A. Spence, Does the Helmholtz boundary element method suffer from the pollution effect?, SIAM Review, vol. 65, no. 3, pages 806–828 (2023), arxiv copy

• R. Hiptmair, A. Moiola, E.A. Spence, Spurious quasi-resonances in boundary integral equations for the Helmholtz transmission problem, SIAM J. Appl. Math., vol. 82, number 4, pages 1446-1469 (2022) arxiv copy

• J. Galkowski, P. Marchand, E.A. Spence, High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem , Integr. Equat. Oper. Th., volume 94, article number 36 (2022) arxiv copy

• P. Marchand, J. Galkowski, A. Spence, E.A. Spence, Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency? Adv. Comp. Math., vol. 48, article number 37 (2022) arxiv copy

• J. Galkowski, E.A. Spence, Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators Integr. Equat. Oper. Th., vol. 91, issue 1, article 6 (2019), arxiv copy

• J. Galkowski, E.H. Müller, E.A. Spence, Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem Numer. Math., vol. 142, issue 2, pages 329-357 (2019) arxiv copy

• E.A. Spence, I.V. Kamotski, V.P. Smyshlyaev, Coercivity of combined boundary integral equations in high-frequency scattering Comm. Pure Appl. Math., vol. 68, issue 9, pages 1587-1639 (2015), local unofficial copy

• I.G. Graham, M. Löhndorf, J.M. Melenk, E.A. Spence, When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k? BIT Num. Math., vol. 55, no. 1, 171-214 (2015), local unofficial copy

• E.A. Spence, Bounding acoustic layer potentials via oscillatory integral techniques BIT Num. Math., vol. 55, no. 1., 279-318 (2015) local unoffical copy

• T. Betcke, J. Phillips, E.A. Spence, Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering, IMA J. Num. Anal., vol. 34, no. 2, 700-731 (2014)

• S.N. Chandler-Wilde, I.G. Graham, S. Langdon, E.A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numerica, vol. 21, 89-305 (2012) local official copy

• T. Betcke, E.A. Spence, Numerical estimation of coercivity constants for boundary integral operators in acoustic scattering, SIAM J. Numer. Anal. vol. 49, issue 4, 1572-1601 (2011) local official copy

• E.A. Spence, S.N. Chandler-Wilde, I.G. Graham, V. P. Smyshlyaev A new frequency-uniform coercive boundary integral equation for acoustic scattering, Comm. Pure Appl. Math. vol. 64, issue 10, 1384-1415, (2011) unofficial copy

Papers about seismic imaging

• S. Downing, S. Gazzola, I.G. Graham, E.A. Spence, Optimising seismic imaging design parameters via bilevel learning, Inverse Probl., vol 40, no. 11, article number 115008 (2024) arxiv copy

Papers about coercive formulations of the Helmholtz equation

• G. Diwan, A. Moiola, E.A. Spence, Can coercive formulations lead to fast and accurate solution of the Helmholtz equation? J. Comp. Appl. Math. vol. 352, 110-131 (2019) arxiv copy

• A. Moiola, E.A. Spence, Is the Helmholtz equation really sign-indefinite? SIAM Review, vol. 56, no. 2, 274-312 (2014) local official copy

Review articles

• I.G. Graham, E.A. Spence, E. Vainikko, Recent Results on Domain Decomposition Preconditioning for the High-frequency Helmholtz equation using absorption in "Modern Solvers for Helmholtz Problems", D. Lahaye, J. Tang, C. Vuik eds., Springer (2017)

• E.A. Spence, "When all else fails, integrate by parts" - an overview of new and old variational formulations for linear elliptic PDEs in "Unified Transform Method for Boundary Value Problems: Applications and Advances", A.S. Fokas and B. Pelloni eds., SIAM (2015)

Papers about asymptotics of integrals

• A. Fernandez, E.A. Spence, A.S. Fokas, Uniform asymptotics as a stationary point approaches an endpoint, IMA J. Appl. Math. vol. 83, issue 1, 202-242 (2018) arxiv copy

Papers about transform methods for linear PDEs

• E.A. Spence, The Watson transformation revisited, (2014)

• E.A. Spence, Transform methods for linear PDEs, in Encyclopedia of Applied and Computational Mathematics, Springer (2015)

• A.S. Fokas, E.A. Spence Synthesis, as opposed to separation, of variables, SIAM Review, vol. 54, no. 2, 291-324 (2012) local official copy

• E.A. Spence, A.S. Fokas, A New Transform Method I: Domain Dependent Fundamental Solutions and Integral Representations. Proc. Roy. Soc. A. vol. 466, 2259-2281 (2010)

• E.A. Spence, A.S. Fokas, A New Transform Method II: the Global Relation, and Boundary Value Problems in Polar Co-ordinates. Proc. Roy. Soc. A. vol 466, 2283-2307 (2010) Corrections

• S.A. Smitheman, E.A. Spence, A.S. Fokas, A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon IMA J. Num. Anal. 30(4): 1184-1205 (2010)

• A.S. Fokas, N. Flyer, S.A. Smitheman, E.A. Spence, A semi-analytical numerical method for solving evolution and elliptic partial differential equations, J. Comp. Appl. Math. Volume 227, Issue 1, 59-74 (2009) (Invited Paper)

• E.A. Spence, Boundary Value Problems for Linear Elliptic PDEs, PhD thesis, Cambridge, submitted 23/03/2009, viva 05/02/2010