S. Larsson, V. Thomée, and L. B. Wahlbin,

Numerical solution of parabolic integro-differential
equations by the discontinuous Galerkin method,

Math. Comp. 67 (1998), 45-71. 

Subjclass: 65M60, 65R20, 45L10 

Keywords: Parabolic integro-differential equation, weakly singular
  kernel, discontinuous Galerkin, variable time step, finite element,
  error estimate, Gronwall lemma

Abstract:
  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.


Address: Stig Larsson,
  Department of Mathematics,
  Chalmers University of Technology and G\"oteborg University,
  S--412 96 G\"oteborg,
  Sweden

E-mail: stig@math.chalmers.se 

Address: Vidar Thom\'ee, 
  Department of Mathematics,
  Chalmers University of Technology and G\"oteborg University,
  S--412 96 G\"oteborg,
  Sweden 

E-mail: thomee@math.chalmers.se 

Address: Lars B. Wahlbin,
  Department of Mathematics,
  Cornell University,
  Ithaca, New York 14853 

E-mail: wahlbin@math.cornell.edu