{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T08:44:04Z","timestamp":1758703444085,"version":"3.40.5"},"reference-count":30,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-10-16332","DMS-16-20273"],"award-info":[{"award-number":["DMS-10-16332","DMS-16-20273"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We study a nonconforming finite element approximation of the vibration modes\nof an acoustic fluid-structure interaction. Displacement variables are used\nfor both the fluid and the solid. The numerical scheme is based on an irrotational\nfluid displacement formulation and hence it is free of spurious eigenmodes.\nThe method uses weakly continuous <jats:inline-formula id=\"j_cmam-2017-0050_ineq_9999_w2aab3b7e1642b1b6b1aab1c14b1b1Aa\">\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-2017-0050_eq_1078.png\"\/>\n                        <jats:tex-math>{P_{1}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> vector fields for the fluid and\nclassical piecewise linear elements for the solid, and it has <jats:inline-formula id=\"j_cmam-2017-0050_ineq_9998_w2aab3b7e1642b1b6b1aab1c14b1b3Aa\">\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>h<\/m:mi>\n                                    <m:mn>2<\/m:mn>\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-2017-0050_eq_1077.png\"\/>\n                        <jats:tex-math>{O(h^{2})}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\nconvergence for the eigenvalues on properly graded meshes. The theoretical results\nare confirmed by numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2017-0050","type":"journal-article","created":{"date-parts":[[2017,12,8]],"date-time":"2017-12-08T22:15:54Z","timestamp":1512771354000},"page":"383-406","source":"Crossref","is-referenced-by-count":7,"title":["A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem"],"prefix":"10.1515","volume":"18","author":[{"given":"Susanne C.","family":"Brenner","sequence":"first","affiliation":[{"name":"Department of Mathematics & Center for Computation and Technology , Louisiana State University , Baton Rouge , LA 70803 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8057-6349","authenticated-orcid":false,"given":"Ay\u00e7\u0131l","family":"\u00c7e\u015fmelio\u011flu","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics , Oakland University , Rochester , MI 48309 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jintao","family":"Cui","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics , The Hong Kong Polytechnic University , Hung Hom , Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li-Yeng","family":"Sung","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Center for Computation and Technology , Louisiana State University , Baton Rouge , LA 70803 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,8]]},"reference":[{"key":"2023033110284051333_j_cmam-2017-0050_ref_001_w2aab3b7e1642b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka and J.  Osborn,\nEigenvalue problems,\nHandbook of Numerical Analysis. Vol. II,\nNorth-Holland, Amsterdam (1991), 641\u2013787.","DOI":"10.1016\/S1570-8659(05)80042-0"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_002_w2aab3b7e1642b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"A.  Berm\u00fadez, R.  Dur\u00e1n, M. A.  Muschietti, R.  Rodr\u00edguez and J.  Solomin,\nFinite element vibration analysis of fluid-solid systems without spurious modes,\nSIAM J. Numer. Anal. 32 (1995), no. 4, 1280\u20131295.","DOI":"10.1137\/0732059"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_003_w2aab3b7e1642b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"A.  Berm\u00fadez, R.  Dur\u00e1n and R.  Rodr\u00edguez,\nFinite element analysis of compressible and incompressible fluid-solid systems,\nMath. Comp. 67 (1998), no. 221, 111\u2013136.","DOI":"10.1090\/S0025-5718-98-00901-6"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_004_w2aab3b7e1642b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"A.  Berm\u00fadez, P.  Gamallo, M. R.  Nogueiras and R.  Rodr\u00edguez,\nApproximation of a structural acoustic vibration problem by hexahedral finite elements,\nIMA J. Numer. Anal. 26 (2006), no. 2, 391\u2013421.","DOI":"10.1093\/imanum\/dri032"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_005_w2aab3b7e1642b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"A.  Berm\u00fadez, L.  Hervella-Nieto and R.  Rodr\u00edguez,\nFinite element computation of three-dimensional elastoacoustic vibrations,\nJ. Sound Vibration 219 (1999), 279\u2013306.","DOI":"10.1006\/jsvi.1998.1873"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_006_w2aab3b7e1642b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"A.  Berm\u00fadez and R.  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Math.,\nAcademic Press, New York, 1983."},{"key":"2023033110284051333_j_cmam-2017-0050_ref_015_w2aab3b7e1642b1b6b1ab2b1c15Aa","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":"2023033110284051333_j_cmam-2017-0050_ref_016_w2aab3b7e1642b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"M.  Crouzeix and P.-A.  Raviart,\nConforming and nonconforming finite element methods for solving the stationary Stokes equations. I,\nRev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. Rouge 7 (1973), no. R-3, 33\u201375.","DOI":"10.1051\/m2an\/197307R300331"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_017_w2aab3b7e1642b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"T.  Dupont and R.  Scott,\nPolynomial approximation of functions in Sobolev spaces,\nMath. Comp. 34 (1980), no. 150, 441\u2013463.","DOI":"10.1090\/S0025-5718-1980-0559195-7"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_018_w2aab3b7e1642b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"R. G.  Dur\u00e1n, L.  Hervella-Nieto, E.  Liberman, R.  Rodr\u00edguez and J.  Solomin,\nFinite element analysis of the vibration problem of a plate coupled with a fluid,\nNumer. Math. 86 (2000), no. 4, 591\u2013616.","DOI":"10.1007\/PL00005411"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_019_w2aab3b7e1642b1b6b1ab2b1c19Aa","unstructured":"P.  Grisvard,\nElliptic Problems in Nonsmooth Domains,\nMonogr. Stud. Math. 24,\nPitman, Boston, 1985."},{"key":"2023033110284051333_j_cmam-2017-0050_ref_020_w2aab3b7e1642b1b6b1ab2b1c20Aa","unstructured":"P.  Grisvard,\nSingularities in Boundary Value Problems,\nRech. Math. 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Feng,\nFluid-structure finite element vibrational analysis,\nAIAA J. 14 (1976), no. 2, 199\u2013203.","DOI":"10.2514\/3.61357"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_024_w2aab3b7e1642b1b6b1ab2b1c24Aa","doi-asserted-by":"crossref","unstructured":"V. A.  Kozlov, V. G.  Maz\u2019ya and J.  Rossmann,\nSpectral Problems Associated with Corner Singularities of Solutions to Elliptic Problems,\nAmerican Mathematical Society, Providence, 2001.","DOI":"10.1090\/surv\/085"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_025_w2aab3b7e1642b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"S.  Meddahi and D.  Mora,\nNonconforming mixed finite element approximation of a fluid-structure interaction spectral problem,\nDiscrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 1, 269\u2013287.","DOI":"10.3934\/dcdss.2016.9.269"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_026_w2aab3b7e1642b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"S.  Meddahi, D.  Mora and R.  Rodr\u00edguez,\nFinite element analysis for a pressure-stress formulation of a fluid-structure interaction spectral problem,\nComput. Math. Appl. 68 (2014), no. 12, 1733\u20131750.","DOI":"10.1016\/j.camwa.2014.10.016"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_027_w2aab3b7e1642b1b6b1ab2b1c27Aa","doi-asserted-by":"crossref","unstructured":"P.  Monk,\nFinite Element Methods for Maxwell\u2019s Equations,\nNumer. Math. Sci. Comput.,\nOxford University Press, New York, 2003.","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_028_w2aab3b7e1642b1b6b1ab2b1c28Aa","unstructured":"H. J.-P.  Morand and R.  Ohayon,\nInteractions Fluides-Structures,\nRech. Math. Appl. 23,\nMasson, Paris, 1992."},{"key":"2023033110284051333_j_cmam-2017-0050_ref_029_w2aab3b7e1642b1b6b1ab2b1c29Aa","doi-asserted-by":"crossref","unstructured":"R.  Rodr\u00edguez and J. E.  Solomin,\nThe order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems,\nMath. Comp. 65 (1996), no. 216, 1463\u20131475.","DOI":"10.1090\/S0025-5718-96-00739-9"},{"key":"2023033110284051333_j_cmam-2017-0050_ref_030_w2aab3b7e1642b1b6b1ab2b1c30Aa","doi-asserted-by":"crossref","unstructured":"A. D.  Russo and A.  Alonso,\nHybrid finite element analysis of fluid-structure systems with coupling on curved interfaces,\nIMA J. Numer. Anal. 31 (2011), no. 4, 1636\u20131682.","DOI":"10.1093\/imanum\/drq051"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/3\/article-p383.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0050\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0050\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:56:43Z","timestamp":1680263803000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0050\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,8]]},"references-count":30,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,5,16]]},"published-print":{"date-parts":[[2018,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0050"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0050","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2017,12,8]]}}}