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.
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