{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T02:49:36Z","timestamp":1751942976288,"version":"3.40.3"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T00:00:00Z","timestamp":1740960000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T00:00:00Z","timestamp":1740960000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006764","name":"Technische Universit\u00e4t Berlin","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006764","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The perfectly matched layer method (PML method) is a truncation technique well known for the numerical treatment of wave scattering problems in unbounded domains. In this paper, we study the convergence of the PML method for the wave scattering from an open waveguide in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\mathbb {R}^2_+=\\{x\\in \\mathbb {R}^2:x_2&gt;0\\}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mrow>\n                        <mml:mi>R<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mo>+<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msubsup>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>{<\/mml:mo>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>\u2208<\/mml:mo>\n                      <mml:msup>\n                        <mml:mrow>\n                          <mml:mi>R<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msup>\n                      <mml:mo>:<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>x<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>&gt;<\/mml:mo>\n                      <mml:mn>0<\/mml:mn>\n                      <mml:mo>}<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, where the refractive index is assumed to be a local perturbation of a function which is periodic with respect to <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$x_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and equal to one above a finite height. The problem is challenging from the theoretical, and also from the numerical, point of view due to the existence of guided waves. A typical way to deal with this difficulty is to apply the limiting absorption principle. Based on the Floquet-Bloch transform and a curve deformation theory, the solution, derived from the limiting absorption principle, is rewritten as the line integral (with respect to the Floquet-Bloch parameter) of the solution of a system of quasi-periodic problems. By comparing the Dirichlet-to-Neumann maps on a straight line above the locally perturbed periodic layer, we finally show that the PML method converges exponentially with respect to the PML parameter. Finally, some numerical examples are shown to illustrate the theoretical results.<\/jats:p>","DOI":"10.1007\/s00211-025-01456-9","type":"journal-article","created":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T15:23:54Z","timestamp":1741015434000},"page":"717-748","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The PML-method for a scattering problem for a local perturbation of an open periodic waveguide"],"prefix":"10.1007","volume":"157","author":[{"given":"Andreas","family":"Kirsch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruming","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,3]]},"reference":[{"issue":"3","key":"1456_CR1","doi-asserted-by":"crossref","first-page":"2536","DOI":"10.1137\/17M1118920","volume":"50","author":"A Kirsch","year":"2018","unstructured":"Kirsch, A., Lechleiter, A.: The limiting absorption principle and a radiation condition for the scattering by a periodic layer. SIAM J. Math. Anal. 50(3), 2536\u20132565 (2018)","journal-title":"SIAM J. Math. Anal."},{"key":"1456_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-015-0897-3","author":"S Fliss","year":"2015","unstructured":"Fliss, S., Joly, P.: Solutions of the time-harmonic wave equation in periodic waveguides: asymptotic behaviour and radiation condition. Arch. Rational Mech. Anal. (2015). https:\/\/doi.org\/10.1007\/s00205-015-0897-3","journal-title":"Arch. Rational Mech. Anal."},{"issue":"10","key":"1456_CR3","doi-asserted-by":"crossref","first-page":"3955","DOI":"10.1002\/mma.4879","volume":"41","author":"A Kirsch","year":"2018","unstructured":"Kirsch, A., Lechleiter, A.: A radiation condition arising from the limiting absorption principle for a closed full- or half-waveguide problem. Math. Meth. Appl. Sci. 41(10), 3955\u20133975 (2018)","journal-title":"Math. Meth. Appl. Sci."},{"issue":"3","key":"1456_CR4","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1137\/100791798","volume":"71","author":"V Hoang","year":"2011","unstructured":"Hoang, V.: The limiting absorption principle for a periodic semin-infinite waveguide. SIAM J. Appl. Math. 71(3), 791\u2013810 (2011)","journal-title":"SIAM J. Appl. Math."},{"issue":"1","key":"1456_CR5","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.matpur.2012.10.013","volume":"100","author":"T Hohage","year":"2013","unstructured":"Hohage, T., Soussi, S.: Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides. J. Math. Pures Appl. 100(1), 113\u2013135 (2013)","journal-title":"J. Math. Pures Appl."},{"issue":"1","key":"1456_CR6","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1137\/19M1290942","volume":"81","author":"R Zhang","year":"2021","unstructured":"Zhang, R.: Spectrum decomposition of translation operators in periodic waveguide. SIAM J. Appl. Math. 81(1), 233\u2013257 (2021). https:\/\/doi.org\/10.1137\/19M1290942","journal-title":"SIAM J. Appl. Math."},{"key":"1456_CR7","first-page":"945","volume":"1","author":"P Joly","year":"2006","unstructured":"Joly, P., Li, J.-R., Fliss, S.: Exact boundary conditions for periodic waveguides containing a local perturbation. Commun. Comput. Phys. 1, 945\u2013973 (2006)","journal-title":"Commun. Comput. Phys."},{"key":"1456_CR8","doi-asserted-by":"publisher","first-page":"2155","DOI":"10.1016\/j.apnum.2008.12.013","volume":"59","author":"S Fliss","year":"2009","unstructured":"Fliss, S., Joly, P.: Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media. Appl. Numer. Math. 59, 2155\u20132178 (2009). https:\/\/doi.org\/10.1016\/j.apnum.2008.12.013","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"1456_CR9","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1137\/12086697X","volume":"35","author":"S Fliss","year":"2013","unstructured":"Fliss, S.: A dirichlet-to-neumann approach for the exact computation of guided modes in photonic crystal waveguides. SIAM J. Sci. Comput. 35(2), 438\u2013461 (2013). https:\/\/doi.org\/10.1137\/12086697X","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"1456_CR10","doi-asserted-by":"crossref","first-page":"487","DOI":"10.2140\/paa.2021.3.487","volume":"3","author":"S Fliss","year":"2021","unstructured":"Fliss, S., Joly, P., Lescarret, V.: A DtN approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity. Pure Appl. Anal. 3(3), 487\u2013526 (2021)","journal-title":"Pure Appl. Anal."},{"key":"1456_CR11","doi-asserted-by":"crossref","first-page":"3649","DOI":"10.1109\/JLT.2007.907742","volume":"25","author":"L Yuan","year":"2007","unstructured":"Yuan, L., Lu, Y.Y.: A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides. J. Lightwave Technol. 25, 3649\u20133656 (2007)","journal-title":"J. Lightwave Technol."},{"issue":"14","key":"1456_CR12","doi-asserted-by":"crossref","first-page":"6877","DOI":"10.1016\/j.jcp.2008.03.042","volume":"227","author":"M Ehrhardt","year":"2008","unstructured":"Ehrhardt, M., Zheng, C.: Exact artificial boundary conditions for problems with periodic structures. J. Comput. Phys. 227(14), 6877\u20136894 (2008)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"1456_CR13","doi-asserted-by":"crossref","first-page":"347","DOI":"10.4310\/CMS.2009.v7.n2.a4","volume":"7","author":"M Ehrhardt","year":"2009","unstructured":"Ehrhardt, M., Sun, J., Zheng, C.: Evaluation of scattering operators for semi-infinite periodic arrays. Commun. Math. Sci. 7(2), 347\u2013364 (2009)","journal-title":"Commun. Math. Sci."},{"key":"1456_CR14","first-page":"849","volume":"5","author":"M Ehrhardt","year":"2009","unstructured":"Ehrhardt, M., Han, H., Zheng, C.: Numerical simulation of waves in periodic structures. Commun. Comput. Phys. 5, 849\u2013870 (2009)","journal-title":"Commun. Comput. Phys."},{"issue":"10","key":"1456_CR15","volume":"35","author":"A Kirsch","year":"2019","unstructured":"Kirsch, A.: Scattering by a periodic tube in $$\\mathbb{R} ^{3}$$: part i. the limiting absorption principle. Inverse Prob. 35(10), 104004 (2019)","journal-title":"Inverse Prob."},{"issue":"10","key":"1456_CR16","volume":"35","author":"A Kirsch","year":"2019","unstructured":"Kirsch, A.: Scattering by a periodic tube in $$\\mathbb{R} ^{3}$$: part i. the radiation condition. Inverse Prob. 35(10), 104005 (2019)","journal-title":"Inverse Prob."},{"issue":"10","key":"1456_CR17","doi-asserted-by":"crossref","first-page":"5737","DOI":"10.1002\/mma.8137","volume":"45","author":"A Kirsch","year":"2022","unstructured":"Kirsch, A.: A scattering problem for a local perturbation of an open periodic waveguide. Math. Methods Appl. Sci. 45(10), 5737\u20135773 (2022)","journal-title":"Math. Methods Appl. Sci."},{"issue":"9","key":"1456_CR18","doi-asserted-by":"crossref","first-page":"10698","DOI":"10.1002\/mma.9147","volume":"46","author":"A Kirsch","year":"2023","unstructured":"Kirsch, A.: On the scattering of a plane wave by a perturbed open periodic waveguide. Math. Methods Appl. Sci. 46(9), 10698\u201310718 (2023)","journal-title":"Math. Methods Appl. Sci."},{"issue":"2","key":"1456_CR19","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1137\/040615523","volume":"37","author":"SN Chandler-Wilde","year":"2005","unstructured":"Chandler-Wilde, S.N., Monk, P.: Existence, uniqueness and variational methods for scattering by unbounded rough surfaces. SIAM J. Math. Appl. 37(2), 598\u2013618 (2005). https:\/\/doi.org\/10.1137\/040615523","journal-title":"SIAM J. Math. Appl."},{"key":"1456_CR20","doi-asserted-by":"publisher","first-page":"1675","DOI":"10.1016\/j.jcp.2011.10.022","volume":"231","author":"J Coatl\u00e9ven","year":"2012","unstructured":"Coatl\u00e9ven, J.: Helmholtz equation in periodic media with a line defect. J. Comp. Phys. 231, 1675\u20131704 (2012). https:\/\/doi.org\/10.1016\/j.jcp.2011.10.022","journal-title":"J. Comp. Phys."},{"issue":"1","key":"1456_CR21","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1080\/00036811.2016.1221942","volume":"96","author":"H Haddar","year":"2016","unstructured":"Haddar, H., Nguyen, T.P.: A volume integral method for solving scattering problems from locally perturbed infinite periodic layers. Appl. Anal. 96(1), 130\u2013158 (2016)","journal-title":"Appl. Anal."},{"key":"1456_CR22","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1017\/S0308210515000335","volume":"231","author":"A Lechleiter","year":"2015","unstructured":"Lechleiter, A., Nguyen, D.-L.: Scattering of Herglotz waves from periodic structures and mapping properties of the Bloch transform. Proc. Roy. Soc. Edinburgh Sect. A 231, 1283\u20131311 (2015)","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A"},{"issue":"1","key":"1456_CR23","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.jmaa.2016.08.055","volume":"446","author":"A Lechleiter","year":"2017","unstructured":"Lechleiter, A.: The Floquet-Bloch transform and scattering from locally perturbed periodic surfaces. J. Math. Anal. Appl. 446(1), 605\u2013627 (2017)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"1456_CR24","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1137\/16M1067524","volume":"55","author":"A Lechleiter","year":"2017","unstructured":"Lechleiter, A., Zhang, R.: A convergent numerical scheme for scattering of aperiodic waves from periodic surfaces based on the Floquet-Bloch transform. SIAM J. Numer. Anal. 55(2), 713\u2013736 (2017)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"1456_CR25","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1137\/16M1104111","volume":"39","author":"A Lechleiter","year":"2017","unstructured":"Lechleiter, A., Zhang, R.: A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces. SIAM J. Sci. Comput. 39(5), 819\u2013839 (2017)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"1456_CR26","doi-asserted-by":"crossref","first-page":"2286","DOI":"10.1137\/17M1144945","volume":"40","author":"R Zhang","year":"2018","unstructured":"Zhang, R.: A high order numerical method for scattering from locally perturbed periodic surfaces. SIAM J. Sci. Comput. 40(4), 2286\u20132314 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"1456_CR27","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1137\/21M1439043","volume":"60","author":"R Zhang","year":"2022","unstructured":"Zhang, R.: Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces. SIAM J. Numer. Math. 60(2), 804\u2013823 (2022)","journal-title":"SIAM J. Numer. Math."},{"issue":"6","key":"1456_CR28","doi-asserted-by":"crossref","first-page":"2569","DOI":"10.1137\/19M1301679","volume":"81","author":"G Hu","year":"2021","unstructured":"Hu, G., Lu, W., Rathsfeld, A.: Time-harmonic acoustic scattering from locally perturbed periodic curves. SIAM J. Appl. Math. 81(6), 2569\u20132595 (2021)","journal-title":"SIAM J. Appl. Math."},{"issue":"5","key":"1456_CR29","doi-asserted-by":"crossref","first-page":"2592","DOI":"10.1137\/21M1439705","volume":"60","author":"X Yu","year":"2022","unstructured":"Yu, X., Hu, G., Lu, W., Rathsfeld, A.: PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface. SIAM J. Numer. Anal. 60(5), 2592\u20132625 (2022)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"1456_CR30","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1006\/jcph.1994.1159","volume":"114","author":"J-P Berenger","year":"1994","unstructured":"Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185\u2013200 (1994). https:\/\/doi.org\/10.1006\/jcph.1994.1159","journal-title":"J. Comput. Phys."},{"issue":"3","key":"1456_CR31","doi-asserted-by":"publisher","first-page":"799","DOI":"10.1137\/S0036142902400901","volume":"41","author":"Z Chen","year":"2003","unstructured":"Chen, Z., Wu, H.: An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41(3), 799\u2013826 (2003). https:\/\/doi.org\/10.1137\/S0036142902400901","journal-title":"SIAM J. Numer. Anal."},{"key":"1456_CR32","doi-asserted-by":"publisher","first-page":"2131","DOI":"10.1016\/j.apnum.2008.12.007","volume":"59","author":"SN Chandler-Wilde","year":"2009","unstructured":"Chandler-Wilde, S.N., Monk, P.: The PML for rough surface scattering. Appl. Numer. Math. 59, 2131\u20132154 (2009). https:\/\/doi.org\/10.1016\/j.apnum.2008.12.007","journal-title":"Appl. Numer. Math."},{"issue":"6","key":"1456_CR33","doi-asserted-by":"crossref","first-page":"2917","DOI":"10.1137\/22M1532615","volume":"61","author":"R Zhang","year":"2023","unstructured":"Zhang, R.: Higher-order convergence of perfectly matched layers in three-dimensional biperiodic surface scattering problems. SIAM J. Numer. Anal. 61(6), 2917\u20132939 (2023)","journal-title":"SIAM J. Numer. Anal."},{"key":"1456_CR34","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1007\/s00211-021-01229-0","volume":"148","author":"R Zhang","year":"2021","unstructured":"Zhang, R.: Numerical method for scattering problems in periodic waveguides. Numer. Math. 148, 959\u2013996 (2021)","journal-title":"Numer. Math."},{"issue":"5","key":"1456_CR35","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1137\/21M1421532","volume":"44","author":"R Zhang","year":"2022","unstructured":"Zhang, R.: High order complex contour discretization methods to simulate scattering problems in locally perturbed periodic waveguides. SIAM J. Sci. Comput. 44(5), 1257\u20131281 (2022)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1456_CR36","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmaa.2020.124149","volume":"489","author":"T Furuya","year":"2020","unstructured":"Furuya, T.: Scattering by the local perturbation of an open periodic waveguide in the half plane. J. Math. Anal. Appl. 489(1), 124149 (2020)","journal-title":"J. Math. Anal. Appl."},{"key":"1456_CR37","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-30351-8","volume-title":"Inverse acoustic and electromagnetic scattering theory","author":"DL Colton","year":"2019","unstructured":"Colton, D.L., Kress, R.: Inverse acoustic and electromagnetic scattering theory, 4th edn. Springer, Berlin (2019)","edition":"4"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01456-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01456-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01456-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T20:54:07Z","timestamp":1743281647000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01456-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,3]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["1456"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01456-9","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"type":"print","value":"0029-599X"},{"type":"electronic","value":"0945-3245"}],"subject":[],"published":{"date-parts":[[2025,3,3]]},"assertion":[{"value":"18 March 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"This work does not have any Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}