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 discrete stability property and a priori error estimates are proved. Finally, the numerical simulation agrees well with theoretical results.
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