A poro-thermoelastic problem with dissipative heat conduction

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.

Keywords:

• A priori error estimates
• discrete stability
• dissipative mechanism
• finite elements
• porosity
• thermoelasticity

Acknowledgments

The work of J.R. Fernández has been partially supported by Ministerio de Ciencia, Innovación y Universidades under the research project PGC2018-096696-B-I00 (FEDER, UE). The work of A. Magaña and R. Quintanilla is supported by Ministerio de Economía y Competitividad under the research project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE) and Ministerio de Ciencia, Innovación y Universidades under the research project “Análisis matemático aplicado a la termomecánica” (PID2019-105118GB-I00), (AEI/FEDER, UE).