Abstract In this work, we study from the mathematical and numerical points of view a poro-thermoelastic problem. A long-term memory is assumed on the heat equation. Under some assumptions on the constitutive tensors, the resulting linear system is composed of hyperbolic partial differential equations with a dissipative mechanism in the temperature equation. An existence and uniqueness result is proved using the theory of contractive semigroups. Then, a fully discrete approximation is introduced applying the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A discrete stability property is obtained. A priori error estimates are also shown, from which the linear convergence of the approximation is derived under suitable additional regularity conditions. Finally, one- and two-numerical simulations are presented to demonstrate the accuracy of the algorithm and the behavior of the solution.

中文翻译：

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具有耗散热传导的多孔热弹性问题

摘要 在这项工作中，我们从数学和数值的角度研究了多孔热弹性问题。假定热方程具有长期记忆。在对本构张量的一些假设下，所得线性系统由双曲偏微分方程组成，在温度方程中具有耗散机制。使用收缩半群理论证明了一个存在唯一性结果。然后，引入完全离散近似，应用有限元方法近似空间变量和隐式欧拉方案来离散时间导数。获得了离散的稳定性特性。还显示了先验误差估计，从中可以在合适的附加规则条件下导出近似的线性收敛。最后，