Discrete energy behavior of Timoshenko system with Cattaneo’s law

In this paper, we consider a one-dimensional thermoelastic Timoshenko system where the thermal coupling is acting on both the shear force and the bending moment, and the heat flux is given by Cattaneo’s law. Our contribution will consist in studying the numerical stability of a Timoshenko system with Cattaneo’s law. We introduce a \(P_{1}\) $P_1$ finite element method for space discretization and implicit Euler scheme for time discretization. Then we prove that the associated discrete energy decreases and we establish a priori error estimates. Finally, we obtain some numerical simulations.