Discretization of Integro-Differential Equations Modeling Dynamic Fractional
Order Viscoelasticity

K. Adolfsson(1), M. Enelund(1), S. Larsson(2), and M. Racheva(3)

I. Lirkov, S. Margenov, and J. Wasniewski (Eds.) ''Proceedings of
Large-Scale Scientific Computations, 2005, Sozopol,
Bulgaria'', LNCS vol. 3743, Springer 2006, pp. 76-83.


1. Department of Applied Mechanics,
Chalmers University of Technology,
SE-412 96 Göteborg, Sweden 

2. Department of Mathematics,
Chalmers University of Technology,
SE-412 96 Göteborg, Sweden

3. Department of Mathematics,
Technical University of Gabrovo,
5300 Gabrovo, Bulgaria

Abstract

We study a dynamic model for viscoelastic materials based on a constitutive
equation of fractional order.  This results in an integro-differential
equation with a weakly singular convolution kernel.  We discretize in the
spatial variable by a standard Galerkin finite element method.  We prove
stability and regularity estimates which show how the convolution term
introduces dissipation into the equation of motion.  These are then used to
prove a priori error estimates. A numerical experiment is included.