{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T00:51:43Z","timestamp":1776905503995,"version":"3.51.2"},"reference-count":19,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of convection-diffusion type.\nBased on a regularity assumption for the exact solution, we prove almost <jats:inline-formula id=\"j_cmam-2022-0176_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>O<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:msup>\n                                    <m:mi>N<\/m:mi>\n                                    <m:mrow>\n                                       <m:mo>-<\/m:mo>\n                                       <m:mrow>\n                                          <m:mo stretchy=\"false\">(<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>k<\/m:mi>\n                                             <m:mo>+<\/m:mo>\n                                             <m:mfrac>\n                                                <m:mn>1<\/m:mn>\n                                                <m:mn>2<\/m:mn>\n                                             <\/m:mfrac>\n                                          <\/m:mrow>\n                                          <m:mo stretchy=\"false\">)<\/m:mo>\n                                       <\/m:mrow>\n                                    <\/m:mrow>\n                                 <\/m:msup>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0176_eq_0280.png\"\/>\n                        <jats:tex-math>{O(N^{-(k+\\frac{1}{2})})}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter.\nHere, <jats:inline-formula id=\"j_cmam-2022-0176_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>k<\/m:mi>\n                              <m:mo>\u2265<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0176_eq_0414.png\"\/>\n                        <jats:tex-math>{k\\geq 0}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is the maximum degree of piecewise polynomials\nused in discrete space, and <jats:italic>N<\/jats:italic> is the number of mesh elements.\nThe results are valid for the three types of layer-adapted meshes:\nShishkin-type, Bakhvalov\u2013Shishkin-type, and Bakhvalov-type.\nNumerical experiments are conducted to test the theoretical results.<\/jats:p>","DOI":"10.1515\/cmam-2022-0176","type":"journal-article","created":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T23:44:36Z","timestamp":1670370276000},"page":"751-766","source":"Crossref","is-referenced-by-count":6,"title":["Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type"],"prefix":"10.1515","volume":"23","author":[{"given":"Li","family":"Yan","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences , Suzhou University of Science and Technology , Jiangsu , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhoufeng","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Henan University of Science and Technology , Henan , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9500-9398","authenticated-orcid":false,"given":"Yao","family":"Cheng","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , Suzhou University of Science and Technology , Jiangsu , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,12,6]]},"reference":[{"key":"2023070413382858376_j_cmam-2022-0176_ref_001","doi-asserted-by":"crossref","unstructured":"Y.  Cheng,\nOn the local discontinuous Galerkin method for singularly perturbed problem with two parameters,\nJ. Comput. Appl. Math. 392 (2021), Paper No. 113485.","DOI":"10.1016\/j.cam.2021.113485"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_002","doi-asserted-by":"crossref","unstructured":"Y.  Cheng, Y.  Mei and H.-G.  Roos,\nThe local discontinuous Galerkin method on layer-adapted meshes for time-dependent singularly perturbed convection-diffusion problems,\nComput. Math. Appl. 117 (2022), 245\u2013256.","DOI":"10.1016\/j.camwa.2022.05.004"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_003","doi-asserted-by":"crossref","unstructured":"Y.  Cheng, L.  Yan, X.  Wang and Y.  Liu,\nOptimal maximum-norm estimate of the LDG method for singularly perturbed convection-diffusion problem,\nAppl. Math. Lett. 128 (2022), Paper No. 107947.","DOI":"10.1016\/j.aml.2022.107947"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_004","doi-asserted-by":"crossref","unstructured":"Y.  Cheng and Q.  Zhang,\nLocal analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem,\nJ. Sci. Comput. 72 (2017), no. 2, 792\u2013819.","DOI":"10.1007\/s10915-017-0378-y"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_005","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_006","doi-asserted-by":"crossref","unstructured":"B.  Cockburn and C.-W.  Shu,\nThe local discontinuous Galerkin method for time-dependent convection-diffusion systems,\nSIAM J. Numer. Anal. 35 (1998), no. 6, 2440\u20132463.","DOI":"10.1137\/S0036142997316712"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_007","doi-asserted-by":"crossref","unstructured":"F. A.  Howes,\nNonlinear dispersive systems: theory and examples,\nStud. Appl. Math. 69 (1983), no. 1, 75\u201397.","DOI":"10.1002\/sapm198369175"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_008","doi-asserted-by":"crossref","unstructured":"F. A.  Howes,\nThe asymptotic solution of a class of third-order boundary value problems arising in the theory of thin film flows,\nSIAM J. Appl. Math. 43 (1983), no. 5, 993\u20131004.","DOI":"10.1137\/0143065"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_009","doi-asserted-by":"crossref","unstructured":"T.  Lin\u00df,\nThe necessity of Shishkin decompositions,\nAppl. Math. Lett. 14 (2001), no. 7, 891\u2013896.","DOI":"10.1016\/S0893-9659(01)00061-1"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_010","doi-asserted-by":"crossref","unstructured":"T.  Lin\u00df,\nLayer-adapted Meshes for Reaction-Convection-Diffusion Problems,\nLecture Notes in Math. 1985,\nSpringer, Berlin, 2010.","DOI":"10.1007\/978-3-642-05134-0"},{"key":"2023070413382858376_j_cmam-2022-0176_ref_011","unstructured":"R. E.  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