{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T08:01:20Z","timestamp":1777622480226,"version":"3.51.4"},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["No.2017R1D1A1B03032765"],"award-info":[{"award-number":["No.2017R1D1A1B03032765"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The purpose of this paper is to develop a reduced Crouzeix\u2013Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia\u2013Stenberg finite element method (Kouhia et al. 1995) and Crouzeix\u2013Raviart IFEM (CRIFEM) (Kwak et al. 2017).\nWe use a <jats:inline-formula id=\"j_cmam-2019-0046_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>P<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0046_eq_0313.png\"\/>\n                        <jats:tex-math>{P_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-conforming like element for one of the components of the displacement vector, and a <jats:inline-formula id=\"j_cmam-2019-0046_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>P<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0046_eq_0313.png\"\/>\n                        <jats:tex-math>{P_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-nonconforming like element for the other component.\nThe number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM.\nFurthermore, we can choose penalty parameters independent of the Poisson ratio.\nOne of the penalty parameters depends on Lam\u00e9\u2019s second constant \u03bc, and the other penalty parameter is independent of both \u03bc and \u03bb.\nWe prove the optimal order error estimates in piecewise <jats:inline-formula id=\"j_cmam-2019-0046_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>H<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0046_eq_0294.png\"\/>\n                        <jats:tex-math>{H^{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norm, which is independent of the Poisson ratio.\nNumerical experiments show optimal order of convergence both in <jats:inline-formula id=\"j_cmam-2019-0046_ineq_9996\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0046_eq_0306.png\"\/>\n                        <jats:tex-math>{L^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and piecewise <jats:inline-formula id=\"j_cmam-2019-0046_ineq_9995\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>H<\/m:mi>\n                              <m:mn>1<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0046_eq_0294.png\"\/>\n                        <jats:tex-math>{H^{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norms for all problems including nearly incompressible cases.<\/jats:p>","DOI":"10.1515\/cmam-2019-0046","type":"journal-article","created":{"date-parts":[[2019,9,18]],"date-time":"2019-09-18T09:02:34Z","timestamp":1568797354000},"page":"501-516","source":"Crossref","is-referenced-by-count":5,"title":["A Reduced Crouzeix\u2013Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0635-2897","authenticated-orcid":false,"given":"Gwanghyun","family":"Jo","sequence":"first","affiliation":[{"name":"Department of Mathematics , Kunsan National University , Gunsan-si, Jeollabuk-do , Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5743-1501","authenticated-orcid":false,"given":"Do Young","family":"Kwak","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences , Korea Advanced Institute of Science and Technology , Daejeon , Republic of Korea"}]}],"member":"374","published-online":{"date-parts":[[2019,9,18]]},"reference":[{"key":"2023033110434206186_j_cmam-2019-0046_ref_001","doi-asserted-by":"crossref","unstructured":"C.  Bacuta and J. H.  Bramble,\nRegularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains,\nZ. Angew. Math. Phys. 54 (2003), no. 5, 874\u2013878.","DOI":"10.1007\/s00033-003-3211-4"},{"key":"2023033110434206186_j_cmam-2019-0046_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Becker, E.  Burman and P.  Hansbo,\nA Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity,\nComput. Methods Appl. Mech. Engrg. 198 (2009), no. 41\u201344, 3352\u20133360.","DOI":"10.1016\/j.cma.2009.06.017"},{"key":"2023033110434206186_j_cmam-2019-0046_ref_003","doi-asserted-by":"crossref","unstructured":"T.  Belytschko and T.  Black,\nElastic crack growth in finite elements with minimal remeshing,\nInt. J. Numer. 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