A thermoviscoelastic problem with damage

This work studies, from the numerical point of view, a problem which models the dynamic evolution of damage in a thermoviscoelastic body. Material damage which results from tension or compression is taken into account in the constitutive law, and its evolution is described by a hyperbolic partial differential inclusion. The variational problem is formulated as a coupled system of nonlinear evolutionary equations, for which the existence of a unique weak solution is recalled. A fully discrete numerical scheme is introduced, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the numerical scheme is obtained. Finally, the results of simulations of three two-dimensional examples are presented.