{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T09:18:28Z","timestamp":1773393508884,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2019,1,1]],"date-time":"2019-01-01T00:00:00Z","timestamp":1546300800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s00211-018-1012-0","type":"journal-article","created":{"date-parts":[[2019,1,1]],"date-time":"2019-01-01T04:42:29Z","timestamp":1546317749000},"page":"917-966","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Complete radiation boundary conditions for the Helmholtz equation I: waveguides"],"prefix":"10.1007","volume":"141","author":[{"given":"Thomas","family":"Hagstrom","sequence":"first","affiliation":[]},{"given":"Seungil","family":"Kim","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,1,1]]},"reference":[{"issue":"4","key":"1012_CR1","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1145\/1268776.1268779","volume":"33","author":"W Bangerth","year":"2007","unstructured":"Bangerth, W., Hartmann, R., Kanschat, G.: deal. II\u2014a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33(4), 24 (2007)","journal-title":"ACM Trans. Math. Softw."},{"issue":"1","key":"1012_CR2","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1137\/S0036142903420984","volume":"42","author":"E B\u00e9cache","year":"2004","unstructured":"B\u00e9cache, E., Dhia, A.-S.B.-B., Legendre, G.: Perfectly matched layers for the convected Helmholtz equation. SIAM J. Numer. Anal. 42(1), 409\u2013433 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"225","key":"1012_CR3","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1090\/S0025-5718-99-01016-9","volume":"68","author":"A Bendali","year":"1999","unstructured":"Bendali, A., Guillaume, P.: Non-reflecting boundary conditions for waveguides. Math. Comput. 68(225), 123\u2013144 (1999)","journal-title":"Math. Comput."},{"issue":"258","key":"1012_CR4","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1090\/S0025-5718-06-01930-2","volume":"76","author":"JH Bramble","year":"2007","unstructured":"Bramble, J.H., Pasciak, J.E.: Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comput. 76(258), 597\u2013614 (2007)","journal-title":"Math. Comput."},{"issue":"6","key":"1012_CR5","doi-asserted-by":"publisher","first-page":"2158","DOI":"10.1137\/090750603","volume":"48","author":"Z Chen","year":"2010","unstructured":"Chen, Z., Zheng, W.: Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media. SIAM J. Numer. Anal. 48(6), 2158\u20132185 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"1012_CR6","volume-title":"Methods of Mathematical Physics","author":"R Courant","year":"1953","unstructured":"Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Wiley, New York (1953)"},{"issue":"1","key":"1012_CR7","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1137\/140966927","volume":"58","author":"V Druskin","year":"2016","unstructured":"Druskin, V., G\u00fcttel, S., Knizhnerman, L.: Near-optimal perfectly matched layers for indefinite Helmholtz problems. SIAM Rev. 58(1), 90\u2013116 (2016)","journal-title":"SIAM Rev."},{"issue":"5","key":"1012_CR8","doi-asserted-by":"publisher","first-page":"2485","DOI":"10.1137\/090774082","volume":"32","author":"V Druskin","year":"2010","unstructured":"Druskin, V., Lieberman, C., Zaslavsky, M.: On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems. SIAM J. Sci. Comput. 32(5), 2485\u20132496 (2010)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1012_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0021-9991(91)90135-8","volume":"94","author":"D Givoli","year":"1991","unstructured":"Givoli, D.: Nonreflecting boundary conditions. J. Comput. Phys. 94(1), 1\u201329 (1991)","journal-title":"J. Comput. Phys."},{"issue":"29\u201332","key":"1012_CR10","doi-asserted-by":"publisher","first-page":"3666","DOI":"10.1016\/j.cma.2005.01.021","volume":"195","author":"D Givoli","year":"2006","unstructured":"Givoli, D., Hagstrom, T., Patlashenko, I.: Finite element formulation with high-order absorbing boundary conditions for time-dependent waves. Comput. Methods Appl. Mech. Eng. 195(29\u201332), 3666\u20133690 (2006)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"1012_CR11","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/S0021-9991(03)00005-6","volume":"186","author":"D Givoli","year":"2003","unstructured":"Givoli, D., Neta, B.: High-order non-reflecting boundary scheme for time-dependent waves. J. Comput. Phys. 186(1), 24\u201346 (2003)","journal-title":"J. Comput. Phys."},{"issue":"13","key":"1012_CR12","doi-asserted-by":"publisher","first-page":"1955","DOI":"10.1002\/nme.842","volume":"58","author":"D Givoli","year":"2003","unstructured":"Givoli, D., Neta, B., Patlashenko, I.: Finite element analysis of time-dependent semi-infinite wave-guides with high-order boundary treatment. Internat J. Numer. Methods Eng. 58(13), 1955\u20131983 (2003)","journal-title":"Internat J. Numer. Methods Eng."},{"issue":"160","key":"1012_CR13","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1090\/S0025-5718-1982-0669632-7","volume":"39","author":"CI Goldstein","year":"1982","unstructured":"Goldstein, C.I.: A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. Comput. 39(160), 309\u2013324 (1982)","journal-title":"Math. Comput."},{"issue":"3","key":"1012_CR14","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1007\/s10543-013-0420-x","volume":"53","author":"S G\u00fcttel","year":"2013","unstructured":"G\u00fcttel, S., Knizhnerman, L.: A black-box rational Arnoldi variant for Cauchy\u2013Stieltjes matrix functions. BIT 53(3), 595\u2013616 (2013)","journal-title":"BIT"},{"key":"1012_CR15","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1017\/S0962492900002890","volume":"8","author":"T Hagstrom","year":"1999","unstructured":"Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47\u2013106 (1999)","journal-title":"Acta Numer."},{"issue":"6","key":"1012_CR16","doi-asserted-by":"publisher","first-page":"3322","DOI":"10.1016\/j.jcp.2007.11.040","volume":"227","author":"T Hagstrom","year":"2008","unstructured":"Hagstrom, T., Mar-Or, A., Givoli, D.: High-order local absorbing conditions for the wave equation: extensions and improvements. J. Comput. Phys. 227(6), 3322\u20133357 (2008)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"1012_CR17","doi-asserted-by":"publisher","first-page":"3678","DOI":"10.1137\/090745477","volume":"47","author":"T Hagstrom","year":"2009","unstructured":"Hagstrom, T., Warburton, T.: Complete radiation boundary conditions: minimizing the long time error growth of local methods. SIAM J. Numer. Anal. 47(5), 3678\u20133704 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"1012_CR18","doi-asserted-by":"publisher","first-page":"1988","DOI":"10.1016\/j.cam.2009.08.050","volume":"234","author":"T Hagstrom","year":"2010","unstructured":"Hagstrom, T., Warburton, T., Givoli, D.: Radiation boundary conditions for time-dependent waves based on complete plane wave expansions. J. Comput. Appl. Math. 234(6), 1988\u20131995 (2010)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"1012_CR19","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1006\/jcph.1998.5960","volume":"143","author":"I Harari","year":"1998","unstructured":"Harari, I., Patlashenko, I., Givoli, D.: Dirichlet-to-Neumann maps for unbounded wave guides. J. Comput. Phys. 143(1), 200\u2013223 (1998)","journal-title":"J. Comput. Phys."},{"issue":"176","key":"1012_CR20","first-page":"437","volume":"47","author":"RL Higdon","year":"1986","unstructured":"Higdon, R.L.: Absorbing boundary conditions for difference approximations to the multidimensional wave equation. Math. Comput. 47(176), 437\u2013459 (1986)","journal-title":"Math. Comput."},{"issue":"179","key":"1012_CR21","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1090\/S0025-5718-1987-0890254-1","volume":"49","author":"RL Higdon","year":"1987","unstructured":"Higdon, R.L.: Numerical absorbing boundary conditions for the wave equation. Math. Comput. 49(179), 65\u201390 (1987)","journal-title":"Math. Comput."},{"key":"1012_CR22","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1002\/1097-0312(200008)53:8<1039::AID-CPA4>3.0.CO;2-I","volume":"53","author":"D Ingerman","year":"2000","unstructured":"Ingerman, D., Druskin, V., Knizherman, L.: Optimal finite difference grids and rational approximationsof the square root. I. Elliptic functions. Commun. Pure Appl. Math. 53, 1039\u20131066 (2000)","journal-title":"Commun. Pure Appl. Math."},{"key":"1012_CR23","unstructured":"Kim, S.: Analysis of complete radiation boundary conditions for the Helmholtz equation in perturbed waveguides. (Manuscript)"},{"issue":"1","key":"1012_CR24","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/j.jmaa.2013.08.018","volume":"410","author":"S Kim","year":"2014","unstructured":"Kim, S.: Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions. J. Math. Anal. Appl. 410(1), 275\u2013291 (2014)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1012_CR25","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1016\/j.jmaa.2010.05.006","volume":"370","author":"S Kim","year":"2010","unstructured":"Kim, S., Pasciak, J.E.: Analysis of a Cartesian PML approximation to acoustic scattering problems in \n                    \n                      \n                    \n                    $${\\mathbb{R}}^2$$\n                    \n                      \n                        \n                          \n                            R\n                          \n                          2\n                        \n                      \n                    \n                  . J. Math. Anal. Appl. 370(1), 168\u2013186 (2010)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"1012_CR26","doi-asserted-by":"publisher","first-page":"1537","DOI":"10.1137\/140980491","volume":"53","author":"S Kim","year":"2015","unstructured":"Kim, S., Zhang, H.: Optimized Schwarz method with complete radiation transmission conditions for the Helmholtz equation in waveguides. SIAM J. Numer. Anal. 53(3), 1537\u20131558 (2015)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"1012_CR27","doi-asserted-by":"publisher","first-page":"1573","DOI":"10.1016\/j.camwa.2016.07.024","volume":"72","author":"S Kim","year":"2016","unstructured":"Kim, S., Zhang, H.: Optimized double sweep Schwarz method with complete radiation boundary conditions for the Helmholtz equation in waveguides. Comput. Math. Appl. 72(6), 1573\u20131589 (2016)","journal-title":"Comput. Math. Appl."},{"key":"1012_CR28","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1137\/080715159","volume":"47","author":"L Knizherman","year":"2009","unstructured":"Knizherman, L., Druskin, V., Zaslavsky, M.: On optimal convergence rate of the rational Krylovsubspace reduction for electromagnetic problems in unbounded domains. SIAM J. Numer. Anal. 47, 953\u2013971 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1012_CR29","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1109\/7260.916327","volume":"11","author":"M Koshiba","year":"2001","unstructured":"Koshiba, M., Tsuji, Y., Sasaki, S.: High-performance absorbing boundary conditions for photonic crystal waveguide simulations. Microw. Wirel. Compon. Lett. IEEE 11(4), 152\u2013154 (2001)","journal-title":"Microw. Wirel. Compon. Lett. IEEE"},{"key":"1012_CR30","volume-title":"Rational Approximation of Real Functions, Volume 28 of Encyclopedia of Mathematics","author":"P Petrushev","year":"1987","unstructured":"Petrushev, P., Popov, V.: Rational Approximation of Real Functions, Volume 28 of Encyclopedia of Mathematics. Cambridge University Press, Cambridge (1987)"},{"issue":"10","key":"1012_CR31","doi-asserted-by":"publisher","first-page":"1351","DOI":"10.1002\/cnm.1394","volume":"26","author":"D Rabinovich","year":"2010","unstructured":"Rabinovich, D., Givoli, D., B\u00e9cache, E.: Comparison of high-order absorbing boundary conditions and perfectly matched layers in the frequency domain. Int. J. Numer. Methods Biomed. Eng. 26(10), 1351\u20131369 (2010)","journal-title":"Int. J. Numer. Methods Biomed. Eng."},{"key":"1012_CR32","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970739","volume-title":"Numerical Methods for Large Eigen Value Problems","author":"Y Saad","year":"2011","unstructured":"Saad, Y.: Numerical Methods for Large Eigen Value Problems, vol. 66. Society for Industrial and Applied Mathematics, Philadelphia, PA (2011)"},{"key":"1012_CR33","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, 959\u2013962 (1974)","journal-title":"Math. Comput."},{"issue":"1","key":"1012_CR34","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1137\/S0036144598336745","volume":"41","author":"G Strang","year":"1999","unstructured":"Strang, G.: The discrete cosine transform. SIAM Rev. 41(1), 135\u2013147 (1999)","journal-title":"SIAM Rev."},{"issue":"4","key":"1012_CR35","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1016\/S0168-9274(98)00025-7","volume":"27","author":"SV Tsynkov","year":"1998","unstructured":"Tsynkov, S.V.: Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27(4), 465\u2013532 (1998)","journal-title":"Appl. Numer. Math."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-018-1012-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-018-1012-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-018-1012-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T05:40:48Z","timestamp":1577857248000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-018-1012-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,1]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["1012"],"URL":"https:\/\/doi.org\/10.1007\/s00211-018-1012-0","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1,1]]},"assertion":[{"value":"31 July 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 January 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}