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

Part of:Elliptic equations and systemsPartial differential equations, boundary value problems

Published online by Cambridge University Press:  19 June 2026

Jeffrey GalkowskiOpen the ORCID record for Jeffrey Galkowski[Opens in a new window] and
Euan A. SpenceOpen the ORCID record for Euan A. Spence[Opens in a new window]
Jeffrey Galkowski
Affiliation:
Department of Mathematics, University College London, London, WC1H 0AY, UK E-mail: j.galkowski@ucl.ac.uk
Euan A. Spence
Affiliation:
Department of Mathematics, University of Bath, Bath, BA2 7AY, UK E-mail: e.a.spence@bath.ac.uk
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Abstract

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We consider the numerical solution of high-frequency scattering problems modelled 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 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 optics rays.

The goals of the article are (i) to give a introduction to semiclassical analysis aimed at non-experts and (ii) to showcase some of the numerical analysis results about finite element methods, boundary element methods and domain decomposition methods obtained using semiclassical techniques.

MSC classification

Primary:35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods65N38: Boundary element methods65N55: Multigrid methods; domain decomposition

Information

Type
Research Article
Information
Acta Numerica , Volume 35 , July 2026, pp. 1 - 171
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2026. Published by Cambridge University Press

References

Ainsworth, M. (2004), Discrete dispersion relation for hp-version finite element approximation at high wave number, SIAM J. Numer. Anal. 42, 553–575.10.1137/S0036142903423460CrossRefGoogle Scholar
Alessandrini, G. (2012), Strong unique continuation for general elliptic equations in 2D, J. Math. Anal. Appl. 386, 669–676.10.1016/j.jmaa.2011.08.029CrossRefGoogle Scholar
Amini, S. and Harris, P. J. (1990), A comparison between various boundary integral formulations of the exterior acoustic problem, Comput. Methods Appl. Mech. Engrg 84, 59–75.10.1016/0045-7825(90)90089-5CrossRefGoogle Scholar
Apel, T. and Melenk, J. M. (2015), Interpolation and quasi-interpolation in h- and hp-version finite element spaces (extended version). ASC Report No. 39/2015. Available at https://www.tuwien.at/index.php?eID=dumpFile&t=f&f=190654&token=cf4edf3c7e8601fea9d11b7d3c9cc471e42cc6fa.Google Scholar
Aubin, J. P. (1967), Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods, Ann. Sc. Norm. Super. Pisa Cl. Sci. 21, 599–637.Google Scholar
Averseng, M., Galkowski, J. and Spence, E. A. (2024), Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies, Adv. Comput. Math. 50, art. 112.10.1007/s10444-024-10193-wCrossRefGoogle Scholar
Averseng, M., Galkowski, J. and Spence, E. A. (2025), Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems. Available at arXiv:2506.15630.Google Scholar
Aziz, A. K., Kellogg, R. B. and Stephens, A. B. (1988), A two point boundary value problem with a rapidly oscillating solution, Numer. Math. 53, 107–121.10.1007/BF01395880CrossRefGoogle Scholar
Babuška, I. and Sauter, S. A. (1997), Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal. 34, 2392–2423.10.1137/S0036142994269186CrossRefGoogle Scholar
Babuška, I. and Suri, M. (1987), The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal. 24, 750–776.10.1137/0724049CrossRefGoogle Scholar
Babuška, I. and Suri, M. (1994), The p and h-p versions of the finite element method, basic principles and properties, SIAM Rev. 36, 578–632.10.1137/1036141CrossRefGoogle Scholar
Babuška, I., Craig, A., Mandel, J. and Pitkäranta, J. (1991), Efficient preconditioning for the p-version finite element method in two dimensions, SIAM J. Numer. Anal. 28, 624–661.10.1137/0728034CrossRefGoogle Scholar
Bachman, G., Narici, L. and Beckenstein, E. (2000), Fourier and Wavelet Analysis , Springer.10.1007/978-1-4612-0505-0CrossRefGoogle Scholar
Ball, J. M., Capdeboscq, Y. and Tsering-Xiao, B. (2012), On uniqueness for time harmonic anisotropic Maxwell’s equations with piecewise regular coefficients, Math. Models Methods Appl. Sci. 22, art. 1250036.10.1142/S0218202512500364CrossRefGoogle Scholar
Banjai, L. and Sauter, S. (2007), A refined Galerkin error and stability analysis for highly indefinite variational problems, SIAM J. Numer. Anal. 45, 37–53.10.1137/060654177CrossRefGoogle Scholar
Bardos, C., Lebeau, G. and Rauch, J. (1992), Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30, 1024–1065.10.1137/0330055CrossRefGoogle Scholar
Barucq, H., Chaumont-Frelet, T. and Gout, C. (2017), Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation, Math. Comp. 86, 2129–2157.10.1090/mcom/3165CrossRefGoogle Scholar
Baskin, D., Spence, E. A. and Wunsch, J. (2016), Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations, SIAM J. Math. Anal. 48, 229–267.10.1137/15M102530XCrossRefGoogle Scholar
Bellassoued, M. (2003), Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization, Asymptot. Anal. 35, 257–279.Google Scholar
Berenger, J.-P. (1994), A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114, 185–200.10.1006/jcph.1994.1159CrossRefGoogle Scholar
Bernardi, C. (1989), Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26, 1212–1240.10.1137/0726068CrossRefGoogle Scholar
Bernardi, C. and Maday, Y. (1997), Spectral methods, in Techniques of Scientific Computing 2 (Ciarlet, P. G. and Lions, J. L., eds), Vol. 5 of Handbook of Numerical Analysis, Elsevier, pp. 209–485.Google Scholar
Bernardi, C., Dauge, M. and Maday, Y. (1995), Interpolation of nullspaces for polynomial approximation of divergence-free functions in a cube, in Boundary Value Problems and Integral Equations in Nonsmooth Domains , Vol. 167 of Lecture Notes in Pure and Applied Mathematics, Dekker, pp. 27–46.Google Scholar
Bernkopf, M., Chaumont-Frelet, T. and Melenk, J. M. (2025), Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media, Math. Comp. 94, 73–122.10.1090/mcom/3958CrossRefGoogle Scholar
Bloom, C. O. (1973), Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter, J. Math. Anal. Appl. 44, 310–332.10.1016/0022-247X(73)90062-0CrossRefGoogle Scholar
Bloom, C. O. and Kazarinoff, N. D. (1974), Local energy decay for a class of nonstar-shaped bodies, Arch. Ration. Mech. Anal. 55, 73–85.10.1007/BF00282434CrossRefGoogle Scholar
Bloom, C. O. and Kazarinoff, N. D. (1977), A priori bounds for solutions of the Dirichlet problem for $\left[\Delta +{\lambda}^2n(x)\right]u=f\left(x,\lambda \right)$ on an exterior domain, J. Differential Equations 24, 437–465.10.1016/0022-0396(77)90011-0CrossRefGoogle Scholar
Bony, J.-F., Burq, N. and Ramond, T. (2010), Minoration de la résolvante dans le cas captif, C.R. Math. 348(23–24), 1279–1282.10.1016/j.crma.2010.10.025CrossRefGoogle Scholar
Boubendir, Y. and Turc, C. (2013), Wave-number estimates for regularized combined field boundary integral operators in acoustic scattering problems with Neumann boundary conditions, IMA J. Numer. Anal. 33, 1176–1225.10.1093/imanum/drs038CrossRefGoogle Scholar
Bouziani, N., Nataf, F. and Tournier, P.-H. (2023), A unified framework for double sweep methods for the Helmholtz equation, J. Comput. Phys. 490, art. 112305.10.1016/j.jcp.2023.112305CrossRefGoogle Scholar
Brakhage, H. and Werner, P. (1965), Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung, Archiv Math. 16, 325–329.CrossRefGoogle Scholar
Bramble, J. H. and Pasciak, J. (2007), Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems, Math. Comp. 76, 597–614.10.1090/S0025-5718-06-01930-2CrossRefGoogle Scholar
Bramble, J. H. and Pasciak, J. E. (2013), Analysis of a Cartesian PML approximation to acoustic scattering problems in ℝ2 and ℝ3, J. Comput. Appl. Math. 247, 209–230.10.1016/j.cam.2012.12.022CrossRefGoogle Scholar
Brenner, S. C. and Scott, L. R. (2008), The Mathematical Theory of Finite Element Methods , Vol. 15 of Texts in Applied Mathematics, third edition, Springer.CrossRefGoogle Scholar
Brown, D. L., Gallistl, D. and Peterseim, D. (2017), Multiscale Petrov–Galerkin method for high-frequency heterogeneous Helmholtz equations, in Meshfree Methods for Partial Differential Equations VIII , Springer, pp. 85–115.10.1007/978-3-319-51954-8_6CrossRefGoogle Scholar
Bruno, O., Elling, T. and Turc, C. (2012), Regularized integral equations and fast high-order solvers for sound-hard acoustic scattering problems, Internat. J. Numer. Methods Engrg 91, 1045–1072.10.1002/nme.4302CrossRefGoogle Scholar
Burq, N. (1998), Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math. 180, 1–29.10.1007/BF02392877CrossRefGoogle Scholar
Burq, N. (2002), Semi-classical estimates for the resolvent in nontrapping geometries, Int. Math. Res. Not. 2002, 221–241.10.1155/S1073792802103059CrossRefGoogle Scholar
Burton, A. J. (1976), Numerical Solution of Acoustic Radiation Problems , National Physical Laboratory, Division of Numerical Analysis and Computing.Google Scholar
Burton, A. J. and Miller, G. F. (1971), The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proc. R. Soc. A 323, 201–210.Google Scholar
Calderón, A. P. (1961), Lebesgue spaces of differentiable functions and distributions, in Partial Differential Equations (Morrey, C. B. Jr, ed.), Vol. 4 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, pp. 33–49.10.1090/pspum/004/0143037CrossRefGoogle Scholar
Canuto, C. and Quarteroni, A. (1982), Approximation results for orthogonal polynomials in sobolev spaces, Math. Comp. 38, 67–86.10.1090/S0025-5718-1982-0637287-3CrossRefGoogle Scholar
Cardoso, F. and Vodev, G. (2002), Uniform estimates of the resolvent of the Laplace–Beltrami operator on infinite volume Riemannian manifolds II, Ann. Henri Poincaré 3, 673–691.10.1007/s00023-002-8631-8CrossRefGoogle Scholar
Carleman, T. (1939), Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys. 26, issue 17.Google Scholar
Céa, J. (1964), Approximation variationnelle des problèmes aux limites, Ann. Inst. Fourier 14, 345–444.10.5802/aif.181CrossRefGoogle Scholar
Chandler-Wilde, S. N. and Langdon, S. (2007), A Galerkin boundary element method for high frequency scattering by convex polygons, SIAM J. Numer. Anal. 45, 610–640.10.1137/06065595XCrossRefGoogle Scholar
Chandler-Wilde, S. N. and Monk, P. (2008), Wave-number-explicit bounds in time-harmonic scattering, SIAM J. Math. Anal. 39, 1428–1455.10.1137/060662575CrossRefGoogle Scholar
Chandler-Wilde, S. N. and Sadeghi, S. (2025), The cut-off resolvent can grow arbitrarily fast in obstacle scattering. Available at arXiv:2508.16958.Google Scholar
Chandler-Wilde, S. N., Graham, I. G., Langdon, S. and Lindner, M. (2009), Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equ. Appl. 21, 229–279.10.1216/JIE-2009-21-2-229CrossRefGoogle Scholar
Chandler-Wilde, S. N., Graham, I. G., Langdon, S. and Spence, E. A. (2012), Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer. 21, 89–305.10.1017/S0962492912000037CrossRefGoogle Scholar
Chandler-Wilde, S. N., Hewett, D. P., Langdon, S. and Twigger, A. (2015), A high frequency boundary element method for scattering by a class of nonconvex obstacles, Numer. Math. 129, 647–689.10.1007/s00211-014-0648-7CrossRefGoogle Scholar
Chandler-Wilde, S. N., Spence, E. A., Gibbs, A. and Smyshlyaev, V. P. (2020), High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis, SIAM J. Math. Anal. 52, 845–893.10.1137/18M1234916CrossRefGoogle Scholar
Chaumont Frelet, T. (2015), Approximation par éléments finis de problèmes d’Helmholtz pour la propagation d’ondes sismiques. PhD thesis, Rouen, INSA.Google Scholar
Chaumont-Frelet, T. (2025), A ill-posed scattering problem saturating Weyl’s law. Available at arXiv:2505.10346.Google Scholar
Chaumont-Frelet, T. and Nicaise, S. (2020), Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problem, IMA J. Numer. Anal. 40, 1503–1543.10.1093/imanum/drz020CrossRefGoogle Scholar
Chaumont-Frelet, T. and Spence, E. A. (2023), Scattering by finely layered obstacles: frequency-explicit bounds and homogenization, SIAM J. Math. Anal. 55, 1319–1363.10.1137/21M1450136CrossRefGoogle Scholar
Chaumont-Frelet, T. and Spence, E. A. (2025), 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. Available at doi:10.1093/imanum/draf035.Google Scholar
Chaumont-Frelet, T., Dolean, V. and Ingremeau, M. (2024a), Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states, Numer. Math. 156, 1385–1426.10.1007/s00211-024-01411-0CrossRefGoogle Scholar
Chaumont-Frelet, T., Galkowski, J. and Spence, E. A. (2024b), Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations. Available at arXiv:2408.04507. To appear in Math. Comp. Google Scholar
Chaumont-Frelet, T., Gallistl, S., , D.and Nicaise, and Tomezyk, J. (2022), Wavenumber explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers, Commun. Math. Sci. 20, 1–52.10.4310/CMS.2022.v20.n1.a1CrossRefGoogle Scholar
Cheeger, J. and Taylor, M. (1982a), Diffraction by conical singularities II, Commun. Pure Appl. Math. 35, 487–529.10.1002/cpa.3160350403CrossRefGoogle Scholar
Cheeger, J. and Taylor, M. (1982b), On the diffraction of waves by conical singularities I, Commun. Pure Appl. Math. 35, 275–331.CrossRefGoogle Scholar
Chen, Z. and Xiang, X. (2013), A source transfer domain decomposition method for Helmholtz equations in unbounded domain, SIAM J. Numer. Anal. 51, 2331–2356.10.1137/130917144CrossRefGoogle Scholar
Christianson, H. (2018), High-frequency resolvent estimates on asymptotically Euclidean warped products, Commun. Partial Differential Equations 43, 1306–1362.10.1080/03605302.2018.1517786CrossRefGoogle Scholar
Christianson, H. and Wunsch, J. (2013), Local smoothing for the Schrödinger equation with a prescribed loss, Amer. J. Math. 135, 1601–1632.10.1353/ajm.2013.0047CrossRefGoogle Scholar
Ciarlet, P. G. (2002), The Finite Element Method For Elliptic Problems , Classics in Applied Mathematics, SIAM.10.1137/1.9780898719208CrossRefGoogle Scholar
Colton, D. and Kress, R. (1983), Integral Equation Methods in Scattering Theory , Wiley.Google Scholar
Constantine, G. M. and Savits, T. H. (1996), A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc. 348, 503–520.10.1090/S0002-9947-96-01501-2CrossRefGoogle Scholar
Costabel, M., Dauge, M. and Nicaise, S. (2010), Corner singularities and analytic regularity for linear elliptic systems, I: Smooth domains. Available at https://hal.archives-ouvertes.fr/file/index/docid/453934/filename/CoDaNi_Analytic_Part_I.pdf.Google Scholar
Cummings, P. and Feng, X. (2006), Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci. 16, 139–160.10.1142/S021820250600108XCrossRefGoogle Scholar
Datchev, K. and Vasy, A. (2012a), Propagation through trapped sets and semiclassical resolvent estimates, Ann. Inst. Fourier 62, 2347–2377.10.5802/aif.2751CrossRefGoogle Scholar
Datchev, K. and Vasy, A. (2012b), Semiclassical resolvent estimates at trapped sets, Ann. Inst. Fourier 62, 2379–2384.10.5802/aif.2752CrossRefGoogle Scholar
Demlow, A., Guzmán, J. and Schatz, A. H. (2011), Local energy estimates for the finite element method on sharply varying grids, Math. Comp. 80, 1–9.10.1090/S0025-5718-2010-02353-1CrossRefGoogle Scholar
Deraemaeker, A., Babuška, I. and Bouillard, P. (1999), Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions, Internat. J. Numer. Methods Engrg 46, 471–499.10.1002/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-63.0.CO;2-6>CrossRefGoogle Scholar
Descloux, J. (1975), Interior regularity and local convergence of Galerkin finite element approximations for elliptic equations, in Topics in Numerical Analysis II (Miller, J. J. H., ed.), Academic Press, pp. 27–41.10.1016/B978-0-12-496952-0.50010-6CrossRefGoogle Scholar
Dimassi, M. and Sjöstrand, J. (1999), Spectral Asymptotics in the Semi-Classical Limit , Vol. 268 of London Mathematical Society Lecture Note Series, Cambridge University Press.10.1017/CBO9780511662195CrossRefGoogle Scholar
DLMF (2025), NIST Digital Library of Mathematical Functions (F. W. J. Olver et al., eds). Release 1.2.4 of 2025-03-15. Available at https://dlmf.nist.gov/.Google Scholar
Dolean, V., Jolivet, P. and Nataf, F. (2015), An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation , SIAM.CrossRefGoogle Scholar
Douglas, J. Jr, Santos, J. E., Sheen, D. and Bennethum, L. S. (1993), Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci. 3, 171–194.10.1142/S0218202593000102CrossRefGoogle Scholar
Du, Y. and Wu, H. (2015), Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number, SIAM J. Numer. Anal. 53, 782–804.10.1137/140953125CrossRefGoogle Scholar
Du, Y. and Wu, H. (2020), A pure source transfer domain decomposition method for Helmholtz equations in unbounded domain, J. Sci. Comput. 83, art. 68.10.1007/s10915-020-01249-2CrossRefGoogle Scholar
Dyatlov, S. and Zworski, M. (2019), Mathematical Theory of Scattering Resonances , Vol. 200 of Graduate Studies in Mathematics, American Mathematical Society.10.1090/gsm/200CrossRefGoogle Scholar
Engquist, B. and Majda, A. (1977a), Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. USA 74, 1765–1766.10.1073/pnas.74.5.1765CrossRefGoogle Scholar
Engquist, B. and Majda, A. (1977b), Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31, 629–651.10.1090/S0025-5718-1977-0436612-4CrossRefGoogle Scholar
Engquist, B. and Runborg, O. (2003), Computational high frequency wave propagation, Acta Numer. 12, 181–266.10.1017/S0962492902000119CrossRefGoogle Scholar
Engquist, B. and Ying, L. (2011), Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers, Multiscale. Model. Simul. 9, 686–710.10.1137/100804644CrossRefGoogle Scholar
Engquist, B. and Zhao, H.-K. (1998), Absorbing boundary conditions for domain decomposition, Appl. Numer. Math. 27, 341–365.10.1016/S0168-9274(98)00019-1CrossRefGoogle Scholar
Erlangga, Y. A. (2008), Advances in iterative methods and preconditioners for the Helmholtz equation, Arch. Comput. Methods Engrg 15, 37–66.10.1007/s11831-007-9013-7CrossRefGoogle Scholar
Ern, A. and Guermond, J.-L. (2021), Finite Elements I: Approximation and Interpolation , Vol. 72 of Texts in Applied Mathematics, Springer Nature.10.1007/978-3-030-56341-7CrossRefGoogle Scholar
Ernst, O. G. and Gander, M. J. (2012), Why it is difficult to solve Helmholtz problems with classical iterative methods, in Numerical Analysis of Multiscale Problems (Graham, I. G. et al., eds), Vol. 83 of Lecture Notes in Computational Science and Engineering, Springer, pp. 325–363.10.1007/978-3-642-22061-6_10CrossRefGoogle Scholar
Escauriaza, L., Montaner, S. and Zhang, C. (2017), Analyticity of solutions to parabolic evolutions and applications, SIAM J. Math. Anal. 49, 4064–4092.10.1137/15M1039705CrossRefGoogle Scholar
Esterhazy, S. and Melenk, J. M. (2012), On stability of discretizations of the Helmholtz equation, in Numerical Analysis of Multiscale Problems (Graham, I. G. et al., eds), Springer, pp. 285–324.10.1007/978-3-642-22061-6_9CrossRefGoogle Scholar
Feng, X. and Wu, H. (2009), Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal. 47, 2872–2896.10.1137/080737538CrossRefGoogle Scholar
Feng, X. and Wu, H. (2011), hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp. 80, 1997–2024.10.1090/S0025-5718-2011-02475-0CrossRefGoogle Scholar
Filonov, N. (2001), Second-order elliptic equation of divergence form having a compactly supported solution, J. Math. Sci. 106, 3078–3086.10.1023/A:1011379807662CrossRefGoogle Scholar
Friedrichs, K. (1934), Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109, 465–487.10.1007/BF01449150CrossRefGoogle Scholar
Galkowski, J. (2019), Distribution of Resonances in Scattering by Thin Barriers , Vol. 259 of Memoirs of the American Mathematical Society, American Mathematical Society.Google Scholar
Galkowski, J. (2025), Lower bounds for piecewise polynomial approximations of oscillatory functions, J. Approx. Theory 305, art. 106100.10.1016/j.jat.2024.106100CrossRefGoogle Scholar
Galkowski, J. and Spence, E. A. (2023), Does the Helmholtz boundary element method suffer from the pollution effect?, SIAM Rev. 65, 806–828.10.1137/22M1474199CrossRefGoogle Scholar
Galkowski, J. and Spence, E. A. (2025), Sharp preasymptotic error bounds for the Helmholtz h-FEM, SIAM J. Numer. Anal. 63, 1–23.10.1137/23M1546178CrossRefGoogle Scholar
Galkowski, J. and Spence, E. A. (2026), Convergence theory for two-level hybrid Schwarz preconditioners for high-frequency Helmholtz problems, SIAM J. Numer. Anal. 64, 29–54.10.1137/25M1726972CrossRefGoogle Scholar
Galkowski, J., Gong, S., Graham, I. G., Lafontaine, D. and Spence, E. A. (2024a), Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems. Available at arXiv:2404.02156.Google Scholar
Galkowski, J., Lafontaine, D. and Spence, E. A. (2023a), Perfectly-matched-layer truncation is exponentially accurate at high frequency, SIAM J. Math. Anal. 55, 3344–3394.10.1137/21M1443716CrossRefGoogle Scholar
Galkowski, J., Lafontaine, D. and Spence, E. A. (2024b), Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves, IMA J. Numer. Anal. 44, 1946–2069.10.1093/imanum/drad058CrossRefGoogle Scholar
Galkowski, J., Lafontaine, D., Spence, E. A. and Wunsch, J. (2023b), Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method, SIAM J. Math. Anal. 55, 3903–3958.10.1137/21M1409160CrossRefGoogle Scholar
Galkowski, J., Lafontaine, D., Spence, E. A. and Wunsch, J. (2024c), The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect, Commun. Math. Sci. 22, 1761–1816.10.4310/CMS.240918021620CrossRefGoogle Scholar
Galkowski, J., Marchand, P. and Spence, E. A. (2022), High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem, Integral Equations Operator Theory 94, art. 36.10.1007/s00020-022-02715-2CrossRefGoogle Scholar
Galkowski, J., Müller, E. H. and Spence, E. A. (2019), Wavenumber-explicit analysis for the Helmholtz h-BEM: Error estimates and iteration counts for the Dirichlet problem, Numer. Math. 142, 329–357.10.1007/s00211-019-01032-yCrossRefGoogle Scholar
Galkowski, J., Rachh, M. and Spence, E. A. (2025), Helmholtz boundary integral methods and the pollution effect. Available at arXiv:2507.22797.Google Scholar
Galkowski, J., Spence, E. A. and Wunsch, J. (2020), Optimal constants in nontrapping resolvent estimates, Pure Appl. Anal. 2, 157–202.10.2140/paa.2020.2.157CrossRefGoogle Scholar
Gander, M. J. (2006), Optimized Schwarz methods, SIAM J. Numer. Anal. 44, 699–731.10.1137/S0036142903425409CrossRefGoogle Scholar
Gander, M. J. and Zhang, H. (2019), A class of iterative solvers for the Helmholtz equation: Factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods, SIAM Rev. 61, 3–76.10.1137/16M109781XCrossRefGoogle Scholar
Gander, M. J. and Zhang, H. (2022), Schwarz methods by domain truncation, Acta Numer. 31, 1–134.10.1017/S0962492922000034CrossRefGoogle Scholar
Ganesh, M. and Morgenstern, C. (2020), A coercive heterogeneous media Helmholtz model: Formulation, wavenumber-explicit analysis, and preconditioned high-order FEM, Numer. Algorithms 83, 1441–1487.10.1007/s11075-019-00732-8CrossRefGoogle Scholar
Graham, I. G. and Sauter, S. A. (2020), Stability and finite element error analysis for the Helmholtz equation with variable coefficients, Math. Comp. 89, 105–138.10.1090/mcom/3457CrossRefGoogle Scholar
Graham, I. G., Löhndorf, M., Melenk, J. M. and Spence, E. A. (2015), When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k?, BIT Numer. Math. 55, 171–214.10.1007/s10543-014-0501-5CrossRefGoogle Scholar
Graham, I. G., Pembery, O. R. and Spence, E. A. (2019), The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances, J. Differential Equations 266, 2869–2923.10.1016/j.jde.2018.08.048CrossRefGoogle Scholar
Grubb, G. (2008), Distributions and Operators , Vol. 252 of Graduate Texts in Mathematics, Springer.Google Scholar
Guo, B. and Zhang, J. (2009), Stable and compatible polynomial extensions in three dimensions and applications to the p and h-p finite element method, SIAM J. Numer. Anal. 47, 1195–1225.10.1137/070688006CrossRefGoogle Scholar
Harari, I. and Hughes, T. J. R. (1991), Finite element methods for the Helmholtz equation in an exterior domain: Model problems, Comput. Methods Appl. Mech. Engrg 87, 59–96.10.1016/0045-7825(91)90146-WCrossRefGoogle Scholar
Hetmaniuk, U. (2007), Stability estimates for a class of Helmholtz problems, Commun. Math. Sci. 5, 665–678.CrossRefGoogle Scholar
Hiptmair, R., Moiola, A. and Perugia, I. (2016), A survey of Trefftz methods for the Helmholtz equation, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations , Springer, pp. 237–279.10.1007/978-3-319-41640-3_8CrossRefGoogle Scholar
Hohage, T., Schmidt, F. and Zschiedrich, L. (2003), Solving time-harmonic scattering problems based on the pole condition II: Convergence of the PML method, SIAM J. Math. Anal. 35, 547–560.10.1137/S0036141002406485CrossRefGoogle Scholar
Hörmander, L. (1959), On the uniqueness of the Cauchy problem II, Math. Scand. 7, 177–190.10.7146/math.scand.a-10571CrossRefGoogle Scholar
Hörmander, L. (1985), The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators , Springer.Google Scholar
Ihlenburg, F. (1998), Finite Element Analysis of Acoustic Scattering , Springer.10.1007/b98828CrossRefGoogle Scholar
Ihlenburg, F. and Babuška, I. (1995a), Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Internat. J. Numer. Methods Engrg 38, 3745–3774.10.1002/nme.1620382203CrossRefGoogle Scholar
Ihlenburg, F. and Babuška, I. (1995b), Finite element solution of the Helmholtz equation with high wave number I: The h-version of the FEM, Comput. Math. Appl. 30, 9–37.10.1016/0898-1221(95)00144-NCrossRefGoogle Scholar
Ihlenburg, F. and Babuška, I. (1997), Finite element solution of the Helmholtz equation with high wave number II: The hp version of the FEM, SIAM J. Numer. Anal. 34, 315–358.10.1137/S0036142994272337CrossRefGoogle Scholar
Jerison, D. and Kenig, C. E. (1985), Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. 121, 463–488.10.2307/1971205CrossRefGoogle Scholar
Kim, S. and Pasciak, J. E. (2010), Analysis of a Cartesian PML approximation to acoustic scattering problems in ℝ2, J. Math. Anal. Appl. 370, 168–186.10.1016/j.jmaa.2010.05.006CrossRefGoogle Scholar
Kohn, J. J. and Nirenberg, L. (1965), An algebra of pseudo-differential operators, Commun. Pure Appl. Math. 18, 269–305.10.1002/cpa.3160180121CrossRefGoogle Scholar
Lafontaine, D., Spence, E. A. and Wunsch, J. (2021), For most frequencies, strong trapping has a weak effect in frequency-domain scattering, Commun. Pure Appl. Math. 74, 2025–2063.10.1002/cpa.21932CrossRefGoogle Scholar
Lafontaine, D., Spence, E. A. and Wunsch, J. (2022a), A sharp relative-error bound for the Helmholtz h-FEM at high frequency, Numer. Math. 150, 137–178.10.1007/s00211-021-01253-0CrossRefGoogle Scholar
Lafontaine, D., Spence, E. A. and Wunsch, J. (2022b), Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients, Comput. Math. Appl. 113, 59–69.10.1016/j.camwa.2022.03.007CrossRefGoogle Scholar
Lahaye, D., Tang, J. and Vuik, K., eds (2017), Modern Solvers for Helmholtz Problems , Geosystems Mathematics, Birkhäuser.CrossRefGoogle Scholar
Lassas, M. and Somersalo, E. (1998), On the existence and convergence of the solution of PML equations, Computing 60, 229–241.CrossRefGoogle Scholar
Lassas, M. and Somersalo, E. (2001), Analysis of the PML equations in general convex geometry, Proc. Roy. Soc. Edinburgh Sect. A 131, 1183–1207.10.1017/S0308210500001335CrossRefGoogle Scholar
Lax, P. D. and Phillips, R. S. (1989), Scattering Theory , second edition, Academic Press.Google Scholar
Leis, R. (1965), Zur Dirichletschen Randwertaufgabe des Aussenraumes der Schwingungsgleichung, Math. Z. 90, 205–211.10.1007/BF01119203CrossRefGoogle Scholar
Leng, W. and Ju, L. (2019), An additive overlapping domain decomposition method for the Helmholtz equation, SIAM J. Sci. Comput. 41, A1252–A1277.10.1137/18M1196170CrossRefGoogle Scholar
Leng, W. and Ju, L. (2021), A diagonal sweeping domain decomposition method with source transfer for the Helmholtz equation, Commun. Comput. Phys. 29, 357–398.10.4208/cicp.OA-2020-0169CrossRefGoogle Scholar
Leng, W. and Ju, L. (2022), Trace transfer-based diagonal sweeping domain decomposition method for the Helmholtz equation: Algorithms and convergence analysis, J. Comput. Phys. 455, art. 110980.10.1016/j.jcp.2022.110980CrossRefGoogle Scholar
Lenoir, M. (1986), Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal. 23, 562–580.10.1137/0723036CrossRefGoogle Scholar
Lerner, N. (2019), Carleman Inequalities: An Introduction and More , Vol. 353 of Grundlehren der mathematischen Wissenschaften, Springer.10.1007/978-3-030-15993-1CrossRefGoogle Scholar
Li, Y. and Wu, H. (2019), FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation, SIAM J. Numer. Anal. 57, 96–126.10.1137/17M1140522CrossRefGoogle Scholar
Löhndorf, M. and Melenk, J. M. (2011), Wavenumber-explicit hp-BEM for high frequency scattering, SIAM J. Numer. Anal. 49, 2340–2363.10.1137/100786034CrossRefGoogle Scholar
Makridakis, C. H., Ihlenburg, F. and Babuška, I. (1996), Analysis and finite element methods for a fluid–solid interaction problem in one dimension, Math. Models Methods Appl. Sci. 6, 1119–1141.10.1142/S0218202596000468CrossRefGoogle Scholar
Marburg, S. (2002), Six boundary elements per wavelength: Is that enough?, J. Comput. Acous. 10, 25–51.10.1142/S0218396X02001401CrossRefGoogle Scholar
Marburg, S. (2016), Numerical damping in the acoustic boundary element method, Acta Acust. 102, 415–418.10.3813/AAA.918958CrossRefGoogle Scholar
Marburg, S. (2017), Benchmark problem identifying a pollution effect in boundary element method, J. Acoust. Soc. Am. 141, 3975–3975.10.1121/1.4989067CrossRefGoogle Scholar
Marchand, P., Galkowski, J., Spence, A. and Spence, E. A. (2022), Applying GMRES to the Helmholtz equation with strong trapping: How does the number of iterations depend on the frequency?, Adv. Comput. Math. 48, 1–63.10.1007/s10444-022-09931-9CrossRefGoogle Scholar
McLean, W. (2000), Strongly Elliptic Systems and Boundary Integral Equations , Cambridge University Press.Google Scholar
Melenk, J. M. (1995), On generalized finite element methods. PhD thesis, The University of Maryland.Google Scholar
Melenk, J. M. (2004), hp-Finite Element Methods for Singular Perturbations , Springer.Google Scholar
Melenk, J. M. and Sauter, S. (2010), Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp 79(272), 1871–1914.10.1090/S0025-5718-10-02362-8CrossRefGoogle Scholar
Melenk, J. M. and Sauter, S. (2011), Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation, SIAM J. Numer. Anal. 49, 1210–1243.10.1137/090776202CrossRefGoogle Scholar
Melenk, J. M. and Sauter, S. A. (2021), Wavenumber-explicit hp-FEM analysis for Maxwell’s equations with transparent boundary conditions, Found. Comput. Math. 21, 125–241.10.1007/s10208-020-09452-1CrossRefGoogle Scholar
Melenk, J. M., Parsania, A. and Sauter, S. (2013), General DG-methods for highly indefinite Helmholtz problems, J. Sci. Comput. 57, 536–581.10.1007/s10915-013-9726-8CrossRefGoogle Scholar
Melrose, R. B. (1979), Singularities and energy decay in acoustical scattering, Duke Math. J. 46, 43–59.10.1215/S0012-7094-79-04604-0CrossRefGoogle Scholar
Melrose, R. B. and Sjöstrand, J. (1978), Singularities of boundary value problems I, Commun. Pure Appl. Math. 31, 593–617.10.1002/cpa.3160310504CrossRefGoogle Scholar
Melrose, R. B. and Sjöstrand, J. (1982), Singularities of boundary value problems II, Commun. Pure Appl. Math. 35, 129–168.10.1002/cpa.3160350202CrossRefGoogle Scholar
Melrose, R. B. and Taylor, M. E. (1985), Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. Math. 55, 242–315.10.1016/0001-8708(85)90093-3CrossRefGoogle Scholar
Miller, K. (1974), Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Ration. Mech. Anal. 54, 105–117.10.1007/BF00247634CrossRefGoogle Scholar
Moiola, A. and Spence, E. A. (2014), Is the Helmholtz equation really sign-indefinite?, SIAM Rev. 56, 274–312.10.1137/120901301CrossRefGoogle Scholar
Moiola, A. and Spence, E. A. (2019), Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions, Math. Models Methods Appl. Sci. 29, 317–354.10.1142/S0218202519500106CrossRefGoogle Scholar
Morawetz, C. S. (1975), Decay for solutions of the exterior problem for the wave equation, Commun. Pure Appl. Math. 28, 229–264.10.1002/cpa.3160280204CrossRefGoogle Scholar
Morawetz, C. S. and Ludwig, D. (1968), An inequality for the reduced wave operator and the justification of geometrical optics, Commun. Pure Appl. Math. 21, 187–203.10.1002/cpa.3160210206CrossRefGoogle Scholar
Morawetz, C. S., Ralston, J. V. and Strauss, W. A. (1977), Decay of solutions of the wave equation outside nontrapping obstacles, Commun. Pure Appl. Math. 30, 447–508.10.1002/cpa.3160300405CrossRefGoogle Scholar
Muñoz-Sola, R. (1997), Polynomial liftings on a tetrahedron and applications to the hp version of the finite element method in three dimensions, SIAM J. Numer. Anal. 34, 282–314.10.1137/S0036142994267552CrossRefGoogle Scholar
Nataf, F., Rogier, F. and de Sturler, E. (1994), Optimal interface conditions for domain decomposition methods. Technical Report 301, CMAP Ecole Polytechnique. Available at hal-02194208.Google Scholar
Nitsche, J. (1968), Ein Kriterium für die Quasi-Optimalität des ritzschen Verfahrens, Numer. Math. 11, 346–348.10.1007/BF02166687CrossRefGoogle Scholar
Nitsche, J. A. and Schatz, A. H. (1974), Interior estimates for Ritz–Galerkin methods, Math. Comp. 28, 937–958.10.1090/S0025-5718-1974-0373325-9CrossRefGoogle Scholar
Ohlberger, M. and Verfurth, B. (2018), A new heterogeneous multiscale method for the Helmholtz equation with high contrast, Multiscale Model. Simul. 16, 385–411.10.1137/16M1108820CrossRefGoogle Scholar
Olver, P. (1993), Applications of Lie Groups to Differential Equations , second edition, Springer.10.1007/978-1-4612-4350-2CrossRefGoogle Scholar
Panich, O. I. (1965), On the question of the solvability of exterior boundary-value problems for the wave equation and for a system of Maxwell’s equations (in Russian), Uspekhi Mat. Nauk 20, 221–226.Google Scholar
Pembery, O. R. (2020), The Helmholtz equation in heterogeneous and random media:Analysis and numerics. PhD thesis, University of Bath. Available at https://research-portal.bath.ac.uk/en/studentTheses/the-helmholtz-equation-in-heterogeneous-and-random-media-analysis.Google Scholar
Perthame, B. and Vega, L. (1999), Morrey–Campanato estimates for Helmholtz equations, J. Funct. Anal. 164, 340–355.10.1006/jfan.1999.3391CrossRefGoogle Scholar
Plis, A. (1963), On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys 11, 95–100.Google Scholar
Reed, M. and Simon, B. (1980), Methods of Modern Mathematical Physics I: Functional Analysis , second edition, Academic Press.Google Scholar
Rellich, F. (1940), Darstellung der Eigenwerte von Δu + λu = 0 durch ein Randintegral, Math. Z. 46, 635–636.10.1007/BF01181459CrossRefGoogle Scholar
Rellich, F. (1943), Über das asymptotische Verhalten der Lösungen von Δu + λu = 0 in unendlichen Gebieten, Jahresber. Dtsch. Math.-Ver. 53, 57–65.Google Scholar
Royer, A., Geuzaine, C., Béchet, E. and Modave, A. (2022), A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg 395, art. 115006.10.1016/j.cma.2022.115006CrossRefGoogle Scholar
Saad, Y. and Schultz, M. H. (1986), GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7, 856–869.10.1137/0907058CrossRefGoogle Scholar
Sauter, S. A. (2006), A refined finite element convergence theory for highly indefinite Helmholtz problems, Computing 78, 101–115.10.1007/s00607-006-0177-zCrossRefGoogle Scholar
Sauter, S. A. and Schwab, C. (2011), Boundary Element Methods , Springer.10.1007/978-3-540-68093-2CrossRefGoogle Scholar
Schatz, A. H. (1974), An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp. 28, 959–962.10.1090/S0025-5718-1974-0373326-0CrossRefGoogle Scholar
Schatz, A. H. and Wahlbin, L. B. (1977), Interior maximum norm estimates for finite element methods, Math. Comp. 31, 414–442.10.1090/S0025-5718-1977-0431753-XCrossRefGoogle Scholar
Schatz, A. H. and Wahlbin, L. B. (1982), On the quasi-optimality in L ∞ of the ${\overset{\circ }{H}}_1$ -projection into finite element spaces, Math. Comp. 38, 1–22.Google Scholar
Schatz, A. H. and Wang, J. (1996), Some new error estimates for Ritz–Galerkin methods with minimal regularity assumptions, Math. Comp. 65, 19–27.10.1090/S0025-5718-96-00649-7CrossRefGoogle Scholar
Schwab, C. (1998), p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics , Numerical Mathematics and Scientific Computation, Clarendon Press.Google Scholar
Scott, L. R. (1973), Finite element techniques for curved boundaries. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Shannon, C. E. (1949), Communication in the presence of noise, Proc. IRE 37, 10–21.10.1109/JRPROC.1949.232969CrossRefGoogle Scholar
Sjöstrand, J. and Zworski, M. (1991), Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4, 729–769.10.1090/S0894-0347-1991-1115789-9CrossRefGoogle Scholar
Spence, E. A. (2014), Wavenumber-explicit bounds in time-harmonic acoustic scattering, SIAM J. Math. Anal. 46, 2987–3024.10.1137/130932855CrossRefGoogle Scholar
Spence, E. A. (2023), A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation, Adv. Comput. Math. 49, art. 27.10.1007/s10444-023-10025-3CrossRefGoogle Scholar
Spence, E. A., Kamotski, I. V. and Smyshlyaev, V. P. (2015), Coercivity of combined boundary integral equations in high-frequency scattering, Commun. Pure Appl. Math. 68, 1587–1639.10.1002/cpa.21543CrossRefGoogle Scholar
Stolk, C. C. (2013), A rapidly converging domain decomposition method for the Helmholtz equation, J. Comput. Phys. 241, 240–252.10.1016/j.jcp.2013.01.039CrossRefGoogle Scholar
Taus, M., Zepeda-Núñez, L., Hewett, R. J. and Demanet, L. (2019), L-sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation, J. Comput. Phys. 420, art. 109706.Google Scholar
Taylor, M. E. (1996), Partial Differential Equations II: Qualitative Studies of Linear Equations , Vol. 116 of Applied Mathematical Sciences, Springer.Google Scholar
Taylor, M. E. (2023), Partial Differential Equations III: Nonlinear Equations , Vol. 117 of Applied Mathematical Sciences, third edition, Springer.10.1007/978-3-031-33928-8CrossRefGoogle Scholar
Toselli, A. (1998), Overlapping method with perfectly matched layers for the solution of the Helmholtz equation, in Proceedings of the 11th International Conference on Domain Decomposition Methods , pp. 551–557. Available at http://www.ddm.org/ DD11/DD11proc.pdf.Google Scholar
Vainberg, B. R. (1975), On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t → ∞ of solutions of non-stationary problems, Russian Math. Surveys 30, 1–58.10.1070/RM1975v030n02ABEH001406CrossRefGoogle Scholar
Vico, F., Greengard, L. and Gimbutas, Z. (2014), Boundary integral equation analysis on the sphere, Numer. Math. 128, 463–487.10.1007/s00211-014-0619-zCrossRefGoogle Scholar
Vodev, G. (2000), On the exponential bound of the cutoff resolvent, Serdica Math. J. 26, 49–58.Google Scholar
Wahlbin, L. B. (1991), Local behavior in finite element methods, in Finite Element Methods 1 (Ciarlet, P. G. and Lions, J. L., eds), Vol. II of Handbook of Numerical Analysis, Elsevier, pp. 353–522.Google Scholar
Whittaker, E. T. (1915), On the functions which are represented by the expansions of the interpolation-theory, Proc. Roy. Soc. Edinburgh 35, 181–194.10.1017/S0370164600017806CrossRefGoogle Scholar
Wolff, T. (1993), Recent work on sharp estimates in second-order elliptic unique continuation problems, J. Geom. Anal. 3, 621–650.10.1007/BF02921325CrossRefGoogle Scholar
Wolff, T. H. (1992), A property of measures in ℝN and an application to unique continuation, Geom. Funct. Anal. 2, 225–284.10.1007/BF01896975CrossRefGoogle Scholar
Wu, H. (2014), Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number I: Linear version, IMA J. Numer. Anal. 34, 1266–1288.10.1093/imanum/drt033CrossRefGoogle Scholar
Zepeda-Núñez, L. and Demanet, L. (2016), The method of polarized traces for the 2D Helmholtz equation, J. Comput. Phys. 308, 347–388.10.1016/j.jcp.2015.11.040CrossRefGoogle Scholar
Zlámal, M. (1973), Curved elements in the finite element method I, SIAM J. Numer. Anal. 10, 229–240.10.1137/0710022CrossRefGoogle Scholar
Zworski, M. (2012), Semiclassical Analysis , Vol. 138 of Graduate Studies in Mathematics, American Mathematical Society.10.1090/gsm/138CrossRefGoogle Scholar