Volume 60, pp. 405-420, 2024.

Analysis of a one-dimensional nonlocal thermoelastic problem

Noelia Bazarra, José R. Fernández, and Ramón Quintanilla

Abstract

In this paper we study a one-dimensional dynamic thermoelastic problem assuming that the elastic coefficient is negative. Following the ideas proposed by Eringen in the 80s, a nonlocal term is introduced into the constitutive equation for the displacements, leading to a hyperbolic problem. Then, we consider the numerical approximation of the problem by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A discrete stability property is proved, and an a priori error analysis is done, from which we can conclude the linear convergence of the approximations under suitable regularity for the continuous solution. Finally, some numerical simulations are presented, including the demonstration of the numerical convergence and the behavior of the discrete energy for several choices of the constitutive parameters.

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Key words

nonlocal thermoelasticity, finite elements, discrete stability, a priori error estimates, numerical simulations

AMS subject classifications

65M15, 65M60, 65M12

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