{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:21Z","timestamp":1747198041267,"version":"3.40.5"},"reference-count":38,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11571237"],"award-info":[{"award-number":["11571237"]}],"id":[{"id":"10.13039\/501100001809","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>A robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the <jats:inline-formula id=\"j_cmam-2018-0197_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-2018-0197_eq_0248.png\"\/>\n                        <jats:tex-math>{P_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex\u2013Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. <jats:inline-formula id=\"j_cmam-2018-0197_ineq_9998\">\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-2018-0197_eq_0245.png\"\/>\n                        <jats:tex-math>{L^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> norm) is bounded by <jats:inline-formula id=\"j_cmam-2018-0197_ineq_9997\">\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:mrow>\n                                    <m:mi>h<\/m:mi>\n                                    <m:mo>+<\/m:mo>\n                                    <m:mi>k<\/m:mi>\n                                 <\/m:mrow>\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-2018-0197_eq_0246.png\"\/>\n                        <jats:tex-math>{O(h+k)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> (resp. <jats:inline-formula id=\"j_cmam-2018-0197_ineq_9996\">\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:mrow>\n                                    <m:msup>\n                                       <m:mi>h<\/m:mi>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:mi>k<\/m:mi>\n                                 <\/m:mrow>\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-2018-0197_eq_0247.png\"\/>\n                        <jats:tex-math>{O(h^{2}+k)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>), where <jats:italic>h<\/jats:italic> and <jats:italic>k<\/jats:italic> denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lam\u00e9 coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison.<\/jats:p>","DOI":"10.1515\/cmam-2018-0197","type":"journal-article","created":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T09:59:41Z","timestamp":1566986381000},"page":"481-500","source":"Crossref","is-referenced-by-count":0,"title":["A Robust Finite Element Method for Elastic Vibration Problems"],"prefix":"10.1515","volume":"20","author":[{"given":"Yuling","family":"Guo","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences , and MOE-LSC , Shanghai Jiao Tong University , Shanghai 200240 , P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0825-3056","authenticated-orcid":false,"given":"Jianguo","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , and MOE-LSC , Shanghai Jiao Tong University , Shanghai 200240 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2019,8,28]]},"reference":[{"key":"2023033110434174387_j_cmam-2018-0197_ref_001","unstructured":"R. A.  Adams,\nSobolev Spaces,\nPure Appl. Math. 65,\nAcademic Press, New York, 1975."},{"key":"2023033110434174387_j_cmam-2018-0197_ref_002","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold, F.  Brezzi and J.  Douglas, Jr.,\nPEERS: A new mixed finite element for plane elasticity,\nJapan J. Appl. Math. 1 (1984), no. 2, 347\u2013367.","DOI":"10.1007\/BF03167064"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_003","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka and M.  Suri,\nLocking effects in the finite element approximation of elasticity problems,\nNumer. Math. 62 (1992), no. 4, 439\u2013463.","DOI":"10.1007\/BF01396238"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_004","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka and M.  Suri,\nOn locking and robustness in the finite element method,\nSIAM J. Numer. Anal. 29 (1992), no. 5, 1261\u20131293.","DOI":"10.1137\/0729075"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_005","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka and B.  Szabo,\nOn the rates of convergence of the finite element method,\nInternat. J. Numer. Methods Engrg. 18 (1982), no. 3, 323\u2013341.","DOI":"10.1002\/nme.1620180302"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_006","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner,\nForty years of the Crouzeix\u2013Raviart element,\nNumer. Methods Partial Differential Equations 31 (2015), no. 2, 367\u2013396.","DOI":"10.1002\/num.21892"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_007","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_008","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L.-Y.  Sung,\nLinear finite element methods for planar linear elasticity,\nMath. Comp. 59 (1992), no. 200, 321\u2013338.","DOI":"10.1090\/S0025-5718-1992-1140646-2"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_009","doi-asserted-by":"crossref","unstructured":"G.  Chen and X.  Xie,\nA robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses,\nComput. Methods Appl. 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R-3, 33\u201375.","DOI":"10.1051\/m2an\/197307R300331"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_013","doi-asserted-by":"crossref","unstructured":"T.  Dupont,\n\n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     L^{2}\n                  \n               -estimates for Galerkin methods for second order hyperbolic equations,\nSIAM J. Numer. Anal. 10 (1973), 880\u2013889.","DOI":"10.1137\/0710073"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_014","doi-asserted-by":"crossref","unstructured":"D.  Estep and D.  French,\nGlobal error control for the continuous Galerkin finite element method for ordinary differential equations,\nRAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 28 (1994), no. 7, 815\u2013852.","DOI":"10.1051\/m2an\/1994280708151"},{"key":"2023033110434174387_j_cmam-2018-0197_ref_015","unstructured":"L. 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