Space-time discretization of an integro-differential equation modeling
quasi-static fractional order viscoelasticity

K. Adolfsson, M. Enelund, and S. Larsson

preprint 2006, submitted to J. Vib. Control


Abstract.

We study a quasi-static model for viscoelastic materials
based on a constitutive equation of fractional order.  In the
quasi-static case this results in a Volterra integral equation of the
second kind with a weakly singular kernel in the time variable
involving also partial derivatives of second order in the spatial
variables.  We discretize by means of a discontinuous Galerkin finite
element method in time and a standard continuous Galerkin finite
element method in space.  To overcome the problem of the growing
amount of data that has to be stored and used in each time step, we
introduce sparse quadrature in the convolution integral.  We prove a
priori and a posteriori error estimates, and develop an adaptive
strategy based on the a posteriori error estimate.