{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:35:00Z","timestamp":1760236500868,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801396,11802193,12172241"],"award-info":[{"award-number":["11801396,11802193,12172241"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20170374"],"award-info":[{"award-number":["BK20170374"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Nature Science Research Program for Colleges and Universities of Jiangsu Province","award":["17KJB110016"],"award-info":[{"award-number":["17KJB110016"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction\u2013diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and \u201cbalanced\u201d norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the \u201cbalanced\u201d norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss\u2013Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.<\/jats:p>","DOI":"10.3390\/sym13122291","type":"journal-article","created":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T02:56:14Z","timestamp":1638413774000},"page":"2291","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems"],"prefix":"10.3390","volume":"13","author":[{"given":"Yanjie","family":"Mei","sequence":"first","affiliation":[{"name":"International Education School, Suzhou University of Science and Technology, Suzhou 215009, China"}]},{"given":"Sulei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China"}]},{"given":"Zhijie","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4312-8034","authenticated-orcid":false,"given":"Chuanjing","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China"}]},{"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, Suzhou 215009, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,1]]},"reference":[{"key":"ref_1","unstructured":"Roos, H.G., Stynes, M., and Tobiska, L. 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