On the fully discrete approximations of the MGT two-temperatures thermoelastic problem

We consider a one-dimensional two-temperatures thermoelastic model. The corresponding variational problem leads to a coupled system which is written in terms of the mechanical velocity, the temperature speed and the inductive temperature. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A priori error estimates are proved and the linear convergence of the approximations is deduced under suitable additional regularity conditions. Finally, some numerical simulations are shown to demonstrate the accuracy of the proposed algorithm and the behavior of the discrete energy.