In this work, we numerically consider a thermoelastic problem where the thermal law is modeled using the so-called Moore–Gibson–Thompson equation. This thermomechanical problem is written as a coupled system composed of a hyperbolic partial differential equation for a transformation of the displacement field and a parabolic partial differential equation for a transformation of the temperature. Its variational formulation is written in terms of the derivatives of the above transformed functions, leading to a coupled linear system made of two first-order variational equations. Then, a fully discrete algorithm is introduced and a discrete stability property is proved. A priori error estimates are also provided, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical results are shown, including the numerical convergence of the approximations, comparisons with the Lord–Shulman and type III Green–Naghdi theories, and two-dimensional examples which demonstrate the behavior of the solution.

在这项工作中，我们在数值上考虑了热弹性问题，其中使用所谓的Moore-Gibson-Thompson方程对热定律进行建模。这个热力学问题被写成一个耦合系统，该系统由用于位移场转换的双曲型偏微分方程和用于温度转换的抛物型偏微分方程组成。它的变分公式是根据上述变换函数的导数编写的，从而导致由两个一阶变分方程组成的耦合线性系统。然后，介绍了一种完全离散算法，并证明了离散稳定性。还提供了先验误差估计，可以在适当的规则性条件下从中得出线性收敛。最后，显示了一些数值结果，