Adaptive discretization of an integro-differential equation with a
weakly singular convolution kernel

Comput. Methods Appl. Mech. Engrg. 192 (2003), 5285-5304. 

Klas Adolfsson

  Department of Applied Mechanics,
  Chalmers University of Technology,
  SE--412 96 G\"oteborg,
  Sweden
  klas.adolfsson@me.chalmers.se

Mikael Enelund

  Department of Applied Mechanics,
  Chalmers University of Technology,
  SE--412 96 G\"oteborg,
  Sweden
  mikael.enelund@me.chalmers.se

Stig Larsson

  Department of Computational Mathematics,
  Chalmers University of Technology and G\"oteborg University,
  SE--412 96 G\"oteborg,
  Sweden
  stig@math.chalmers.se

Abstract.  
  An integro-differential equation involving a convolution integral
  with a weakly singular kernel is considered. The kernel can be that
  of a fractional integral. The integro-differential equation is
  discretized using the discontinuous Galerkin method with piecewise
  constant basis functions.  Sparse quadrature is introduced for the
  convolution term to overcome the problem with the growing amount of
  data that has to be stored and used in each time-step.  A priori
  and a posteriori error estimates are proved. An adaptive strategy
  based on the a posteriori error estimate is developed. Finally,
  the precision and effectiveness of the algorithm are demonstrated in
  the case that the convolution is a fractional integral. This is done
  by comparing the numerical solutions with analytical solutions.