A second gradient thermoelastic problem with the Coleman-Gurtin model

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Abstract

This article studies the Coleman-Gurtin thermoelastic model with second-gradient effects on the temperature from both analytical and numerical perspectives. Semigroup theory is used to obtain the existence, uniqueness, and continuous dependence of the solutions on the initial parameters. In the one-dimensional case, the polynomial decay of the solutions is obtained. The numerical analysis of the problem is then considered. Employing the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations of a weak form of the thermoelastic problem are introduced. A discrete stability property and an a priori error analysis are also provided. Finally, some numerical simulations are presented to show the numerical behavior of the approximations and the discrete energy.

Keywords

Coleman-Gurtin model / second gradient / existence and uniqueness / stability and energy decay / finite elements / error estimates

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