Analysis of two thermoelastic problems within the second gradient theory

In this article, we analyze two thermoelastic problems arising in the second gradient theory. The existence and uniqueness of solutions are proved for each of them by using the theory of linear semigroups. The energy decay is also shown. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. The discrete stability and a priori error estimates are obtained. The linear convergence of both approximations is derived under suitable regularity on the continuous solutions. Finally, some simulations are presented to demonstrate the numerical convergence and the exponential decay of the discrete energy.