{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T21:00:46Z","timestamp":1757624446465,"version":"3.44.0"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"S1","license":[{"start":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T00:00:00Z","timestamp":1745280000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T00:00:00Z","timestamp":1745280000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,9]]},"DOI":"10.1007\/s12190-025-02495-z","type":"journal-article","created":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T09:59:21Z","timestamp":1745315961000},"page":"197-215","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence analysis of the LDG method on two Duran-type meshes for reaction-diffusion problems"],"prefix":"10.1007","volume":"71","author":[{"given":"You","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9500-9398","authenticated-orcid":false,"given":"Yao","family":"Cheng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,22]]},"reference":[{"key":"2495_CR1","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1016\/j.camwa.2022.01.031","volume":"109","author":"C. Bacuta","year":"2022","unstructured":"Bacuta, C., Hayes, D., Jacavage, J.: Efficient discretization and preconditioning of the singularly perturbed reaction-diffusion problem. Comput. Math. Appl. 109, 270\u2013279 (2022)","journal-title":"Comput. Math. Appl."},{"doi-asserted-by":"crossref","unstructured":"Bacuta, C., Hayes, D., O\u2019Grady, T.: Results on a mixed finite element approach for a model convection-diffusion problem. arXiv:2402.03314 (2023)","key":"2495_CR2","DOI":"10.2139\/ssrn.4368193"},{"key":"2495_CR3","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/0041-5553(69)90038-X","volume":"9","author":"N.S Bakhvalov","year":"1969","unstructured":"Bakhvalov, N.S.: The optimalization of method of solving boundary value problems with a boundary layer. USSR Comput. Phys. 9, 139\u2013166 (1969)","journal-title":"USSR Comput. Phys."},{"issue":"238","key":"2495_CR4","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1090\/S0025-5718-01-01317-5","volume":"71","author":"P. Castillo","year":"2002","unstructured":"Castillo, P., Cockburn, B., Sch\u00f6tzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comp. 71(238), 455\u2013478 (2002)","journal-title":"Math. Comp."},{"key":"2495_CR5","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.cam.2021.113485","volume":"392","author":"Y Cheng","year":"2021","unstructured":"Cheng, Y.: On the local discontinuous Galerkin method for singularly perturbed problem with two parameters. J. Comput. Appl. Math. 392(Paper No. 113485), 22 (2021)","journal-title":"J. Comput. Appl. Math."},{"issue":"343","key":"2495_CR6","doi-asserted-by":"publisher","first-page":"2065","DOI":"10.1090\/mcom\/3844","volume":"92","author":"Y. Cheng","year":"2023","unstructured":"Cheng, Y., Jiang, S., Stynes, M.: Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem. Math. Comp. 92(343), 2065\u20132095 (2023)","journal-title":"Math. Comp."},{"issue":"1-2","key":"2495_CR7","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s00211-023-01361-z","volume":"154","author":"Y. Cheng","year":"2023","unstructured":"Cheng, Y., Stynes, M.: The local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem with characteristic and exponential layers. Numer. Math. 154(1-2), 283\u2013318 (2023)","journal-title":"Numer. Math."},{"key":"2495_CR8","first-page":"69","volume":"9","author":"B Cockburn","year":"1999","unstructured":"Cockburn, B.: Discontinuous galerkin methods for convection-dominated problems. Springer Verlag Berlin. 9, 69\u2013224 (1999)","journal-title":"Springer Verlag Berlin"},{"issue":"3","key":"2495_CR9","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1023\/A:1012873910884","volume":"16","author":"B. Cockburn","year":"2001","unstructured":"Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173\u2013261 (2001)","journal-title":"J. Sci. Comput."},{"issue":"10-11","key":"2495_CR10","doi-asserted-by":"publisher","first-page":"1314","DOI":"10.1016\/j.apnum.2006.03.029","volume":"56","author":"R.G. Dur\u00e1n","year":"2006","unstructured":"Dur\u00e1n, R.G., Lombardi, A.L.: Finite element approximation of convection diffusion problems using graded meshes. Appl. Numer. Math. 56(10-11), 1314\u20131325 (2006)","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"2495_CR11","doi-asserted-by":"publisher","first-page":"935","DOI":"10.3934\/era.2020049","volume":"28","author":"J. Shan","year":"2020","unstructured":"Shan, J., Li, L., Meiling, S., Fang, S.: Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electron. Res. Arch. 28(2), 935\u2013949 (2020)","journal-title":"Electron. Res. Arch."},{"key":"2495_CR12","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1016\/j.matcom.2021.01.006","volume":"185","author":"A. Kaushik","year":"2021","unstructured":"Kaushik, A., Kumar, V., Sharma, M., Sharma, N.: A modified graded mesh and higher order finite element method for singularly perturbed reaction-diffusion problems. Math. Comput. Simul. 185, 486\u2013496 (2021)","journal-title":"Math. Comput. Simul."},{"key":"2495_CR13","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/j.jcp.2019.04.073","volume":"395","author":"A. Kaushik","year":"2019","unstructured":"Kaushik, A., Vashishth, A.K., Kumar, V., Sharma, M.: A modified graded mesh and higher order finite element approximation for singular perturbation problems. J. Comput. Phys. 395, 275\u2013285 (2019)","journal-title":"J. Comput. Phys."},{"key":"2495_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05134-0","volume-title":"Layer-adapted meshes for reaction-convection-diffusion problems, volume 1985 of Lecture Notes in Mathematics","author":"T Lin\u00df","year":"2010","unstructured":"Lin\u00df, T.: Layer-adapted meshes for reaction-convection-diffusion problems, vol. 1985 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2010)"},{"unstructured":"Reed, W.H., Hill, T,R.: Triangular mesh methods for the neutron transport equation. Number LA-UR-73-479. (1973)","key":"2495_CR15"},{"key":"2495_CR16","volume-title":"Robust Numerical Methods for Singularly Perturbed Differential Equations, Volume 24 of Springer Series in Computational Mathematics","author":"H,-G, Roos","year":"2008","unstructured":"Roos, H,-G,, Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, vol. 24 of Springer Series in Computational Mathematics. 2nd edn. Springer-Verlag, Berlin (2008). Convection-diffusion-reaction and flow problems.","edition":"2nd"},{"issue":"1","key":"2495_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4208\/jcm.1405-m4362","volume":"33","author":"H.-G. Roos","year":"2015","unstructured":"Roos, H.-G., Teofanov, L., Uzelac, Z.: Graded meshes for higher order FEM. J. Comput. Math. 33(1), 1\u201316 (2015)","journal-title":"J. Comput. Math."},{"issue":"4","key":"2495_CR18","first-page":"542","volume":"35","author":"G.I Shishkin","year":"1992","unstructured":"Shishkin, G.I.: Discrete approximation of singularly perturbed elliptic and parabolic equations. Russ. Acad. Sci. Ural Sect. Ekaterinburg. 35(4), 542\u2013564 (1992)","journal-title":"Russ. Acad. Sci. Ural Sect. Ekaterinburg"},{"issue":"28","key":"2495_CR19","doi-asserted-by":"publisher","first-page":"3527","DOI":"10.1016\/S0045-7825(00)00271-1","volume":"190","author":"M. Stynes","year":"2001","unstructured":"Stynes, M., Lin\u00df, T.: Numerical methods on Shishkin meshes for linear convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 190(28), 3527\u20133542 (2001)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"2495_CR20","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1007\/s11075-023-01721-8","volume":"97","author":"S. Toprakseven","year":"2024","unstructured":"Toprakseven, S., Kaushik, A., Sharma, M.: A weak Galerkin finite element method for singularly perturbed problems with two small parameters on Bakhvalov-type meshes. Numer. Algorithms. 97(2), 727\u2013751 (2024)","journal-title":"Numer. Algorithms"},{"issue":"11","key":"2495_CR21","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1002\/zamm.202300238","volume":"103","author":"L. Yan","year":"2023","unstructured":"Yan, L., Cheng, Y.: Local discontinuous Galerkin method on graded meshes for a third-order singularly perturbed problem. Zamm Z. Angew. Math. Mech. 103(11), Paper No. e202300238, 15 (2023)","journal-title":"Zamm Z. Angew. Math. Mech."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02495-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02495-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02495-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T13:14:48Z","timestamp":1757423688000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02495-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,22]]},"references-count":21,"journal-issue":{"issue":"S1","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["2495"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02495-z","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2025,4,22]]},"assertion":[{"value":"11 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 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":"Not Applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"No conflict of interest.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}