Stability to double Timoshenko thermoelastic beam

In this work, we study, from both analytical and numerical points of view, a thermoelastic problem involving two elastic beams, which are modeled by using the classical Timoshenko model. The resulting problem is written in terms of the transverse displacements, the angles of rotation, and the temperatures of the beams. The existence of a unique solution, as well as the exponential stability of the problem, is proved by using the theory of linear semigroups and energetic arguments. Then, we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. The discrete stability of its solutions and a priori error estimates are shown, from which we can conclude the linear convergence of the approximations, assuming an additional regularity on the continuous solution. Finally, numerical simulations are performed to demonstrate the numerical convergence and the behavior of the discrete energy.