{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T18:43:07Z","timestamp":1778611387881,"version":"3.51.4"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T00:00:00Z","timestamp":1745452800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T00:00:00Z","timestamp":1745452800000},"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":["Numer Algor"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s11075-025-02061-5","type":"journal-article","created":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T06:37:16Z","timestamp":1745476636000},"page":"1701-1738","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An ADI type operator splitting WG-FEM for 2D nonlinear unsteady singularly perturbed problem"],"prefix":"10.1007","volume":"101","author":[{"given":"Naresh","family":"Kumar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jasbir","family":"Singh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u015euayip","family":"Toprakseven","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ram","family":"Jiwari","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,24]]},"reference":[{"key":"2061_CR1","doi-asserted-by":"publisher","first-page":"988","DOI":"10.1007\/s10915-015-0115-3","volume":"67","author":"N Ahmed","year":"2016","unstructured":"Ahmed, N., Matthies, G.: Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent linear convection-diffusion-reaction equations. J. Sci. Comput. 67, 988\u20131018 (2016)","journal-title":"J. Sci. Comput."},{"issue":"17\u201320","key":"2061_CR2","doi-asserted-by":"publisher","first-page":"1114","DOI":"10.1016\/j.cma.2009.11.023","volume":"199","author":"E Burman","year":"2010","unstructured":"Burman, E.: Consistent SUPG-method for transient transport problems: Stability and convergence. Comput. Methods Appl. Mech. Eng. 199(17\u201320), 1114\u20131123 (2010)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"343","key":"2061_CR3","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. Comput. 92(343), 2065\u20132095 (2023)","journal-title":"Math. Comput."},{"issue":"4","key":"2061_CR4","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1007\/s10092-021-00445-2","volume":"58","author":"Y Cheng","year":"2021","unstructured":"Cheng, Y., Mei, Y.: Analysis of generalised alternating local discontinuous Galerkin method on layer-adapted mesh for singularly perturbed problems. Calcolo 58(4), 52 (2021)","journal-title":"Calcolo"},{"key":"2061_CR5","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1016\/j.camwa.2022.05.004","volume":"117","author":"Y Cheng","year":"2022","unstructured":"Cheng, Y., Mei, Y., Roos, H.-G.: The local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems. Comput. Math. Appl. 117, 245\u2013256 (2022)","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"2061_CR6","doi-asserted-by":"publisher","first-page":"1597","DOI":"10.1007\/s11075-022-01316-9","volume":"91","author":"Y Cheng","year":"2022","unstructured":"Cheng, Y., Yan, L., Mei, Y.: Balanced-norm error estimate of the local discontinuous Galerkin method on layer-adapted meshes for reaction-diffusion problems. Numer. Algorithms 91(4), 1597\u20131626 (2022)","journal-title":"Numer. Algorithms"},{"key":"2061_CR7","unstructured":"Cheng, Y., Zhang, Q., Wang, H.: Local analysis of the local discontinuous Galerkin method with the generalized alternating numerical flux for two-dimensional singularly perturbed problem. Int. J. Numer. Anal. Model. 15(6) (2018)"},{"key":"2061_CR8","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/j.apnum.2020.12.003","volume":"162","author":"B Deka","year":"2021","unstructured":"Deka, B., Kumar, N.: Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions. Appl. Numer. Math. 162, 81\u2013105 (2021)","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"2061_CR9","doi-asserted-by":"publisher","first-page":"2444","DOI":"10.1002\/num.22973","volume":"39","author":"B Deka","year":"2023","unstructured":"Deka, B., Kumar, N.: A systematic study on weak Galerkin finite element method for second-order parabolic problems. Numer. Meth. Part Differ. Equ. 39(3), 2444\u20132474 (2023)","journal-title":"Numer. Meth. Part Differ. Equ."},{"issue":"4","key":"2061_CR10","first-page":"477","volume":"14","author":"G Fu","year":"2017","unstructured":"Fu, G., Shu, C.-W.: Analysis of an embedded discontinuous Galerkin method with implicit-explicit time-marching for convection-diffusion problems. Int. J. Numer. Anal. Model. 14(4), 477\u2013499 (2017)","journal-title":"Int. J. Numer. Anal. Model."},{"issue":"6","key":"2061_CR11","doi-asserted-by":"publisher","first-page":"1447","DOI":"10.1051\/m2an\/2012012","volume":"46","author":"S Ganesan","year":"2012","unstructured":"Ganesan, S.: An operator-splitting Galerkin\/SUPG finite element method for population balance equations: stability and convergence. SAIM Math. Model. Numer. Anal. 46(6), 1447\u20131465 (2012)","journal-title":"SAIM Math. Model. Numer. Anal."},{"issue":"11","key":"2061_CR12","doi-asserted-by":"publisher","first-page":"2738","DOI":"10.1002\/aic.10228","volume":"50","author":"R Gunawan","year":"2004","unstructured":"Gunawan, R., Fusman, I., Braatz, R.D.: High resolution algorithms for multidimensional population balance equations. AIChE J. 50(11), 2738\u20132749 (2004)","journal-title":"AIChE J."},{"key":"2061_CR13","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jnnfm.2014.06.005","volume":"220","author":"R Huilgol","year":"2015","unstructured":"Huilgol, R., Kefayati, G.: Natural convection problem in a bingham fluid using the operator-splitting method. J. Nonnewton. Fluid Mech. 220, 22\u201332 (2015)","journal-title":"J. Nonnewton. Fluid Mech."},{"issue":"3\u20134","key":"2061_CR14","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1016\/j.cma.2008.08.016","volume":"198","author":"V John","year":"2008","unstructured":"John, V., Schmeyer, E.: Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 198(3\u20134), 475\u2013494 (2008)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"2061_CR15","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1016\/j.jmaa.2008.08.017","volume":"348","author":"J Ka\u010dur","year":"2008","unstructured":"Ka\u010dur, J., Malengier, B., Reme\u0161\u00edkov\u00e1, M.: Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system. J. Math. Anal. Appl. 348(2), 894\u2013914 (2008)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"2061_CR16","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s002110050291","volume":"77","author":"KH Karlsen","year":"1997","unstructured":"Karlsen, K.H., Risebro, N.H.: An operator splitting method for nonlinear convection-diffusion equations. Numer. Math. 77(3), 365\u2013382 (1997)","journal-title":"Numer. Math."},{"issue":"6","key":"2061_CR17","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1051\/m2an\/2009034","volume":"43","author":"DJ Knezevic","year":"2009","unstructured":"Knezevic, D.J., S\u00fcli, E.: A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model. ESAIM Math. Model. Numer. Anal. 43(6), 1117\u20131156 (2009)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"4","key":"2061_CR18","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/s10092-023-00541-5","volume":"60","author":"N Kumar","year":"2023","unstructured":"Kumar, N., Deka, B.: A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes. Calcolo 60(4), 47 (2023)","journal-title":"Calcolo"},{"key":"2061_CR19","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.apnum.2023.05.009","volume":"192","author":"N Kumar","year":"2023","unstructured":"Kumar, N., Dutta, J., Deka, B.: $${L}^2$$ estimates for weak Galerkin finite element methods for second-order wave equations with polygonal meshes. Appl. Numer. Math. 192, 84\u2013103 (2023)","journal-title":"Appl. Numer. Math."},{"key":"2061_CR20","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.camwa.2023.06.011","volume":"145","author":"N Kumar","year":"2023","unstructured":"Kumar, N., Singh, J., Jiwari, R.: Convergence analysis of weak Galerkin finite element method for semilinear parabolic convection dominated diffusion equations on polygonal meshes. Comput. Math. Appl. 145, 141\u2013158 (2023)","journal-title":"Comput. Math. Appl."},{"key":"2061_CR21","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1016\/j.jcp.2013.08.031","volume":"255","author":"L Li","year":"2013","unstructured":"Li, L., Xu, D., Luo, M.: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255, 471\u2013485 (2013)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2061_CR22","doi-asserted-by":"publisher","first-page":"1482","DOI":"10.1137\/17M1152528","volume":"56","author":"R Lin","year":"2018","unstructured":"Lin, R., Ye, X., Zhang, S., Zhu, P.: A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems. SIAM J. Numer. Anal. 56(3), 1482\u20131497 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"2061_CR23","first-page":"453","volume":"313","author":"A Majumdar","year":"2017","unstructured":"Majumdar, A., Natesan, S.: Alternating direction numerical scheme for singularly perturbed 2d degenerate parabolic convection-diffusion problems. Appl. Math. Comput. 313, 453\u2013473 (2017)","journal-title":"Appl. Math. Comput."},{"issue":"9","key":"2061_CR24","doi-asserted-by":"publisher","first-page":"3232","DOI":"10.1016\/j.jcp.2009.01.030","volume":"228","author":"NC Nguyen","year":"2009","unstructured":"Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys. 228(9), 3232\u20133254 (2009)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2061_CR25","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1137\/0103003","volume":"3","author":"DW Peaceman","year":"1955","unstructured":"Peaceman, D.W., Rachford, H.H., Jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28\u201341 (1955)","journal-title":"J. Soc. Ind. Appl. Math."},{"issue":"5","key":"2061_CR26","doi-asserted-by":"publisher","first-page":"1304","DOI":"10.1016\/j.ces.2007.07.049","volume":"63","author":"MA Pinto","year":"2008","unstructured":"Pinto, M.A., Immanuel, C.D., Doyle\u00a0Iii, F.J.: A two-level discretisation algorithm for the efficient solution of higher-dimensional population balance models. Chem. Eng. Sci. 63(5), 1304\u20131314 (2008)","journal-title":"Chem. Eng. Sci."},{"key":"2061_CR27","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s10492-006-0005-y","volume":"51","author":"H-G Roos","year":"2006","unstructured":"Roos, H.-G.: Error estimates for linear finite elements on Bakhvalov-type meshes. Appl. Math. 51, 63\u201372 (2006)","journal-title":"Appl. Math."},{"key":"2061_CR28","unstructured":"Roos, H.-G.: Robust numerical methods for singularly perturbed differential equations. Springer (2008)"},{"issue":"1\u201316","key":"2061_CR29","first-page":"62","volume":"32","author":"F Schieweck","year":"2008","unstructured":"Schieweck, F.: On the role of boundary conditions for cip stabilization of higher order finite elements. Electron. Trans. Numer. Anal. 32(1\u201316), 62 (2008)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"2061_CR30","unstructured":"Toprakseven, \u015e.: Optimal order uniform convergence of weak Galerkin finite element method on bakhvalov-type meshes for singularly perturbed convection dominated problems. Hacet. J. Math. Stat., pp 1\u201326"},{"key":"2061_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.apnum.2021.05.021","volume":"168","author":"\u015e Toprakseven","year":"2021","unstructured":"Toprakseven, \u015e: A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients. Appl. Numer. Math. 168, 1\u201312 (2021)","journal-title":"Appl. Numer. Math."},{"issue":"8","key":"2061_CR32","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/s40314-022-02090-z","volume":"41","author":"\u015e Toprakseven","year":"2022","unstructured":"Toprakseven, \u015e: Optimal order uniform convergence in energy and balanced norms of weak Galerkin finite element method on bakhvalov-type meshes for nonlinear singularly perturbed problems. Comput. Appl. Math. 41(8), 377 (2022)","journal-title":"Comput. Appl. Math."},{"issue":"12","key":"2061_CR33","doi-asserted-by":"publisher","first-page":"2314","DOI":"10.1016\/j.camwa.2014.03.021","volume":"68","author":"C Wang","year":"2014","unstructured":"Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math. Appl. 68(12), 2314\u20132330 (2014)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"2061_CR34","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1007\/s10915-017-0496-6","volume":"74","author":"J Wang","year":"2018","unstructured":"Wang, J., Wang, R., Zhai, Q., Zhang, R.: A systematic study on weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 74(3), 1369\u20131396 (2018)","journal-title":"J. Sci. Comput."},{"key":"2061_CR35","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2012.10.003","volume":"241","author":"J Wang","year":"2013","unstructured":"Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103\u2013115 (2013)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"2061_CR36","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s10444-015-9415-2","volume":"42","author":"J Wang","year":"2016","unstructured":"Wang, J., Ye, X.: A weak Galerkin finite element method for the stokes equations. Adv. Comput. Math. 42(1), 155\u2013174 (2016)","journal-title":"Adv. Comput. Math."},{"issue":"2","key":"2061_CR37","doi-asserted-by":"publisher","first-page":"530","DOI":"10.1108\/HFF-12-2015-0521","volume":"27","author":"X Xiao","year":"2017","unstructured":"Xiao, X., Gui, D., Feng, X.: A highly efficient operator-splitting finite element method for 2D\/3D nonlinear Allen-Cahn equation. Int. J. Numer. Method. H. 27(2), 530\u2013542 (2017)","journal-title":"Int. J. Numer. Method. H."},{"key":"2061_CR38","doi-asserted-by":"crossref","unstructured":"Yadav, N.S., Mukherjee, K.: Efficient parameter-robust numerical methods for singularly perturbed semilinear parabolic pdes of convection-diffusion type. Numer. Algorithms, pp. 1\u201349 (2023)","DOI":"10.1007\/s11075-023-01670-2"},{"issue":"10","key":"2061_CR39","doi-asserted-by":"publisher","first-page":"2243","DOI":"10.1016\/j.camwa.2017.07.009","volume":"74","author":"Q Zhai","year":"2017","unstructured":"Zhai, Q., Ye, X., Wang, R., Zhang, R.: A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems. Comput. Math. Appl. 74(10), 2243\u20132252 (2017)","journal-title":"Comput. Math. Appl."},{"key":"2061_CR40","doi-asserted-by":"publisher","first-page":"440","DOI":"10.1016\/j.ijheatmasstransfer.2015.01.028","volume":"84","author":"S Zhai","year":"2015","unstructured":"Zhai, S., Weng, Z., Gui, D., Feng, X.: High-order compact operator splitting method for three-dimensional fractional equation with subdiffusion. Int. J. Heat Mass Transf. 84, 440\u2013447 (2015)","journal-title":"Int. J. Heat Mass Transf."},{"issue":"1","key":"2061_CR41","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1007\/s10915-020-01312-y","volume":"85","author":"J Zhang","year":"2020","unstructured":"Zhang, J., Liu, X.: Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes. J. Sci. Comput. 85(1), 2 (2020)","journal-title":"J. Sci. Comput."},{"key":"2061_CR42","doi-asserted-by":"crossref","unstructured":"Zhu P., Xie, S.: A uniformly convergent weak Galerkin finite element method on Shishkin mesh for 1d convection-diffusion problem. J. Sci. Comput. 85(2):Paper No. 34, 22 (2020)","DOI":"10.1007\/s10915-020-01345-3"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02061-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-025-02061-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02061-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T09:33:39Z","timestamp":1772098419000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-025-02061-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,24]]},"references-count":42,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2061"],"URL":"https:\/\/doi.org\/10.1007\/s11075-025-02061-5","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,24]]},"assertion":[{"value":"13 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}