{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T10:41:52Z","timestamp":1762252912950,"version":"3.37.3"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,1,18]],"date-time":"2019-01-18T00:00:00Z","timestamp":1547769600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,1,18]],"date-time":"2019-01-18T00:00:00Z","timestamp":1547769600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Nature Science Foundation of China","doi-asserted-by":"crossref","award":["11601026","11471031"],"award-info":[{"award-number":["11601026","11471031"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Nature Science Foundation of China","doi-asserted-by":"crossref","award":["91430216"],"award-info":[{"award-number":["91430216"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"NASF","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["1419040"],"award-info":[{"award-number":["1419040"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s10915-019-00906-5","type":"journal-article","created":{"date-parts":[[2019,1,18]],"date-time":"2019-01-18T12:12:41Z","timestamp":1547813561000},"page":"1713-1736","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Supercloseness of Linear DG-FEM and Its Superconvergence Based on the Polynomial Preserving Recovery for Helmholtz Equation"],"prefix":"10.1007","volume":"79","author":[{"given":"Yu","family":"Du","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,1,18]]},"reference":[{"issue":"2","key":"906_CR1","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1137\/S0036142903423460","volume":"42","author":"M Ainsworth","year":"2004","unstructured":"Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553\u2013575 (2004)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR2","first-page":"93","volume":"1","author":"AK Aziz","year":"1979","unstructured":"Aziz, A.K., Kellogg, R.B.: A scattering problem for the Helmholtz equation. Adv. Comput. Methods Partial Diff. Equ.-III 1, 93\u201395 (1979)","journal-title":"Adv. Comput. Methods Partial Diff. Equ.-III"},{"issue":"3","key":"906_CR3","first-page":"451","volume":"42","author":"I Babu\u0161ka","year":"2000","unstructured":"Babu\u0161ka, I., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42(3), 451\u2013484 (2000)","journal-title":"SIAM Rev."},{"key":"906_CR4","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0045-7825(95)00890-X","volume":"128","author":"I Babu\u0161ka","year":"1995","unstructured":"Babu\u0161ka, I., Ihlenburg, F., Paik, E.T., Sauter, S.A.: A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Methods Appl. Mech. Eng. 128, 325\u2013359 (1995)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"906_CR5","doi-asserted-by":"publisher","first-page":"2294","DOI":"10.1137\/S003614290139874X","volume":"41","author":"RE Bank","year":"2003","unstructured":"Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part I: grid with superconvergence. SIAM J. Numer. Anal. 41, 2294\u20132312 (2003)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR6","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01389427","volume":"49","author":"H Blum","year":"1986","unstructured":"Blum, H., Rannacher, R.: Asymptotic error expansion and richardson extrapolation for linear finite elements. Numer. Math. 49, 11\u201338 (1986)","journal-title":"Numer. Math."},{"issue":"194","key":"906_CR7","first-page":"463","volume":"56","author":"J\u00a0H Bramble","year":"1991","unstructured":"Bramble, J\u00a0.H., Xu, J.: Some estimates for a weighted L$$^2$$ projection. Math. Comput. 56(194), 463\u2013476 (1991)","journal-title":"Math. Comput."},{"key":"906_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Element Methods","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)","edition":"3"},{"key":"906_CR9","unstructured":"Burman, E., Wu, H., Zhu, L.: Continuous interior penalty finite element method for Helmholtz equation with high wave number: one dimensional analysis. arXiv:1211.1424"},{"key":"906_CR10","doi-asserted-by":"publisher","first-page":"2331","DOI":"10.1137\/130917144","volume":"51","author":"Z Chen","year":"2013","unstructured":"Chen, Z., Xiang, X.: A source transfer domain decomposition method for Helmholtz equations in unbounded domain. SIAM J. Numer. Anal. 51, 2331\u20132356 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR11","volume-title":"Adaptive Computations: Theory and Algorithms","author":"L Chen","year":"2007","unstructured":"Chen, L., Xu, J.: Topics on Adaptive Finite Element Methods. In: Tang, T., Xu, J. (eds.) Adaptive Computations: Theory and Algorithms. Science Press, Beijing (2007)"},{"key":"906_CR12","volume-title":"The Finite Element Method for Elliptic Problems","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Pub. Co., New York (1978)"},{"key":"906_CR13","unstructured":"COMSOL AB.: Comsol Multiphysics User\u2019s Guide, 3.5a ed. (2008)"},{"key":"906_CR14","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1002\/(SICI)1097-0207(19991010)46:4<471::AID-NME684>3.0.CO;2-6","volume":"46","author":"A Deraemaeker","year":"1999","unstructured":"Deraemaeker, A., Babu\u0161ka, I., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Internat. J. Numer. Methods Eng. 46, 471\u2013499 (1999)","journal-title":"Internat. J. Numer. Methods Eng."},{"key":"906_CR15","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1142\/S0218202594000297","volume":"4","author":"J Douglas Jr","year":"1994","unstructured":"Douglas Jr., J., Santos, J.E., Sheen, D.: Approximation of scalar waves in the space\u2013frequency domain. Math. Models Methods Appl. Sci. 4, 509\u2013531 (1994)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"906_CR16","unstructured":"Du, Y., Wu, H., Zhang, Z.: Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number. arXiv:1703.00156 (2017)"},{"issue":"2","key":"906_CR17","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":"906_CR18","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1007\/s10915-015-0074-8","volume":"67","author":"Y Du","year":"2015","unstructured":"Du, Y., Zhu, L.: Preasymptotic error analysis of high order interior penalty discontinuous Galerkin methods for the Helmholtz equation with high wave number. J. Sci. Comput. 67, 130\u2013152 (2015)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"906_CR19","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1002\/cpa.3160320303","volume":"32","author":"B Engquist","year":"1979","unstructured":"Engquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Commun. Pure Appl. Math. 32(3), 313\u2013357 (1979)","journal-title":"Commun. Pure Appl. Math."},{"issue":"4","key":"906_CR20","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 numbers. SIAM J. Numer. Anal. 47(4), 2872\u20132896 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"276","key":"906_CR21","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."},{"key":"906_CR22","doi-asserted-by":"crossref","unstructured":"Grisvard, P.: Elliptic Problems in Nonsmooth Domains, vol. 69. SIAM (2011)","DOI":"10.1137\/1.9781611972030"},{"issue":"1","key":"906_CR23","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/S0045-7825(96)01034-1","volume":"140","author":"I Harari","year":"1997","unstructured":"Harari, I.: Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Eng. 140(1), 39\u201358 (1997)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"906_CR24","first-page":"11","volume":"158","author":"P Helfrich","year":"1983","unstructured":"Helfrich, P.: Asymptotic expansion for the finite element approximations of parabolic problems. Bonner Math. Schriften 158, 11\u201330 (1983)","journal-title":"Bonner Math. Schriften"},{"issue":"263","key":"906_CR25","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1090\/S0025-5718-08-02051-6","volume":"77","author":"Y Huang","year":"2008","unstructured":"Huang, Y., Jinchao, X.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comput. 77(263), 1253\u20131268 (2008)","journal-title":"Math. Comput."},{"issue":"9","key":"906_CR26","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. I. The $$h$$-version of the FEM. Comput. Math. Appl. 30(9), 9\u201337 (1995)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"906_CR27","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1137\/S0036142994272337","volume":"34","author":"F Ihlenburg","year":"1997","unstructured":"Ihlenburg, F., Babu\u0161ka, I.: Finite element solution of the Helmholtz equation with high wave number. II. The $$h$$-$$p$$ version of the FEM. SIAM J. Numer. Anal. 34(1), 315\u2013358 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0045-7825(99)00281-9","volume":"189","author":"AM Lakhany","year":"2000","unstructured":"Lakhany, A.M., Marek, I., Whiteman, J.R.: Superconvergence results on mildly structured triangulations. Comput. Methods Appl. Mech. Eng. 189, 1\u201375 (2000)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"906_CR29","first-page":"57","volume":"16","author":"Q Lin","year":"1998","unstructured":"Lin, Q., Zhang, S., Yan, N.: Asymptotic error expansion and defect correction for Sobolev and viscoelasticity type equations. J. Comput. Math. 16, 57\u201362 (1998)","journal-title":"J. Comput. Math."},{"key":"906_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8224-9","volume-title":"Difference Methods and Their Extrapolation","author":"G Marchuk","year":"1983","unstructured":"Marchuk, G., Shaidurov, V.: Difference Methods and Their Extrapolation. Springer, New York (1983)"},{"issue":"272","key":"906_CR31","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.A.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comput. 79(272), 1871\u20131914 (2010)","journal-title":"Math. Comput."},{"issue":"3","key":"906_CR32","doi-asserted-by":"publisher","first-page":"1210","DOI":"10.1137\/090776202","volume":"49","author":"JM Melenk","year":"2011","unstructured":"Melenk, J.M., Sauter, S.A.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49(3), 1210\u20131243 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR33","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, 536\u2013581 (2013)","journal-title":"J. Sci. Comput."},{"key":"906_CR34","doi-asserted-by":"publisher","first-page":"1780","DOI":"10.1137\/S0036142903413002","volume":"42","author":"A Naga","year":"2004","unstructured":"Naga, A., Zhang, Z.: A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42, 1780\u20131800 (2004)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR35","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717440","volume-title":"Discontinous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation","author":"B Rivi\u00e8re","year":"2008","unstructured":"Rivi\u00e8re, B.: Discontinous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)"},{"key":"906_CR36","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\u2013Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959\u2013962 (1974)","journal-title":"Math. Comput."},{"key":"906_CR37","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/BF01396046","volume":"55","author":"J Wang","year":"1989","unstructured":"Wang, J.: Asymptotic expansions and $$l^\\infty $$-error estimates for mixed finite element methods for second order elliptic problems. Numer. Math. 55, 401\u2013430 (1989)","journal-title":"Numer. Math."},{"key":"906_CR38","doi-asserted-by":"publisher","first-page":"1701","DOI":"10.1137\/060661430","volume":"45","author":"H Wu","year":"2007","unstructured":"Wu, H., Zhang, Z.: Can we have superconvergent gradient recovery under adaptive meshes? SIAM J. Numer. Anal. 45, 1701\u20131722 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"906_CR39","doi-asserted-by":"publisher","first-page":"1139","DOI":"10.1090\/S0025-5718-03-01600-4","volume":"73","author":"J Xu","year":"2003","unstructured":"Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139\u20131152 (2003)","journal-title":"Math. Comput."},{"key":"906_CR40","doi-asserted-by":"publisher","first-page":"4289","DOI":"10.1016\/S0045-7825(00)00319-4","volume":"190","author":"N Yan","year":"2001","unstructured":"Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Comput. Methods Appl. Mech. Eng. 190, 4289\u20134299 (2001)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"906_CR41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4208\/ijnam.OA-2004-1101","volume":"1","author":"Z Zhang","year":"2004","unstructured":"Zhang, Z.: Polynomial preserving gradient recovery and a posteriori estimate for bilinear element on irregular quadrilaterals. Int. J. Numer. Anal. Model. 1, 1\u201324 (2004)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"906_CR42","first-page":"331","volume":"22","author":"Z Zhang","year":"2004","unstructured":"Zhang, Z.: Polynomial preserving recovery for anisotropic and irregular grids. J. Comput. Math. 22, 331\u2013340 (2004)","journal-title":"J. Comput. Math."},{"key":"906_CR43","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1002\/(SICI)1098-2426(199903)15:2<151::AID-NUM2>3.0.CO;2-O","volume":"15","author":"Z Zhang","year":"1999","unstructured":"Zhang, Z., Li, B.: Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numer. Methods Partial Diff. Equ. 15, 151\u2013167 (1999)","journal-title":"Numer. Methods Partial Diff. Equ."},{"key":"906_CR44","doi-asserted-by":"publisher","first-page":"1192","DOI":"10.1137\/S1064827503402837","volume":"26","author":"Z Zhang","year":"2005","unstructured":"Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26, 1192\u20131213 (2005)","journal-title":"SIAM J. Sci. Comput."},{"key":"906_CR45","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1016\/j.apnum.2014.06.013","volume":"85","author":"T Zhang","year":"2014","unstructured":"Zhang, T., Yu, S.: The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method. Appl. Numer. Math. 85, 128\u2013141 (2014)","journal-title":"Appl. Numer. Math."},{"key":"906_CR46","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1016\/j.camwa.2015.06.007","volume":"70","author":"L Zhu","year":"2015","unstructured":"Zhu, L., Du, Y.: Pre-asymptotic error analysis of $$hp$$-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number. Comput. Math. Appl. 70, 917\u2013933 (2015)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"906_CR47","doi-asserted-by":"publisher","first-page":"1828","DOI":"10.1137\/120874643","volume":"51","author":"L Zhu","year":"2013","unstructured":"Zhu, L., Wu, H.: Pre-asymptotic 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":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-00906-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00906-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00906-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,10]],"date-time":"2022-09-10T16:51:42Z","timestamp":1662828702000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-00906-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,18]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["906"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-00906-5","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2019,1,18]]},"assertion":[{"value":"28 May 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 January 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 January 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}