{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:36:57Z","timestamp":1776724617829,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretization in space is also studied.<\/p>","DOI":"10.1090\/s0025-5718-98-00883-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"45-71","source":"Crossref","is-referenced-by-count":105,"title":["Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method"],"prefix":"10.1090","volume":"67","author":[{"given":"Stig","family":"Larsson","sequence":"first","affiliation":[]},{"given":"Vidar","family":"Thom\u00e9e","sequence":"additional","affiliation":[]},{"given":"Lars","family":"Wahlbin","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/BF02575943","article-title":"Nonclassical \ud835\udc3b\u00b9 projection and Galerkin methods for nonlinear parabolic integro-differential equations","volume":"25","author":"Cannon, J. R.","year":"1988","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"issue":"198","key":"2","doi-asserted-by":"publisher","first-page":"587","DOI":"10.2307\/2153204","article-title":"Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel","volume":"58","author":"Chen, C.","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/BF01596912","article-title":"Equipresence and constitutive equations for rigid heat conductors","volume":"18","author":"Coleman, Bernard D.","year":"1967","journal-title":"Z. Angew. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-2275","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1137\/0728003","article-title":"Adaptive finite element methods for parabolic problems. I. A linear model problem","volume":"28","author":"Eriksson, Kenneth","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1051\/m2an\/1985190406111","article-title":"Time discretization of parabolic problems by the discontinuous Galerkin method","volume":"19","author":"Eriksson, Kenneth","year":"1985","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"6","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","volume-title":"Real analysis","author":"Folland, Gerald B.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0471809586"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1016\/0022-0396(83)90076-1","article-title":"Analytic resolvent operators for integral equations in Banach space","volume":"50","author":"Grimmer, R. C.","year":"1983","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"1047","DOI":"10.1137\/0728056","article-title":"Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations","volume":"28","author":"Lin, Yan Ping","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"12","key":"9","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1016\/0362-546X(88)90080-6","article-title":"An inverse problem in the theory of materials with memory","volume":"12","author":"Lorenzi, A.","year":"1988","journal-title":"Nonlinear Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-546X","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1016\/0022-0396(86)90056-2","article-title":"\ud835\udc36^{\ud835\udefc}-regularity for nonautonomous linear integro-differential equations of parabolic type","volume":"63","author":"Lunardi, Alessandra","year":"1986","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1216\/jiea\/1181075713","article-title":"Numerical methods for hyperbolic and parabolic integro-differential equations","volume":"4","author":"Pani, A. K.","year":"1992","journal-title":"J. Integral Equations Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0897-3962","issn-type":"print"},{"issue":"5","key":"12","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1137\/0723073","article-title":"Time discretization of an integro-differential equation of parabolic type","volume":"23","author":"Sloan, I. H.","year":"1986","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"206","key":"13","doi-asserted-by":"publisher","first-page":"477","DOI":"10.2307\/2153519","article-title":"Long-time numerical solution of a parabolic equation with memory","volume":"62","author":"Thom\u00e9e, Vidar","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"187","key":"14","doi-asserted-by":"publisher","first-page":"121","DOI":"10.2307\/2008352","article-title":"Error estimates for semidiscrete finite element methods for parabolic integro-differential equations","volume":"53","author":"Thom\u00e9e, Vidar","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1142\/9789812798886_0029","article-title":"Backward Euler type methods for parabolic integro-differential equations with nonsmooth data","author":"Thom\u00e9e, Vidar","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/9810214375"},{"issue":"8","key":"16","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1016\/0362-546X(88)90039-9","article-title":"Finite element methods for parabolic and hyperbolic partial integro-differential equations","volume":"12","author":"Yanik, Elizabeth G.","year":"1988","journal-title":"Nonlinear Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-546X","issn-type":"print"},{"issue":"201","key":"17","doi-asserted-by":"publisher","first-page":"133","DOI":"10.2307\/2153159","article-title":"On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type","volume":"60","author":"Zhang, Nai Ying","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00883-7\/S0025-5718-98-00883-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00883-7\/S0025-5718-98-00883-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:39:45Z","timestamp":1776721185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-221\/S0025-5718-98-00883-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":17,"journal-issue":{"issue":"221","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["S0025-5718-98-00883-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00883-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1998]]}}}