{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:03:09Z","timestamp":1776790989646,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"268","license":[{"start":{"date-parts":[[2010,2,23]],"date-time":"2010-02-23T00:00:00Z","timestamp":1266883200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n s<\/mml:mi>\n \n )<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n C(t-s)^{\\alpha -1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n 0<\/mml:mn>\n ><\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 0>\\alpha >1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most\u00a0\n \n \n \n \n q<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n q-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , for\n \n \n \n \n q<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n q=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\u00a0\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order\n \n \n \n \n \n k<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msup>\n +<\/mml:mo>\n \n h<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2113\n \n <\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n k^q+h^2\\ell (k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are the mesh sizes in time and space, respectively, and\n \n \n \n \n \n \u2113\n \n <\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n max<\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n k<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\ell (k)=\\max (1,\\log k^{-1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In the case\u00a0\n \n \n \n \n q<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n q=2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we prove a higher convergence rate of order\n \n \n \n \n \n k<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \n h<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2113\n \n <\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n k^3+h^2\\ell (k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at\u00a0\n \n \n \n \n t<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n t=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , necessitating the use of non-uniform time steps. We compare our theoretical error bounds with the results of numerical computations.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02234-0","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T10:39:02Z","timestamp":1246358342000},"page":"1975-1995","source":"Crossref","is-referenced-by-count":56,"title":["Discontinuous Galerkin method for an evolution equation with a memory term of positive type"],"prefix":"10.1090","volume":"78","author":[{"given":"Kassem","family":"Mustapha","sequence":"first","affiliation":[]},{"given":"William","family":"McLean","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,2,23]]},"reference":[{"issue":"51-52","key":"1","doi-asserted-by":"publisher","first-page":"5285","DOI":"10.1016\/j.cma.2003.09.001","article-title":"Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel","volume":"192","author":"Adolfsson, Klas","year":"2003","journal-title":"Comput. 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