{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T08:51:38Z","timestamp":1781859098476,"version":"3.54.5"},"reference-count":213,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T00:00:00Z","timestamp":1781827200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Acta Numerica"],"published-print":{"date-parts":[[2026,7]]},"abstract":"<jats:p>\n                    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\n                    <jats:italic>semiclassical analysis<\/jats:italic>\n                    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.\n                  <\/jats:p>\n                  <jats:p>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.<\/jats:p>","DOI":"10.1017\/s0962492925100275","type":"journal-article","created":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T07:58:58Z","timestamp":1781855938000},"page":"1-171","source":"Crossref","is-referenced-by-count":0,"title":["Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis"],"prefix":"10.1017","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5228-4998","authenticated-orcid":false,"given":"Jeffrey","family":"Galkowski","sequence":"first","affiliation":[{"name":"University College London"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1236-4592","authenticated-orcid":false,"given":"Euan A.","family":"Spence","sequence":"additional","affiliation":[{"name":"University of Bath"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,6,19]]},"reference":[{"key":"S0962492925100275_r115","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3457"},{"key":"S0962492925100275_r116","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-014-0501-5"},{"key":"S0962492925100275_r117","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2018.08.048"},{"key":"S0962492925100275_r170","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994267552"},{"key":"S0962492925100275_r107","doi-asserted-by":"publisher","DOI":"10.1007\/s00020-022-02715-2"},{"key":"S0962492925100275_r142","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.OA-2020-0169"},{"key":"S0962492925100275_r73","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2010-02353-1"},{"key":"S0962492925100275_r183","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115006"},{"key":"S0962492925100275_r78","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974065"},{"key":"S0962492925100275_r32","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-06-01930-2"},{"key":"S0962492925100275_r149","doi-asserted-by":"publisher","DOI":"10.1142\/S0218396X02001401"},{"key":"S0962492925100275_r68","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-96-01501-2"},{"key":"S0962492925100275_r185","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-006-0177-z"},{"key":"S0962492925100275_r109","unstructured":"Galkowski, J. , Rachh, M. and Spence, E. A. (2025), Helmholtz boundary integral methods and the pollution effect. Available at arXiv:2507.22797."},{"key":"S0962492925100275_r29","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drs038"},{"key":"S0962492925100275_r140","doi-asserted-by":"publisher","DOI":"10.1007\/BF01119203"},{"key":"S0962492925100275_r85","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492902000119"},{"key":"S0962492925100275_r17","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3165"},{"key":"S0962492925100275_r120","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(91)90146-W"},{"key":"S0962492925100275_r19","first-page":"257","article-title":"Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization","volume":"35","author":"Bellassoued","year":"2003","journal-title":"Asymptot. Anal."},{"key":"S0962492925100275_r94","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2011-02475-0"},{"key":"S0962492925100275_r49","doi-asserted-by":"publisher","DOI":"10.1216\/JIE-2009-21-2-229"},{"key":"S0962492925100275_r202","unstructured":"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\u2013557. Available at http:\/\/www.ddm.org\/ DD11\/DD11proc.pdf."},{"key":"S0962492925100275_r74","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-6"},{"key":"S0962492925100275_r125","volume-title":"The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators","author":"H\u00f6rmander","year":"1985"},{"key":"S0962492925100275_r35","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-51954-8_6"},{"key":"S0962492925100275_r114","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-019-00732-8"},{"key":"S0962492925100275_r162","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160350202"},{"key":"S0962492925100275_r82","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/200"},{"key":"S0962492925100275_r12","doi-asserted-by":"publisher","DOI":"10.1137\/0728034"},{"key":"S0962492925100275_r186","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68093-2"},{"key":"S0962492925100275_r13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0505-0"},{"key":"S0962492925100275_r104","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drad058"},{"key":"S0962492925100275_r4","unstructured":"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."},{"key":"S0962492925100275_r199","article-title":"L-sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation","volume":"420","author":"Taus","year":"2019","journal-title":"J. Comput. Phys."},{"key":"S0962492925100275_r147","doi-asserted-by":"publisher","DOI":"10.1137\/100786034"},{"key":"S0962492925100275_r23","first-page":"27","volume-title":"Boundary Value Problems and Integral Equations in Nonsmooth Domains","volume":"167","author":"Bernardi","year":"1995"},{"key":"S0962492925100275_r41","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/004\/0143037"},{"key":"S0962492925100275_r188","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431753-X"},{"key":"S0962492925100275_r156","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-10-02362-8"},{"key":"S0962492925100275_r201","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-33928-8"},{"key":"S0962492925100275_r178","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.1999.3391"},{"key":"S0962492925100275_r44","article-title":"Sur un probl\u00e8me d\u2019unicit\u00e9 pour les syst\u00e8mes d\u2019\u00e9quations aux d\u00e9riv\u00e9es partielles \u00e0 deux variables ind\u00e9pendantes","volume":"26","author":"Carleman","year":"1939","journal-title":"Ark. Mat., Astr. Fys."},{"key":"S0962492925100275_r80","doi-asserted-by":"publisher","DOI":"10.1137\/140953125"},{"key":"S0962492925100275_r37","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392877"},{"key":"S0962492925100275_r40","first-page":"201","article-title":"The application of integral equation methods to the numerical solution of some exterior boundary-value problems","volume":"323","author":"Burton","year":"1971","journal-title":"Proc. R. Soc. A"},{"key":"S0962492925100275_r42","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1982-0637287-3"},{"key":"S0962492925100275_r65","doi-asserted-by":"publisher","DOI":"10.1353\/ajm.2013.0047"},{"key":"S0962492925100275_r145","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-15993-1"},{"key":"S0962492925100275_r63","doi-asserted-by":"publisher","DOI":"10.1137\/130917144"},{"key":"S0962492925100275_r47","doi-asserted-by":"publisher","DOI":"10.1137\/060662575"},{"key":"S0962492925100275_r208","doi-asserted-by":"publisher","DOI":"10.1007\/BF02921325"},{"key":"S0962492925100275_r144","doi-asserted-by":"publisher","DOI":"10.1137\/0723036"},{"key":"S0962492925100275_r193","doi-asserted-by":"publisher","DOI":"10.1109\/JRPROC.1949.232969"},{"key":"S0962492925100275_r64","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2018.1517786"},{"key":"S0962492925100275_r135","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2022.03.007"},{"key":"S0962492925100275_r192","volume-title":"Finite element techniques for curved boundaries","author":"Scott","year":"1973"},{"key":"S0962492925100275_r180","volume-title":"Methods of Modern Mathematical Physics I: Functional Analysis","author":"Reed","year":"1980"},{"key":"S0962492925100275_r60","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2022.v20.n1.a1"},{"key":"S0962492925100275_r210","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drt033"},{"key":"S0962492925100275_r99","doi-asserted-by":"publisher","DOI":"10.1137\/22M1474199"},{"key":"S0962492925100275_r106","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.240918021620"},{"key":"S0962492925100275_r161","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160310504"},{"key":"S0962492925100275_r112","doi-asserted-by":"publisher","DOI":"10.1137\/16M109781X"},{"key":"S0962492925100275_r172","doi-asserted-by":"publisher","DOI":"10.1007\/BF02166687"},{"key":"S0962492925100275_r70","doi-asserted-by":"publisher","DOI":"10.1142\/S021820250600108X"},{"key":"S0962492925100275_r189","first-page":"1","article-title":"On the quasi-optimality in L\u221e of the \n\n\n${\\overset{\\circ }{H}}_1$\n\n\n\n-projection into finite element spaces","volume":"38","author":"Schatz","year":"1982","journal-title":"Math. Comp."},{"key":"S0962492925100275_r184","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"S0962492925100275_r166","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202519500106"},{"key":"S0962492925100275_r118","volume-title":"Distributions and Operators","volume":"252","author":"Grubb","year":"2008"},{"key":"S0962492925100275_r6","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-024-10193-w"},{"key":"S0962492925100275_r198","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2013.01.039"},{"key":"S0962492925100275_r153","volume-title":"Strongly Elliptic Systems and Boundary Integral Equations","author":"McLean","year":"2000"},{"key":"S0962492925100275_r197","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21543"},{"key":"S0962492925100275_r61","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160350403"},{"key":"S0962492925100275_r75","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-496952-0.50010-6"},{"key":"S0962492925100275_r175","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"S0962492925100275_r100","doi-asserted-by":"publisher","DOI":"10.1137\/23M1546178"},{"key":"S0962492925100275_r91","doi-asserted-by":"publisher","DOI":"10.1137\/15M1039705"},{"key":"S0962492925100275_r58","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-024-01411-0"},{"key":"S0962492925100275_r71","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2751"},{"key":"S0962492925100275_r196","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-023-10025-3"},{"key":"S0962492925100275_r131","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.05.006"},{"key":"S0962492925100275_r113","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492922000034"},{"key":"S0962492925100275_r7","unstructured":"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."},{"key":"S0962492925100275_r36","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4302"},{"key":"S0962492925100275_r87","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(98)00019-1"},{"key":"S0962492925100275_r128","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(95)00144-N"},{"key":"S0962492925100275_r203","doi-asserted-by":"publisher","DOI":"10.1070\/RM1975v030n02ABEH001406"},{"key":"S0962492925100275_r187","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1974-0373326-0"},{"key":"S0962492925100275_r194","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1991-1115789-9"},{"key":"S0962492925100275_r34","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0"},{"key":"S0962492925100275_r88","doi-asserted-by":"publisher","DOI":"10.1007\/s11831-007-9013-7"},{"key":"S0962492925100275_r167","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160280204"},{"key":"S0962492925100275_r86","doi-asserted-by":"publisher","DOI":"10.1137\/100804644"},{"key":"S0962492925100275_r62","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160350302"},{"key":"S0962492925100275_r24","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3958"},{"key":"S0962492925100275_r148","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202596000468"},{"key":"S0962492925100275_r54","unstructured":"Chaumont-Frelet, T. (2025), A ill-posed scattering problem saturating Weyl\u2019s law. Available at arXiv:2505.10346."},{"key":"S0962492925100275_r207","doi-asserted-by":"publisher","DOI":"10.1017\/S0370164600017806"},{"key":"S0962492925100275_r20","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1994.1159"},{"key":"S0962492925100275_r38","doi-asserted-by":"publisher","DOI":"10.1155\/S1073792802103059"},{"key":"S0962492925100275_r111","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903425409"},{"key":"S0962492925100275_r57","article-title":"The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains","author":"Chaumont-Frelet","year":"2025","journal-title":"IMA J. Numer. Anal."},{"key":"S0962492925100275_r177","volume-title":"The Helmholtz equation in heterogeneous and random media:Analysis and numerics","author":"Pembery","year":"2020"},{"key":"S0962492925100275_r160","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-79-04604-0"},{"key":"S0962492925100275_r164","doi-asserted-by":"publisher","DOI":"10.1007\/BF00247634"},{"key":"S0962492925100275_r10","doi-asserted-by":"publisher","DOI":"10.1137\/0724049"},{"key":"S0962492925100275_r121","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2007.v5.n3.a8"},{"key":"S0962492925100275_r56","doi-asserted-by":"publisher","DOI":"10.1137\/21M1450136"},{"key":"S0962492925100275_r52","doi-asserted-by":"publisher","DOI":"10.1137\/18M1234916"},{"key":"S0962492925100275_r127","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620382203"},{"key":"S0962492925100275_r152","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-022-09931-9"},{"key":"S0962492925100275_r124","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10571"},{"key":"S0962492925100275_r108","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-019-01032-y"},{"key":"S0962492925100275_r138","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500001335"},{"key":"S0962492925100275_r176","first-page":"221","article-title":"On the question of the solvability of exterior boundary-value problems for the wave equation and for a system of Maxwell\u2019s equations (in Russian)","volume":"20","author":"Panich","year":"1965","journal-title":"Uspekhi Mat. Nauk"},{"key":"S0962492925100275_r169","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160300405"},{"key":"S0962492925100275_r98","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2024.106100"},{"key":"S0962492925100275_r173","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1974-0373325-9"},{"key":"S0962492925100275_r101","doi-asserted-by":"publisher","DOI":"10.1137\/25M1726972"},{"key":"S0962492925100275_r21","doi-asserted-by":"publisher","DOI":"10.1137\/0726068"},{"key":"S0962492925100275_r130","doi-asserted-by":"publisher","DOI":"10.2307\/1971205"},{"key":"S0962492925100275_r89","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-56341-7"},{"key":"S0962492925100275_r95","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011379807662"},{"key":"S0962492925100275_r77","unstructured":"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\/."},{"key":"S0962492925100275_r171","unstructured":"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."},{"key":"S0962492925100275_r209","doi-asserted-by":"publisher","DOI":"10.1007\/BF01896975"},{"key":"S0962492925100275_r96","doi-asserted-by":"publisher","DOI":"10.1007\/BF01449150"},{"key":"S0962492925100275_r5","first-page":"599","article-title":"Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin\u2019s and finite difference methods","volume":"21","author":"Aubin","year":"1967","journal-title":"Ann. Sc. Norm. Super. Pisa Cl. Sci."},{"key":"S0962492925100275_r205","first-page":"49","article-title":"On the exponential bound of the cutoff resolvent","volume":"26","author":"Vodev","year":"2000","journal-title":"Serdica Math. J."},{"key":"S0962492925100275_r103","doi-asserted-by":"publisher","DOI":"10.1137\/21M1443716"},{"key":"S0962492925100275_r200","volume-title":"Partial Differential Equations II: Qualitative Studies of Linear Equations","volume":"116","author":"Taylor","year":"1996"},{"key":"S0962492925100275_r84","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0436612-4"},{"key":"S0962492925100275_r69","unstructured":"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."},{"key":"S0962492925100275_r90","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22061-6_10"},{"key":"S0962492925100275_r92","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22061-6_9"},{"key":"S0962492925100275_r16","doi-asserted-by":"publisher","DOI":"10.1137\/0330055"},{"key":"S0962492925100275_r1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903423460"},{"key":"S0962492925100275_r53","volume-title":"Approximation par \u00e9l\u00e9ments finis de probl\u00e8mes d\u2019Helmholtz pour la propagation d\u2019ondes sismiques","author":"Chaumont Frelet","year":"2015"},{"key":"S0962492925100275_r67","volume-title":"Integral Equation Methods in Scattering Theory","author":"Colton","year":"1983"},{"key":"S0962492925100275_r119","doi-asserted-by":"publisher","DOI":"10.1137\/070688006"},{"key":"S0962492925100275_r174","doi-asserted-by":"publisher","DOI":"10.1137\/16M1108820"},{"key":"S0962492925100275_r39","volume-title":"Numerical Solution of Acoustic Radiation Problems","author":"Burton","year":"1976"},{"key":"S0962492925100275_r158","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-020-09452-1"},{"key":"S0962492925100275_r163","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(85)90093-3"},{"key":"S0962492925100275_r204","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-014-0619-z"},{"key":"S0962492925100275_r141","doi-asserted-by":"publisher","DOI":"10.1137\/18M1196170"},{"key":"S0962492925100275_r27","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(77)90011-0"},{"key":"S0962492925100275_r123","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141002406485"},{"key":"S0962492925100275_r2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2011.08.029"},{"key":"S0962492925100275_r66","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208"},{"key":"S0962492925100275_r102","unstructured":"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."},{"key":"S0962492925100275_r157","doi-asserted-by":"publisher","DOI":"10.1137\/090776202"},{"key":"S0962492925100275_r165","doi-asserted-by":"publisher","DOI":"10.1137\/120901301"},{"key":"S0962492925100275_r8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01395880"},{"key":"S0962492925100275_r110","doi-asserted-by":"publisher","DOI":"10.2140\/paa.2020.2.157"},{"key":"S0962492925100275_r18","doi-asserted-by":"publisher","DOI":"10.1137\/15M102530X"},{"key":"S0962492925100275_r159","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-013-9726-8"},{"key":"S0962492925100275_r97","volume-title":"Distribution of Resonances in Scattering by Thin Barriers","volume":"259","author":"Galkowski","year":"2019"},{"key":"S0962492925100275_r83","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.74.5.1765"},{"key":"S0962492925100275_r51","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-014-0648-7"},{"key":"S0962492925100275_r129","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994272337"},{"key":"S0962492925100275_r3","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(90)90089-5"},{"key":"S0962492925100275_r48","unstructured":"Chandler-Wilde, S. N. and Sadeghi, S. (2025), The cut-off resolvent can grow arbitrarily fast in obstacle scattering. Available at arXiv:2508.16958."},{"key":"S0962492925100275_r122","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-41640-3_8"},{"key":"S0962492925100275_r206","first-page":"353","volume-title":"Finite Element Methods 1","volume":"II","author":"Wahlbin","year":"1991"},{"key":"S0962492925100275_r22","first-page":"209","volume-title":"Techniques of Scientific Computing 2","volume":"5","author":"Bernardi","year":"1997"},{"key":"S0962492925100275_r93","doi-asserted-by":"publisher","DOI":"10.1137\/080737538"},{"key":"S0962492925100275_r45","doi-asserted-by":"publisher","DOI":"10.5802\/aif.181"},{"key":"S0962492925100275_r132","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160180121"},{"key":"S0962492925100275_r25","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(73)90062-0"},{"key":"S0962492925100275_r211","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.11.040"},{"key":"S0962492925100275_r31","doi-asserted-by":"publisher","DOI":"10.1007\/BF01220037"},{"key":"S0962492925100275_r151","doi-asserted-by":"publisher","DOI":"10.1121\/1.4989067"},{"key":"S0962492925100275_r212","doi-asserted-by":"publisher","DOI":"10.1137\/0710022"},{"key":"S0962492925100275_r146","doi-asserted-by":"publisher","DOI":"10.1137\/17M1140522"},{"key":"S0962492925100275_r11","doi-asserted-by":"publisher","DOI":"10.1137\/1036141"},{"key":"S0962492925100275_r139","volume-title":"Scattering Theory","author":"Lax","year":"1989"},{"key":"S0962492925100275_r15","doi-asserted-by":"publisher","DOI":"10.1137\/060654177"},{"key":"S0962492925100275_r154","volume-title":"On generalized finite element methods","author":"Melenk","year":"1995"},{"key":"S0962492925100275_r59","unstructured":"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."},{"key":"S0962492925100275_r191","volume-title":"p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics","author":"Schwab","year":"1998"},{"key":"S0962492925100275_r14","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202512500364"},{"key":"S0962492925100275_r30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112305"},{"key":"S0962492925100275_r26","doi-asserted-by":"publisher","DOI":"10.1007\/BF00282434"},{"key":"S0962492925100275_r33","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2012.12.022"},{"key":"S0962492925100275_r168","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160210206"},{"key":"S0962492925100275_r105","doi-asserted-by":"publisher","DOI":"10.1137\/21M1409160"},{"key":"S0962492925100275_r134","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-021-01253-0"},{"key":"S0962492925100275_r143","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.110980"},{"key":"S0962492925100275_r126","doi-asserted-by":"publisher","DOI":"10.1007\/b98828"},{"key":"S0962492925100275_r190","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-96-00649-7"},{"key":"S0962492925100275_r150","doi-asserted-by":"publisher","DOI":"10.3813\/AAA.918958"},{"key":"S0962492925100275_r133","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21932"},{"key":"S0962492925100275_r213","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/138"},{"key":"S0962492925100275_r136","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-28832-1"},{"key":"S0962492925100275_r28","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2010.10.025"},{"key":"S0962492925100275_r195","doi-asserted-by":"publisher","DOI":"10.1137\/130932855"},{"key":"S0962492925100275_r137","doi-asserted-by":"publisher","DOI":"10.1007\/BF02684334"},{"key":"S0962492925100275_r155","volume-title":"hp-Finite Element Methods for Singular Perturbations","author":"Melenk","year":"2004"},{"key":"S0962492925100275_r181","doi-asserted-by":"publisher","DOI":"10.1007\/BF01181459"},{"key":"S0962492925100275_r81","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-020-01249-2"},{"key":"S0962492925100275_r9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994269186"},{"key":"S0962492925100275_r46","doi-asserted-by":"publisher","DOI":"10.1137\/06065595X"},{"key":"S0962492925100275_r79","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202593000102"},{"key":"S0962492925100275_r76","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511662195"},{"key":"S0962492925100275_r43","doi-asserted-by":"publisher","DOI":"10.1007\/s00023-002-8631-8"},{"key":"S0962492925100275_r182","first-page":"57","article-title":"\u00dcber das asymptotische Verhalten der L\u00f6sungen von \u0394u + \u03bbu = 0 in unendlichen Gebieten","volume":"53","author":"Rellich","year":"1943","journal-title":"Jahresber. Dtsch. Math.-Ver."},{"key":"S0962492925100275_r179","first-page":"95","article-title":"On non-uniqueness in Cauchy problem for an elliptic second order differential equation","volume":"11","author":"Plis","year":"1963","journal-title":"Bull. Acad. Polon. Sci. S\u00e9r. Sci. Math. Astronom. Phys"},{"key":"S0962492925100275_r72","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2752"},{"key":"S0962492925100275_r55","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drz020"},{"key":"S0962492925100275_r50","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492912000037"}],"container-title":["Acta Numerica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0962492925100275","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T08:00:06Z","timestamp":1781856006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0962492925100275\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,19]]},"references-count":213,"alternative-id":["S0962492925100275"],"URL":"https:\/\/doi.org\/10.1017\/s0962492925100275","relation":{},"ISSN":["0962-4929","1474-0508"],"issn-type":[{"value":"0962-4929","type":"print"},{"value":"1474-0508","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,19]]}}}