{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T09:12:27Z","timestamp":1721293947159},"reference-count":44,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1173"],"award-info":[{"award-number":["SFB 1173"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["891734"],"award-info":[{"award-number":["891734"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The simulation of the elastodynamics equations at high frequency suffers from the well-known pollution effect.\nWe present a Petrov\u2013Galerkin multiscale sub-grid correction method that remains pollution-free in natural resolution and oversampling regimes.\nThis is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>log<\/m:mi>\n                              <m:mo>\u2061<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>k<\/m:mi>\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-2022-0041_ineq_0001.png\" \/>\n                        <jats:tex-math>\\log(k)<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, \ud835\udc58 being the wave number.\nKey to this method are polynomial-in-\ud835\udc58 bounds for stability constants and related inf-sup constants.\nTo this end, we establish polynomial-in-\ud835\udc58 bounds for the elastodynamics stability constants in general Lipschitz domains with radiation boundary conditions in <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi mathvariant=\"double-struck\">R<\/m:mi>\n                              <m:mn>3<\/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-2022-0041_ineq_0002.png\" \/>\n                        <jats:tex-math>\\mathbb{R}^{3}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nPrevious methods relied on variational techniques, Rellich identities, and geometric constraints.\nIn the context of elastodynamics, these suffer from the need to hypothesize a Korn\u2019s inequality on the boundary.\nThe methods in this work are based on boundary integral operators and estimation of Green\u2019s function\u2019s derivatives dependence on \ud835\udc58 and do not require this extra hypothesis.\nWe also implemented numerical examples in two and three dimensions to show the method eliminates pollution in the natural resolution and oversampling regimes, as well as performs well when compared to standard Lagrange finite elements.<\/jats:p>","DOI":"10.1515\/cmam-2022-0041","type":"journal-article","created":{"date-parts":[[2022,7,13]],"date-time":"2022-07-13T08:53:49Z","timestamp":1657702429000},"page":"65-82","source":"Crossref","is-referenced-by-count":2,"title":["Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wave Number Explicit Bounds"],"prefix":"10.1515","volume":"23","author":[{"given":"Donald L.","family":"Brown","sequence":"first","affiliation":[{"name":"Applied Research and Development Division , The Equity Engineering Group , 20600 Chagrin Blvd #1200 , Shaker Heights , OH 44122 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dietmar","family":"Gallistl","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mathematik , Friedrich-Schiller-Universit\u00e4t Jena , Ernst-Abbe-Platz 2, 07743 Jena , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,3,12]]},"reference":[{"key":"2023033112515663444_j_cmam-2022-0041_ref_001","doi-asserted-by":"crossref","unstructured":"I. 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Methods Partial Differential Equations 27 (2011), no. 1, 31\u201369.","DOI":"10.1002\/num.20643"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_004","doi-asserted-by":"crossref","unstructured":"D. L. Brown, D. Gallistl and D. Peterseim,\nMultiscale Petrov\u2013Galerkin method for high-frequency heterogeneous Helmholtz equations,\nMeshfree Methods for Partial Differential Equations VIII,\nLect. Notes Comput. Sci. Eng. 115,\nSpringer, Cham (2017), 85\u2013115.","DOI":"10.1007\/978-3-319-51954-8_6"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_005","doi-asserted-by":"crossref","unstructured":"D. L. Brown and D. Peterseim,\nA multiscale method for porous microstructures,\nMultiscale Model. Simul. 14 (2016), no. 3, 1123\u20131152.","DOI":"10.1137\/140995210"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_006","doi-asserted-by":"crossref","unstructured":"S. N. Chandler-Wilde, I. G. Graham, S. Langdon and M. Lindner,\nCondition number estimates for combined potential boundary integral operators in acoustic scattering,\nJ. Integral Equations Appl. 21 (2009), no. 2, 229\u2013279.","DOI":"10.1216\/JIE-2009-21-2-229"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_007","doi-asserted-by":"crossref","unstructured":"S. N. Chandler-Wilde, I. G. Graham, S. Langdon and E. A. Spence,\nNumerical-asymptotic boundary integral methods in high-frequency acoustic scattering,\nActa Numer. 21 (2012), 89\u2013305.","DOI":"10.1017\/S0962492912000037"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_008","doi-asserted-by":"crossref","unstructured":"S. N. Chandler-Wilde and P. Monk,\nWave-number-explicit bounds in time-harmonic scattering,\nSIAM J. Math. Anal. 39 (2008), no. 5, 1428\u20131455.","DOI":"10.1137\/060662575"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_009","doi-asserted-by":"crossref","unstructured":"T. Chaumont-Frelet and S. 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Melenk,\nOn generalized finite-element methods,\nPhD thesis, University of Maryland, 1995."},{"key":"2023033112515663444_j_cmam-2022-0041_ref_032","doi-asserted-by":"crossref","unstructured":"J. M. Melenk,\nMapping properties of combined field Helmholtz boundary integral operators,\nSIAM J. Math. Anal. 44 (2012), no. 4, 2599\u20132636.","DOI":"10.1137\/100784072"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_033","doi-asserted-by":"crossref","unstructured":"J. M. Melenk and S. Sauter,\nConvergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,\nMath. Comp. 79 (2010), no. 272, 1871\u20131914.","DOI":"10.1090\/S0025-5718-10-02362-8"},{"key":"2023033112515663444_j_cmam-2022-0041_ref_034","doi-asserted-by":"crossref","unstructured":"J. M. Melenk and S. Sauter,\nWavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation,\nSIAM J. Numer. 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Phys. 230 (2011), no. 7, 2406\u20132432.","DOI":"10.1016\/j.jcp.2010.12.001"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0041\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0041\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:20:38Z","timestamp":1680283238000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0041\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,12]]},"references-count":44,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,10,6]]},"published-print":{"date-parts":[[2023,1,1]]}},"alternative-id":["10.1515\/cmam-2022-0041"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0041","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,12]]}}}