{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T07:17:59Z","timestamp":1779175079455,"version":"3.51.4"},"reference-count":61,"publisher":"American Mathematical Society (AMS)","issue":"351","license":[{"start":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T00:00:00Z","timestamp":1748476800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1245"],"award-info":[{"award-number":["W1245"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1245"],"award-info":[{"award-number":["W1245"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["F65"],"award-info":[{"award-number":["F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1245"],"award-info":[{"award-number":["W1245"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1245"],"award-info":[{"award-number":["W1245"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["F65"],"award-info":[{"award-number":["F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We present a wavenumber-explicit convergence analysis of the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h p\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>h<\/mml:mi>\n      <mml:mi>p<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">hp<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n  <mml:semantics>\n    <mml:mi>k<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.<\/p>","DOI":"10.1090\/mcom\/3958","type":"journal-article","created":{"date-parts":[[2024,3,29]],"date-time":"2024-03-29T20:22:48Z","timestamp":1711743768000},"page":"73-122","source":"Crossref","is-referenced-by-count":8,"title":["Wavenumber-explicit stability and convergence analysis of \u210e\ud835\udc5d finite element discretizations of Helmholtz problems in piecewise smooth media"],"prefix":"10.1090","volume":"94","author":[{"given":"M.","family":"Bernkopf","sequence":"first","affiliation":[]},{"given":"T.","family":"Chaumont-Frelet","sequence":"additional","affiliation":[]},{"given":"J.","family":"Melenk","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,5,29]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1137\/S0036142903423460","article-title":"Discrete dispersion relation for \u210e\ud835\udc5d-version finite element approximation at high wave number","volume":"42","author":"Ainsworth, Mark","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"307","key":"2","doi-asserted-by":"publisher","first-page":"2129","DOI":"10.1090\/mcom\/3165","article-title":"Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation","volume":"86","author":"Barucq, H\u00e9l\u00e8ne","year":"2017","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","unstructured":"[BCFM22] M. Bernkopf, T. Chaumont-Frelet, and J. M. Melenk, Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media (extended version),  arXiv:2209.03601, 2022."},{"key":"4","unstructured":"[Ber21] M. Bernkopf, Finite element analysis of the heterogeneous Helmholtz equation and least squares methods, Ph.D. Thesis, Technische Universit\u00e4t Wien, 2021."},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1137\/0142032","article-title":"Boundary conditions for the numerical solution of elliptic equations in exterior regions","volume":"42","author":"Bayliss, Alvin","year":"1982","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1002\/nme.5172","article-title":"Efficient implementation of high-order finite elements for Helmholtz problems","volume":"106","author":"B\u00e9riot, Hadrien","year":"2016","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1137\/15M102530X","article-title":"Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations","volume":"48","author":"Baskin, Dean","year":"2016","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"9","key":"8","doi-asserted-by":"publisher","first-page":"2203","DOI":"10.1016\/j.camwa.2016.08.026","article-title":"On high order methods for the heterogeneous Helmholtz equation","volume":"72","author":"Chaumont-Frelet, Th\u00e9ophile","year":"2016","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s00211-021-01192-w","article-title":"On the derivation of guaranteed and \ud835\udc5d-robust a posteriori error estimates for the Helmholtz equation","volume":"148","author":"Chaumont-Frelet, T.","year":"2021","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"1803","DOI":"10.1051\/m2an\/2018031","article-title":"High-frequency behaviour of corner singularities in Helmholtz problems","volume":"52","author":"Chaumont-Frelet, T.","year":"2018","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"1503","DOI":"10.1093\/imanum\/drz020","article-title":"Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems","volume":"40","author":"Chaumont-Frelet, T.","year":"2020","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1137\/21M1450136","article-title":"Scattering by finely layered obstacles: frequency-explicit bounds and homogenization","volume":"55","author":"Chaumont-Frelet, T.","year":"2023","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1137\/19M1255616","article-title":"A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains","volume":"58","author":"Chaumont-Frelet, Th\u00e9ophile","year":"2020","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"1774","DOI":"10.1137\/21M1421805","article-title":"Frequency-explicit a posteriori error estimates for finite element discretizations of Maxwell\u2019s equations","volume":"60","author":"Chaumont-Frelet, Th\u00e9ophile","year":"2022","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"15","doi-asserted-by":"publisher","first-page":"2061","DOI":"10.1137\/S1064827596301406","article-title":"The perfectly matched layer in curvilinear coordinates","volume":"19","author":"Collino, Francis","year":"1998","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/0022-247X(85)90118-0","article-title":"A direct boundary integral equation method for transmission problems","volume":"106","author":"Costabel, Martin","year":"1985","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"17","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1017\/S0962492912000037","article-title":"Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering","volume":"21","author":"Chandler-Wilde, Simon N.","year":"2012","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1137\/18M1234916","article-title":"High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis","volume":"52","author":"Chandler-Wilde, S. N.","year":"2020","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/PL00005440","article-title":"Analysis of a coupled finite-infinite element method for exterior Helmholtz problems","volume":"88","author":"Demkowicz, Leszek","year":"2001","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1515\/cmam-2013-0008","article-title":"A posteriori error estimation for highly indefinite Helmholtz problems","volume":"13","author":"D\u00f6rfler, Willy","year":"2013","journal-title":"Comput. Methods Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1609-4840","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"782","DOI":"10.1137\/140953125","article-title":"Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number","volume":"53","author":"Du, Yu","year":"2015","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"139","key":"22","doi-asserted-by":"publisher","first-page":"629","DOI":"10.2307\/2005997","article-title":"Absorbing boundary conditions for the numerical simulation of waves","volume":"31","author":"Engquist, Bjorn","year":"1977","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1002\/cpa.3160320303","article-title":"Radiation boundary conditions for acoustic and elastic wave calculations","volume":"32","author":"Engquist, Bj\u00f6rn","year":"1979","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"24","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1007\/978-3-642-22061-6_9","article-title":"On stability of discretizations of the Helmholtz equation","author":"Esterhazy, S.","year":"2012"},{"key":"25","first-page":"1439","article-title":"Finite element method and natural boundary reduction","author":"Feng, Kang","year":"1984"},{"key":"26","unstructured":"[GLS21] J. Galkowski, D. Lafontaine, and E. A. Spence,  Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves, Preprint,  arXiv:2101.02154, 2021."},{"issue":"4","key":"27","doi-asserted-by":"publisher","first-page":"3344","DOI":"10.1137\/21M1443716","article-title":"Perfectly-matched-layer truncation is exponentially accurate at high frequency","volume":"55","author":"Galkowski, Jeffrey","year":"2023","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"28","unstructured":"[GLSW22] Jeffrey Galkowski, D. Lafontaine, Euan A. Spence, and J. Wunsch. The \u210e\ud835\udc5d-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect, 2022. arXiv:2207.05542v2."},{"issue":"4","key":"29","doi-asserted-by":"publisher","first-page":"3903","DOI":"10.1137\/21M1409160","article-title":"Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method","volume":"55","author":"Galkowski, J.","year":"2023","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"1","key":"30","doi-asserted-by":"publisher","first-page":"157","DOI":"10.2140\/paa.2020.2.157","article-title":"Optimal constants in nontrapping resolvent estimates and applications in numerical analysis","volume":"2","author":"Galkowski, Jeffrey","year":"2020","journal-title":"Pure Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2578-5885","issn-type":"print"},{"issue":"9","key":"31","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1016\/0898-1221(95)00144-N","article-title":"Finite element solution of the Helmholtz equation with high wave number. I. The \u210e-version of the FEM","volume":"30","author":"Ihlenburg, F.","year":"1995","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"1","key":"32","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1137\/S0036142994272337","article-title":"Finite element solution of the Helmholtz equation with high wave number. II. The \u210e-\ud835\udc5d version of the FEM","volume":"34","author":"Ihlenburg, Frank","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"33","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b98828","volume-title":"Finite element analysis of acoustic scattering","volume":"132","author":"Ihlenburg, Frank","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0387983198"},{"issue":"2","key":"34","doi-asserted-by":"publisher","first-page":"113","DOI":"10.5802\/aif.1137","article-title":"Decay of solutions of the wave equation in the exterior of several convex bodies","volume":"38","author":"Ikawa, Mitsuru","year":"1988","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"key":"35","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1090\/S0002-9904-1964-11051-X","article-title":"Scattering theory","volume":"70","author":"Lax, Peter D.","year":"1964","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"issue":"10","key":"36","doi-asserted-by":"publisher","first-page":"2025","DOI":"10.1002\/cpa.21932","article-title":"For most frequencies, strong trapping has a weak effect in frequency-domain scattering","volume":"74","author":"Lafontaine, David","year":"2021","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"37","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.camwa.2022.03.007","article-title":"Wavenumber-explicit convergence of the \u210e\ud835\udc5d-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients","volume":"113","author":"Lafontaine, D.","year":"2022","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"38","isbn-type":"print","volume-title":"Strongly elliptic systems and boundary integral equations","author":"McLean, William","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0521663326"},{"issue":"4","key":"39","first-page":"605","article-title":"Microlocal parametrices for diffractive boundary value problems","volume":"42","author":"Melrose, Richard B.","year":"1975","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"40","volume-title":"On generalized finite-element methods","author":"Melenk, Jens Markus","year":"1995"},{"key":"41","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b84212","volume-title":"$hp$-finite element methods for singular perturbations","volume":"1796","author":"Melenk, Jens M.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/3540442014"},{"issue":"4","key":"42","doi-asserted-by":"publisher","first-page":"2599","DOI":"10.1137\/100784072","article-title":"Mapping properties of combined field Helmholtz boundary integral operators","volume":"44","author":"Melenk, Jens Markus","year":"2012","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"11","key":"43","doi-asserted-by":"publisher","first-page":"2351","DOI":"10.1016\/j.camwa.2020.04.014","article-title":"FEM-BEM mortar coupling for the Helmholtz problem in three dimensions","volume":"80","author":"Mascotto, Lorenzo","year":"2020","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"3","key":"44","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1007\/s10915-013-9726-8","article-title":"General DG-methods for highly indefinite Helmholtz problems","volume":"57","author":"Melenk, J. M.","year":"2013","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"321","key":"45","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1090\/mcom\/3454","article-title":"On commuting \ud835\udc5d-version projection-based interpolation on tetrahedra","volume":"89","author":"Melenk, J. M.","year":"2020","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"272","key":"46","doi-asserted-by":"publisher","first-page":"1871","DOI":"10.1090\/S0025-5718-10-02362-8","article-title":"Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions","volume":"79","author":"Melenk, J. M.","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"47","doi-asserted-by":"publisher","first-page":"1210","DOI":"10.1137\/090776202","article-title":"Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation","volume":"49","author":"Melenk, J. M.","year":"2011","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"48","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1142\/S0218202519500106","article-title":"Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions","volume":"29","author":"Moiola, Andrea","year":"2019","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"1","key":"49","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s10208-020-09452-1","article-title":"Wavenumber-explicit \u210e\ud835\udc5d-FEM analysis for Maxwell\u2019s equations with transparent boundary conditions","volume":"21","author":"Melenk, Jens M.","year":"2021","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"key":"50","doi-asserted-by":"crossref","unstructured":"[MS22] Jens M. Melenk and Stefan A. Sauter. Wavenumber-explicit hp-FEM analysis for Maxwell\u2019s equations with impedance boundary conditions, 2022. arXiv:2201.02602.","DOI":"10.1007\/s10208-023-09626-7"},{"key":"51","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4393-7","volume-title":"Acoustic and electromagnetic equations","volume":"144","author":"N\u00e9d\u00e9lec, Jean-Claude","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0387951555"},{"issue":"2","key":"52","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s00607-006-0177-z","article-title":"A refined finite element convergence theory for highly indefinite Helmholtz problems","volume":"78","author":"Sauter, S. A.","year":"2006","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"53","doi-asserted-by":"publisher","first-page":"959","DOI":"10.2307\/2005357","article-title":"An observation concerning Ritz-Galerkin methods with indefinite bilinear forms","volume":"28","author":"Schatz, Alfred H.","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"54","doi-asserted-by":"publisher","first-page":"2987","DOI":"10.1137\/130932855","article-title":"Wavenumber-explicit bounds in time-harmonic acoustic scattering","volume":"46","author":"Spence, E. A.","year":"2014","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"55","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68093-2","volume-title":"Boundary element methods","volume":"39","author":"Sauter, Stefan A.","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540680925"},{"key":"56","series-title":"Princeton Mathematical Series, No. 30","volume-title":"Singular integrals and differentiability properties of functions","author":"Stein, Elias M.","year":"1970"},{"key":"57","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-68805-3","volume-title":"Numerical approximation methods for elliptic boundary value problems","author":"Steinbach, Olaf","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387313122"},{"key":"58","doi-asserted-by":"crossref","unstructured":"[TZNHD17] M. Taus, L. Zepeda-Nunez, R. Hewett, and L. Demanet, Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation, Proc. SEG Annual Meeting, Houston, 2017.","DOI":"10.1190\/segam2017-17728116.1"},{"key":"59","doi-asserted-by":"crossref","unstructured":"[Vai75] Boris R. Vainberg. On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as \ud835\udc61\u2192\u221e of solutions of non-stationary problems. Russian Math. Surveys, 30(2):1\u201358, 1975.","DOI":"10.1070\/RM1975v030n02ABEH001406"},{"issue":"4","key":"60","doi-asserted-by":"publisher","first-page":"967","DOI":"10.1007\/s00211-021-01195-7","article-title":"Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods","volume":"147","author":"Xu, Liwei","year":"2021","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"61","doi-asserted-by":"publisher","first-page":"1828","DOI":"10.1137\/120874643","article-title":"Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: \u210e\ud835\udc5d version","volume":"51","author":"Zhu, Lingxue","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03958-3\/mcom3958_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03958-3\/S0025-5718-2024-03958-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T17:49:35Z","timestamp":1728582575000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03958-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,29]]},"references-count":61,"journal-issue":{"issue":"351","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["S0025-5718-2024-03958-3"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3958","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,29]]}}}