{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:11:23Z","timestamp":1774437083070,"version":"3.50.1"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T00:00:00Z","timestamp":1681084800000},"content-version":"vor","delay-in-days":9,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R005591\/1"],"award-info":[{"award-number":["EP\/R005591\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In<jats:italic>d<\/jats:italic>dimensions, accurately approximating an arbitrary function oscillating with frequency<jats:inline-formula><jats:alternatives><jats:tex-math>$\\lesssim k$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>\u2272<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>requires<jats:inline-formula><jats:alternatives><jats:tex-math>$\\sim k^{d}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>\u223c<\/mml:mo><mml:msup><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber<jats:italic>k<\/jats:italic>) suffers from the pollution effect if, as<jats:inline-formula><jats:alternatives><jats:tex-math>$k\\rightarrow \\infty $<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the<jats:italic>h<\/jats:italic>-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth<jats:italic>h<\/jats:italic>and keeping the polynomial degree<jats:italic>p<\/jats:italic>fixed) suffers from the pollution effect, the<jats:italic>hp<\/jats:italic>-FEM (where accuracy is increased by decreasing the meshwidth<jats:italic>h<\/jats:italic>and increasing the polynomial degree<jats:italic>p<\/jats:italic>) does not suffer from the pollution effect. The heart of the proof of this result is a PDE result splitting the solution of the Helmholtz equation into \u201chigh\u201d and \u201clow\u201d frequency components. This result for the constant-coefficient Helmholtz equation in full space (i.e. in<jats:inline-formula><jats:alternatives><jats:tex-math>$\\mathbb {R}^{d}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mrow><mml:mi>\u211d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>) was originally proved in Melenk and Sauter (<jats:italic>Math. Comp<\/jats:italic><jats:bold>79<\/jats:bold>(272), 1871\u20131914, 2010). In this paper, we prove this result using<jats:italic>only<\/jats:italic>integration by parts and elementary properties of the Fourier transform. The proof in this paper is motivated by the recent proof in Lafontaine et al. (<jats:italic>Comp. Math. Appl.<\/jats:italic><jats:bold>113<\/jats:bold>, 59\u201369, 2022) of this splitting for the variable-coefficient Helmholtz equation in full space use the more-sophisticated tools of semiclassical pseudodifferential operators.<\/jats:p>","DOI":"10.1007\/s10444-023-10025-3","type":"journal-article","created":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T06:02:38Z","timestamp":1681106558000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation"],"prefix":"10.1007","volume":"49","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1236-4592","authenticated-orcid":false,"given":"E. A.","family":"Spence","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,10]]},"reference":[{"issue":"1","key":"10025_CR1","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/BF01395880","volume":"53","author":"AK Aziz","year":"1988","unstructured":"Aziz, A.K., Kellogg, R.B., Stephens, A.B.: A two point boundary value problem with a rapidly oscillating solution. Numer. Math. 53(1), 107\u2013121 (1988)","journal-title":"Numer. Math."},{"key":"10025_CR2","unstructured":"Babu\u0161ka, I. M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev., 451\u2013484 (2000)"},{"issue":"307","key":"10025_CR3","doi-asserted-by":"publisher","first-page":"2129","DOI":"10.1090\/mcom\/3165","volume":"86","author":"H Barucq","year":"2017","unstructured":"Barucq, H., Chaumont-Frelet, T., Gout, C.: Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation. Math. Comp. 86(307), 2129\u20132157 (2017). https:\/\/doi.org\/10.1090\/mcom\/3165","journal-title":"Math. Comp."},{"key":"10025_CR4","unstructured":"Bernkopf, M., Chaumont-Frelet, T., Melenk, J.M.: Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media. arXiv:2209.03601 (2022)"},{"key":"10025_CR5","doi-asserted-by":"crossref","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer (2008)","DOI":"10.1007\/978-0-387-75934-0"},{"key":"10025_CR6","doi-asserted-by":"crossref","unstructured":"Brown, D.L., Gallistl, D., Peterseim, D.: Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations. In: Meshfree Methods for Partial Differential Equations VIII, pp 85\u2013115. Springer (2017)","DOI":"10.1007\/978-3-319-51954-8_6"},{"key":"10025_CR7","doi-asserted-by":"publisher","first-page":"112590","DOI":"10.1016\/j.cam.2019.112590","volume":"369","author":"H Cao","year":"2020","unstructured":"Cao, H., Wu, H.: IPCDGM and multiscale IPDPGM for the Helmholtz problem with large wave number. J. Comput. Appl. Math. 369, 112590 (2020)","journal-title":"J. Comput. Appl. Math."},{"issue":"5","key":"10025_CR8","doi-asserted-by":"publisher","first-page":"1428","DOI":"10.1137\/060662575","volume":"39","author":"SN Chandler-Wilde","year":"2008","unstructured":"Chandler-Wilde, S.N., Monk, P.: Wave-number-explicit bounds in time-harmonic scattering. SIAM J. Math. Anal. 39(5), 1428\u20131455 (2008)","journal-title":"SIAM J. Math. Anal."},{"issue":"9","key":"10025_CR9","doi-asserted-by":"publisher","first-page":"2203","DOI":"10.1016\/j.camwa.2016.08.026","volume":"72","author":"T Chaumont-Frelet","year":"2016","unstructured":"Chaumont-Frelet, T.: On high order methods for the heterogeneous Helmholtz equation. Comput. Math. Appl. 72(9), 2203\u20132225 (2016)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"10025_CR10","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1137\/19M1255616","volume":"58","author":"T Chaumont-Frelet","year":"2020","unstructured":"Chaumont-Frelet, T., Valentin, F.: A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains. SIAM J. Num. Anal. 58 (2), 1029\u20131067 (2020)","journal-title":"SIAM J. Num. Anal."},{"issue":"2","key":"10025_CR11","doi-asserted-by":"publisher","first-page":"1503","DOI":"10.1093\/imanum\/drz020","volume":"40","author":"T Chaumont-Frelet","year":"2020","unstructured":"Chaumont-Frelet, T., Nicaise, S.: Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problem. IMA J. Numer. Anal. 40(2), 1503\u20131543 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"10025_CR12","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, Vol. II. Handb. Numer. Anal., II, pp 17\u2013351, North-Holland (1991)","DOI":"10.1016\/S1570-8659(05)80039-0"},{"key":"10025_CR13","first-page":"271","volume-title":"Integral Equation Methods in Scattering Theory","author":"DL Colton","year":"1983","unstructured":"Colton, D.L., Kress, R.: Integral Equation Methods in Scattering Theory, p 271. Wiley, New York (1983)"},{"issue":"2","key":"10025_CR14","doi-asserted-by":"publisher","first-page":"782","DOI":"10.1137\/140953125","volume":"53","author":"Y Du","year":"2015","unstructured":"Du, Y., Wu, H.: Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53(2), 782\u2013804 (2015)","journal-title":"SIAM J. Numer. Anal."},{"key":"10025_CR15","doi-asserted-by":"crossref","unstructured":"Dyatlov, S., Zworski, M.: Mathematical Theory of Scattering Resonances. AMS (2019)","DOI":"10.1090\/gsm\/200"},{"key":"10025_CR16","doi-asserted-by":"crossref","unstructured":"Esterhazy, S., Melenk, J.M.: On stability of discretizations of the Helmholtz equation. In: Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R. (eds.) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol. 83, pp 285\u2013324. Springer (2012)","DOI":"10.1007\/978-3-642-22061-6_9"},{"issue":"4","key":"10025_CR17","doi-asserted-by":"publisher","first-page":"2872","DOI":"10.1137\/080737538","volume":"47","author":"X Feng","year":"2009","unstructured":"Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872\u20132896 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"276","key":"10025_CR18","doi-asserted-by":"publisher","first-page":"1997","DOI":"10.1090\/S0025-5718-2011-02475-0","volume":"80","author":"X Feng","year":"2011","unstructured":"Feng, X., Wu, H.: hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80(276), 1997\u20132024 (2011)","journal-title":"Math. Comput."},{"issue":"1","key":"10025_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4310\/CMS.2022.v20.n1.a1","volume":"20","author":"D Gallistl","year":"2022","unstructured":"Gallistl, D., Chaumont-Frelet, T., Nicaise, S., Tomezyk, J.: Wavenumber explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers. Comm. Math. Sci. 20(1), 1\u201352 (2022)","journal-title":"Comm. Math. Sci."},{"key":"10025_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cma.2015.06.017","volume":"295","author":"D Gallistl","year":"2015","unstructured":"Gallistl, D., Peterseim, D.: Stable multiscale Petrov\u2013Galerkin finite element method for high frequency acoustic scattering. Comput. Methods Appl. Mech. Eng. 295, 1\u201317 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10025_CR21","doi-asserted-by":"crossref","unstructured":"Galkowski, J., Lafontaine, D., Spence, E.A., Wunsch, J.: Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method. SIAM J. Math. Anal. to appear. arXiv:2102.13081(2021)","DOI":"10.5802\/slsedp.152"},{"key":"10025_CR22","unstructured":"Galkowski, J., Lafontaine, D., Spence, E.A., Wunsch, J.: The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect. arXiv:2207.05542 (2022)"},{"key":"10025_CR23","unstructured":"Galkowski, J.: Lower bounds for piecewise polynomial approximations of oscillatory functions. arXiv:2211.04757 (2022)"},{"key":"10025_CR24","unstructured":"Galkowski, J., Spence, E.A.: Sharp preasymptotic error bounds for the Helmholtz h-FEM. arXiv:2301.03574 (2023)"},{"issue":"1","key":"10025_CR25","doi-asserted-by":"publisher","first-page":"157","DOI":"10.2140\/paa.2020.2.157","volume":"2","author":"J Galkowski","year":"2020","unstructured":"Galkowski, J., Spence, E.A., Wunsch, J.: Optimal constants in nontrapping resolvent estimates. Pure Appl. Anal. 2(1), 157\u2013202 (2020)","journal-title":"Pure Appl. Anal."},{"key":"10025_CR26","doi-asserted-by":"crossref","unstructured":"Galkowski, J., Spence, E.A.: Does the Helmholtz boundary element method suffer from the pollution effect? SIAM Review to appear (2023)","DOI":"10.1137\/22M1474199"},{"issue":"321","key":"10025_CR27","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1090\/mcom\/3457","volume":"89","author":"IG Graham","year":"2020","unstructured":"Graham, I.G., Sauter, S.: Stability and finite element error analysis for the Helmholtz equation with variable coefficients. Math. Comp. 89(321), 105\u2013138 (2020)","journal-title":"Math. Comp."},{"issue":"6","key":"10025_CR28","doi-asserted-by":"publisher","first-page":"2869","DOI":"10.1016\/j.jde.2018.08.048","volume":"266","author":"IG Graham","year":"2019","unstructured":"Graham, I.G., Pembery, O.R., Spence, E.A.: The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances. J. Differ. Equ. 266(6), 2869\u20132923 (2019)","journal-title":"J. Differ. Equ."},{"issue":"22","key":"10025_CR29","doi-asserted-by":"publisher","first-page":"3745","DOI":"10.1002\/nme.1620382203","volume":"38","author":"F Ihlenburg","year":"1995","unstructured":"Ihlenburg, F., Babu\u0161ka, I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Meth. Eng. 38(22), 3745\u20133774 (1995)","journal-title":"Int. J. Numer. Meth. Eng."},{"issue":"1","key":"10025_CR30","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1137\/S0036142994272337","volume":"34","author":"F Ihlenburg","year":"1997","unstructured":"Ihlenburg, F., Babuska, I.: Finite element solution of the Helmholtz equation with high wave number part II: the hp version of the FEM. SIAM J. Numer. Anal. 34(1), 315\u2013358 (1997)","journal-title":"SIAM J. Numer. Anal."},{"issue":"9","key":"10025_CR31","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1016\/0898-1221(95)00144-N","volume":"30","author":"F Ihlenburg","year":"1995","unstructured":"Ihlenburg, F., Babu\u0161ka, I.: Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM. Comp. Math. Appl. 30(9), 9\u201337 (1995)","journal-title":"Comp. Math. Appl."},{"key":"10025_CR32","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.camwa.2022.03.007","volume":"113","author":"D Lafontaine","year":"2022","unstructured":"Lafontaine, D., Spence, E.A., Wunsch, J.: Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients. Comp. Math. Appl. 113, 59\u201369 (2022)","journal-title":"Comp. Math. Appl."},{"issue":"1","key":"10025_CR33","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1137\/17M1140522","volume":"57","author":"Y Li","year":"2019","unstructured":"Li, Y., Wu, H.: FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation. SIAM J. Numer. Anal. 57(1), 96\u2013126 (2019)","journal-title":"SIAM J. Numer. Anal."},{"issue":"272","key":"10025_CR34","doi-asserted-by":"publisher","first-page":"1871","DOI":"10.1090\/S0025-5718-10-02362-8","volume":"79","author":"JM Melenk","year":"2010","unstructured":"Melenk, J.M., Sauter, S.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comp 79(272), 1871\u20131914 (2010)","journal-title":"Math. Comp"},{"key":"10025_CR35","unstructured":"Melenk, J.M.: On generalized finite element methods. PhD thesis, The University of Maryland (1995)"},{"key":"10025_CR36","doi-asserted-by":"publisher","first-page":"1210","DOI":"10.1137\/090776202","volume":"49","author":"JM Melenk","year":"2011","unstructured":"Melenk, J.M., Sauter, S.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49, 1210\u20131243 (2011)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"10025_CR37","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1007\/s10915-013-9726-8","volume":"57","author":"JM Melenk","year":"2013","unstructured":"Melenk, J.M., Parsania, A., Sauter, S.: General DG-methods for highly indefinite Helmholtz problems. J. Sci. Comput. 57(3), 536\u2013581 (2013)","journal-title":"J. Sci. Comput."},{"key":"10025_CR38","unstructured":"Ma, C., Alber, C., Scheichl, R.: Wavenumber explicit convergence of a multiscale GFEM for heterogeneous Helmholtz problems. arXiv:2112.10544 (2021)"},{"issue":"1","key":"10025_CR39","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s10208-020-09452-1","volume":"21","author":"JM Melenk","year":"2021","unstructured":"Melenk, J.M., Sauter, S.A.: Wavenumber-explicit hp-FEM analysis for Maxwell\u2019s equations with transparent boundary conditions. Found. Comp. Math. 21(1), 125\u2013241 (2021)","journal-title":"Found. Comp. Math."},{"key":"10025_CR40","doi-asserted-by":"crossref","unstructured":"Melenk, J.M., Sauter, S.A.: Wavenumber-explicit hp-FEM analysis for Maxwell\u2019s equations with impedance boundary conditions. arXiv:2201.02602(2022)","DOI":"10.1007\/s10208-023-09626-7"},{"key":"10025_CR41","unstructured":"McLean, W.C.H.: Strongly Elliptic Systems and Boundary Integral Equations. CUP (2000)"},{"key":"10025_CR42","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1002\/cpa.3160210206","volume":"21","author":"CS Morawetz","year":"1968","unstructured":"Morawetz, C.S., Ludwig, D.: An inequality for the reduced wave operator and the justification of geometrical optics. Comm. Pure Appl. Math. 21, 187\u2013203 (1968)","journal-title":"Comm. Pure Appl. Math."},{"issue":"2","key":"10025_CR43","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1002\/cpa.3160280204","volume":"28","author":"CS Morawetz","year":"1975","unstructured":"Morawetz, C.S.: Decay for solutions of the exterior problem for the wave equation. Commun. Pure Appl. Math. 28(2), 229\u2013264 (1975)","journal-title":"Commun. Pure Appl. Math."},{"issue":"6","key":"10025_CR44","doi-asserted-by":"publisher","first-page":"1868","DOI":"10.1002\/num.22508","volume":"36","author":"S Nicaise","year":"2020","unstructured":"Nicaise, S., Tomezyk, J.: Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary. Numer. Methods Partial Differ. Equ. 36(6), 1868\u20131903 (2020)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"1","key":"10025_CR45","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1137\/16M1108820","volume":"16","author":"M Ohlberger","year":"2018","unstructured":"Ohlberger, M., Verfurth, B.: A new heterogeneous multiscale method for the Helmholtz equation with high contrast. Multiscale Model. Simul. 16(1), 385\u2013411 (2018)","journal-title":"Multiscale Model. Simul."},{"issue":"305","key":"10025_CR46","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1090\/mcom\/3156","volume":"86","author":"D Peterseim","year":"2017","unstructured":"Peterseim, D.: Eliminating the pollution effect in Helmholtz problems by local subscale correction. Math. Comput. 86(305), 1005\u20131036 (2017)","journal-title":"Math. Comput."},{"key":"10025_CR47","doi-asserted-by":"publisher","DOI":"10.1201\/9780429507069","volume-title":"Variational Techniques for Elliptic Partial Differential Equations: Theoretical Tools and Advanced Applications","author":"F-J Sayas","year":"2019","unstructured":"Sayas, F. -J., Brown, T.S., Hassell, M.E.: Variational Techniques for Elliptic Partial Differential Equations: Theoretical Tools and Advanced Applications. CRC Press, Boca Raton (2019)"},{"issue":"2","key":"10025_CR48","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s00607-006-0177-z","volume":"78","author":"SA Sauter","year":"2006","unstructured":"Sauter, S.A.: A refined finite element convergence theory for highly indefinite Helmholtz problems. Computing 78(2), 101\u2013115 (2006)","journal-title":"Computing"},{"issue":"128","key":"10025_CR49","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1090\/S0025-5718-1974-0373326-0","volume":"28","author":"AH Schatz","year":"1974","unstructured":"Schatz, A.H.: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28(128), 959\u2013962 (1974)","journal-title":"Math. Comput."},{"issue":"1","key":"10025_CR50","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1109\/JRPROC.1949.232969","volume":"37","author":"CE Shannon","year":"1949","unstructured":"Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10\u201321 (1949)","journal-title":"Proc. IRE"},{"issue":"9","key":"10025_CR51","doi-asserted-by":"publisher","first-page":"1587","DOI":"10.1002\/cpa.21543","volume":"68","author":"EA Spence","year":"2015","unstructured":"Spence, E.A., Kamotski, I.V., Smyshlyaev, V.P.: Coercivity of combined boundary integral equations in high-frequency scattering. Comm. Pure Appl. Math 68(9), 1587\u20131639 (2015)","journal-title":"Comm. Pure Appl. Math"},{"issue":"3","key":"10025_CR52","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1093\/imanum\/drt033","volume":"34","author":"H Wu","year":"2014","unstructured":"Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version. IMA J. Numer. Anal. 34(3), 1266\u20131288 (2014)","journal-title":"IMA J. Numer. Anal."},{"key":"10025_CR53","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1017\/S0370164600017806","volume":"35","author":"ET Whittaker","year":"1915","unstructured":"Whittaker, E.T.: On the functions which are represented by the expansions of the interpolation-theory. Proc. R. Soc. Edinb. 35, 181\u2013194 (1915)","journal-title":"Proc. R. Soc. Edinb."},{"issue":"2","key":"10025_CR54","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01473-4","volume":"87","author":"B Zhu","year":"2021","unstructured":"Zhu, B., Wu, H.: Preasymptotic error analysis of the HDG method for Helmholtz equation with large wave number. J. Sci. Comput. 87(2), 1\u201334 (2021)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"10025_CR55","doi-asserted-by":"publisher","first-page":"1828","DOI":"10.1137\/120874643","volume":"51","author":"L Zhu","year":"2013","unstructured":"Zhu, L., Wu, H.: Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: hp version. SIAM J. Numer. Anal. 51(3), 1828\u20131852 (2013)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10025-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-023-10025-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-023-10025-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,10]],"date-time":"2023-12-10T11:49:30Z","timestamp":1702208970000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-023-10025-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4]]},"references-count":55,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4]]}},"alternative-id":["10025"],"URL":"https:\/\/doi.org\/10.1007\/s10444-023-10025-3","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4]]},"assertion":[{"value":"19 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}],"article-number":"27"}}