{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T14:26:18Z","timestamp":1766067978493,"version":"3.32.0"},"reference-count":57,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this article, we employ discontinuous Galerkin methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first develop a posteriori error estimates in the energy norm wherein, the reliability and efficiency of the proposed a posteriori\nerror estimator is addressed. The suitable construction of the discrete Lagrange multiplier <jats:inline-formula id=\"j_cmam-2023-0015_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>\ud835\udf40<\/m:mi>\n                              <m:mi>\ud835\udc89<\/m:mi>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0015_eq_0416.png\"\/>\n                        <jats:tex-math>{\\boldsymbol{\\lambda_{h}}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and some intermediate operators play a key role in developing a posteriori error analysis. Further, we establish an optimal a priori error estimates under the appropriate regularity assumption on the exact solution <jats:inline-formula id=\"j_cmam-2023-0015_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>\ud835\udc96<\/m:mi>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0015_eq_0479.png\"\/>\n                        <jats:tex-math>{\\boldsymbol{u}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.<\/jats:p>","DOI":"10.1515\/cmam-2023-0015","type":"journal-article","created":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T17:56:39Z","timestamp":1713894999000},"page":"189-213","source":"Crossref","is-referenced-by-count":2,"title":["Quadratic Discontinuous Galerkin Finite Element Methods for the Unilateral Contact Problem"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5713-9498","authenticated-orcid":false,"given":"Kamana","family":"Porwal","sequence":"first","affiliation":[{"name":"Department of Mathematics , 28817 Indian Institute of Technology Delhi , Delhi 110016 , India"}]},{"given":"Tanvi","family":"Wadhawan","sequence":"additional","affiliation":[{"name":"Department of Mathematics , 28817 Indian Institute of Technology Delhi , Delhi 110016 , India"}]}],"member":"374","published-online":{"date-parts":[[2024,4,24]]},"reference":[{"key":"2025010318340349510_j_cmam-2023-0015_ref_001","doi-asserted-by":"crossref","unstructured":"M. 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Math. 54 (2005), no. 3\u20134, 555\u2013576.","DOI":"10.1016\/j.apnum.2004.09.019"},{"key":"2025010318340349510_j_cmam-2023-0015_ref_040","doi-asserted-by":"crossref","unstructured":"R.  Khandelwal, K.  Porwal and T.  Wadhawan,\nAdaptive quadratic finite element method for the unilateral contact problem,\nJ. Sci. Comput. 96 (2023), no. 1, Paper No. 20.","DOI":"10.1007\/s10915-023-02206-5"},{"key":"2025010318340349510_j_cmam-2023-0015_ref_041","doi-asserted-by":"crossref","unstructured":"N.  Kikuchi and J. T.  Oden,\nContact Problem in Elasticity,\nSociety for Industrial and Applied Mathematics, Philadelphia, 1988.","DOI":"10.1137\/1.9781611970845"},{"key":"2025010318340349510_j_cmam-2023-0015_ref_042","doi-asserted-by":"crossref","unstructured":"D.  Kinderlehrer and G.  Stampacchia,\nAn Introduction to Variational Inequalities and Their Applications,\nClass. Appl. 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