{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T21:41:46Z","timestamp":1757540506690,"version":"3.40.5"},"reference-count":24,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a <jats:italic>fading memory<\/jats:italic> kernel.\nThis integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation.\nWe consider such a dynamic linear viscoelastic model problem resulting from using a <jats:italic>Dirichlet\u2013Prony<\/jats:italic> series of decaying exponentials to provide the fading memory in the Volterra kernel.\nWe introduce two types of <jats:italic>internal variable<\/jats:italic> to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and \u2013 in time \u2013 a Crank\u2013Nicolson method to formulate the fully discrete problems: one for each type of internal variable.\nWe present a priori stability and error analyses without using Gr\u00f6nwall\u2019s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially.\nIn this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems.\nWe also carry out a number of numerical experiments using the FEniCS environment (<jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"https:\/\/fenicsproject.org\">https:\/\/fenicsproject.org<\/jats:ext-link>), describe a simulation using \u201creal\u201d material data, and explain how the codes can be obtained and all of the results reproduced.<\/jats:p>","DOI":"10.1515\/cmam-2022-0201","type":"journal-article","created":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T17:03:50Z","timestamp":1676999030000},"page":"647-669","source":"Crossref","is-referenced-by-count":4,"title":["A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2036-558X","authenticated-orcid":false,"given":"Yongseok","family":"Jang","sequence":"first","affiliation":[{"name":"Department of Aerodynamics, Aeroelasticity, Acoustics (DAAA) , ONERA Centre de Chatillon , 29 avenue de la Division Leclerc, 92322 , Chatillon , France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1406-7225","authenticated-orcid":false,"given":"Simon","family":"Shaw","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Brunel University London , Kingston Lane , Uxbridge , UB8 3PH , United Kingdom"}]}],"member":"374","published-online":{"date-parts":[[2023,2,22]]},"reference":[{"key":"2025041712114362641_j_cmam-2022-0201_ref_001","unstructured":"M. Aln\u00e6s, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells,\nThe FEniCS project version 1.5,\nArch. Numer. Softw. 3 (2015), no. 100, 9\u201323."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_002","unstructured":"Bang\u2019s Laboratories,\nMaterial properties of polystyrene and poly(methyl methacrylate) (PMMA) microspheres."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_003","doi-asserted-by":"crossref","unstructured":"S. C. Brenner,\nPoincar\u00e9\u2013Friedrichs inequalities for piecewise \n                  \n                     \n                        \n                           H\n                           1\n                        \n                     \n                     \n                     H^{1}\n                  \n                functions,\nSIAM J. Numer. Anal. 41 (2003), no. 1, 306\u2013324.","DOI":"10.1137\/S0036142902401311"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_004","doi-asserted-by":"crossref","unstructured":"S. C. Brenner,\nKorn\u2019s inequalities for piecewise \n                  \n                     \n                        \n                           H\n                           1\n                        \n                     \n                     \n                     H^{1}\n                  \n                vector fields,\nMath. Comp. 73 (2004), no. 247, 1067\u20131087.","DOI":"10.1090\/S0025-5718-03-01579-5"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_005","doi-asserted-by":"crossref","unstructured":"A. Buffa and C. Ortner,\nCompact embeddings of broken Sobolev spaces and applications,\nIMA J. Numer. Anal. 29 (2009), no. 4, 827\u2013855.","DOI":"10.1093\/imanum\/drn038"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_006","unstructured":"R. M. Christensen,\nTheory of Viscoelasticity: An Introduction,\nAcademic Press, London, 1971."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_007","doi-asserted-by":"crossref","unstructured":"J. Crank and P. Nicolson,\nA practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,\nAdv. Comput. Math. 6 (1996), 207\u2013226.","DOI":"10.1007\/BF02127704"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_008","unstructured":"A. D. Drozdov,\nViscoelastic Structures: Mechanics of Growth and Aging,\nAcademic Press, San Diego, 1998."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_009","unstructured":"J. D. Ferry,\nViscoelastic Properties of Polymers,\nJohn Wiley& Sons, New York, 1970."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_010","unstructured":"W. N. Findley, J. S. Lai and K. Onaran,\nCreep and Relaxation of nonlinear Viscoelastic Materials,\nDover, New York, 1989."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_011","unstructured":"J. M. Golden and G. A. C. Graham,\nBoundary Value Problems in Linear Viscoelasticity,\nSpringer, Berlin, 2013."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_012","doi-asserted-by":"crossref","unstructured":"P. Hansbo and M. G. Larson,\nDiscontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche\u2019s method,\nComput. Methods Appl. Mech. Engrg. 191 (2002), no. 17\u201318, 1895\u20131908.","DOI":"10.1016\/S0045-7825(01)00358-9"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_013","doi-asserted-by":"crossref","unstructured":"P. Houston, D. Sch\u00f6tzau and T. P. Wihler,\nAn \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n               -adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity,\nComput. Methods Appl. Mech. Engrg. 195 (2006), no. 25\u201328, 3224\u20133246.","DOI":"10.1016\/j.cma.2005.06.012"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_014","unstructured":"S. C. Hunter,\nMechanics of Continuous Media,\nMath. Appl.,\nJohn Wiley & Sons, New York, 1976."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_015","unstructured":"Y. Jang,\nSpatially continuous and discontinuous Galerkin finite element approximations for dynamic viscoelastic problems,\nPhD thesis, Brunel University London, 2020."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_016","unstructured":"F. J. Lockett,\nNonlinear Viscoelastic Solids,\nAcademic Press, New York, 1972."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_017","unstructured":"S. Ozisik, B. Riviere and T. Warburton,\nOn the constants in inverse inequalities in \n                  \n                     \n                        \n                           \n                              L\n                              2\n                           \n                        \n                        \n                        L_{2}\n                     \n                  \n               ,\nTechnical report, Rice University, 2010."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_018","doi-asserted-by":"crossref","unstructured":"S. W. Park and R. A. Schapery,\nMethods of interconversion between linear viscoelastic material functions. Part I \u2013 a numerical method based on Prony series,\nInt. J. Solids Struct. 36 (1999), 1653\u20131675.","DOI":"10.1016\/S0020-7683(98)00055-9"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_019","doi-asserted-by":"crossref","unstructured":"B. Rivi\u00e8re,\nDiscontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation,\nFront. Appl. Math. 35,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2008.","DOI":"10.1137\/1.9780898717440"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_020","doi-asserted-by":"crossref","unstructured":"B. Rivi\u00e8re, S. Shaw, M. F. Wheeler and J. R. Whiteman,\nDiscontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity,\nNumer. Math. 95 (2003), no. 2, 347\u2013376.","DOI":"10.1007\/s002110200394"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_021","doi-asserted-by":"crossref","unstructured":"B. Rivi\u00e8re, S. Shaw and J. R. Whiteman,\nDiscontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems,\nNumer. Methods Partial Differential Equations 23 (2007), no. 5, 1149\u20131166.","DOI":"10.1002\/num.20215"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_022","unstructured":"S. Shaw and J. R. Whiteman,\nSome partial differential Volterra equation problems arising in viscoelasticity,\nProceedings of Equadiff. Vol. 9,\nMasaryk University, Brno (1998), 183\u2013200."},{"key":"2025041712114362641_j_cmam-2022-0201_ref_023","doi-asserted-by":"crossref","unstructured":"T. Warburton and J. S. Hesthaven,\nOn the constants in \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n               -finite element trace inverse inequalities,\nComput. Methods Appl. Mech. Engrg. 192 (2003), no. 25, 2765\u20132773.","DOI":"10.1016\/S0045-7825(03)00294-9"},{"key":"2025041712114362641_j_cmam-2022-0201_ref_024","doi-asserted-by":"crossref","unstructured":"T. P. Wihler,\nLocking-free adaptive discontinuous Galerkin FEM for linear elasticity problems,\nMath. Comp. 75 (2006), no. 255, 1087\u20131102.","DOI":"10.1090\/S0025-5718-06-01815-1"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0201\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0201\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,17]],"date-time":"2025-04-17T12:12:04Z","timestamp":1744891924000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2022-0201\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,22]]},"references-count":24,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,2,22]]},"published-print":{"date-parts":[[2023,7,1]]}},"alternative-id":["10.1515\/cmam-2022-0201"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0201","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2023,2,22]]}}}