  In this paper, we consider a nonlinear PDE system governed by a parabolic
heat equation coupled in a nonlinear way with a hyperbolic momentum equation
describing the behavior of a displacement field coupled with a nonlinear
elliptic equation based on an internal damage variable. We present a numerical
scheme based on a low-order Galerkin finite element method (FEM) for the space
discretization of the time-dependent nonlinear PDE system and an implicit
finite difference method (FDM) to discretize in the direction of the time
variable. Moreover, we present a priori estimates for the exact and discrete
solutions for the pointwise-in-time $L^2$-norm. Based on the a priori
estimates, we rigorously prove the convergence of the solutions of the fully
discretized system to the exact solutions. Denoting the properties of the
internal parameters, we find the order of convergence concerning the
discretization parameters.
