An a priori error analysis of the porous thermoelastic Coleman–Gurtin model

Noelia Bazarra; José R. Fernández; Víctor Prego; Ramón Quintanilla; Noelia Bazarra; José R. Fernández; Víctor Prego; Ramón Quintanilla

doi:10.3934/math.20251336

AIMS Mathematics

2025, Volume 10, Issue 12: 30460-30477. doi: 10.3934/math.20251336

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Research article

An a priori error analysis of the porous thermoelastic Coleman–Gurtin model

1.
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
2.
Departament de Matemàtiques, Escuela Superior de Ingenierías Industrial, Aeroespacial y Audiovisual de Terrassa, Universitat Politécnica de Catalunya, Colom 11, 08222 Terrassa, Barcelona, Spain

Received: 31 October 2025 Revised: 13 December 2025 Accepted: 17 December 2025 Published: 25 December 2025
MSC : 65M60, 74F05, 74K10, 74F10, 65M15, 65M12

In this work, we study, from the numerical point of view, a poro-thermoelastic problem where the heat conduction is modeled by using the Coleman–Gurtin law. This is written as a linear system of partial differential equations written in terms of the displacements, the porosity (or volume fraction), and the temperature. Then, we introduce a fully discrete approximation of a weak form of the thermomechanical problem, based on the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. We prove a discrete stability property and a main a priori error estimates result, which allows us to conclude the linear convergence of the approximations under suitable additional regularity. Finally, we present some numerical simulations to demonstrate the convergence and the decay of the discrete energy.
- Coleman–Gurtin model,
- porosity,
- finite history,
- finite elements,
- stability,
- error estimates,
- numerical simulations
Citation: Noelia Bazarra, José R. Fernández, Víctor Prego, Ramón Quintanilla. An a priori error analysis of the porous thermoelastic Coleman–Gurtin model[J]. AIMS Mathematics, 2025, 10(12): 30460-30477. doi: 10.3934/math.20251336

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Abstract

In this work, we study, from the numerical point of view, a poro-thermoelastic problem where the heat conduction is modeled by using the Coleman–Gurtin law. This is written as a linear system of partial differential equations written in terms of the displacements, the porosity (or volume fraction), and the temperature. Then, we introduce a fully discrete approximation of a weak form of the thermomechanical problem, based on the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. We prove a discrete stability property and a main a priori error estimates result, which allows us to conclude the linear convergence of the approximations under suitable additional regularity. Finally, we present some numerical simulations to demonstrate the convergence and the decay of the discrete energy.

References

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