numerical stability

Timoshenko's problem is not a recent problem and many articles exist concerning his study.  New physical problems appear and require a good mathematical understanding of the behavior of this phenomenon.  Our contribution will consist in studying the numerical stability of a Timoshenko system with second sound.  We introduce a finite element approximation and prove that the associated discrete energy decreases and we establish a priori error estimates.  Finally, some numerical simulations are obtained.

In this paper, we consider the Moore–Gibson–Thompson–Fourier system made by coupling the Moore–Gibson–Thompson (MGT) equation with the classical Fourier heat equation known as the MGT–Fourier model.  For $\sigma=\alpha\beta-\gamma>0$, the authors used the semi-group method to prove the existence and uniqueness of global solutions and the exponential stability of total energy.  Our contribution will consist in studying numerical method based on finite element discretization in the spacial variable $x$ and finite difference schema in time of the MGT–Fourier model.  A d