Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel Klas Adolfsson a, * , Mikael Enelund a , Stig Larsson b a Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 G€ oteborg, Sweden b Department of Computational Mathematics, Chalmers University of Technology, SE-412 96 G€ oteborg, Sweden Received 18 April 2003; received in revised form 26 August 2003; accepted 5 September 2003 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 frac- tional integral. This is done by comparing the numerical solutions with analytical solutions. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Integro-differential equation; Weakly singular kernel; Sparse quadrature; Error estimate; Adaptivity 1. Introduction Fractional order operators (integrals and derivatives) have proved to be very suitable for modeling memory effects of various materials and systems of technical interest. In particular, they are very useful when modeling viscoelastic materials, see, e.g., [3,4,10]. The drawback of these models is that, when the response is integrated numerically, the whole previous stress or strain history must be included in each time- step. Rather few algorithms for integrating viscoelastic responses (integral equations with singular kernels) are available. Most of them are based on the Lubich convolution quadrature for fractional order operators, see [11] and, e.g., [8]. The Lubich convolution quadrature requires uniformly distributed time-steps. This is a cumbersome restriction, in particular, when analyzing non-linear viscoelastic responses. Furthermore, it is not possible to use adaptivity and goal oriented error estimates. It also restricts the possibility to use sparse * Corresponding author. E-mail addresses: klas.adolfsson@me.chalmers.se (K. Adolfsson), mikael.enelund@me.chalmers.se (M. Enelund), stig@math.chal- mers.se (S. Larsson). 0045-7825/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2003.09.001 Comput. Methods Appl. Mech. Engrg. 192 (2003) 5285–5304 www.elsevier.com/locate/cma