Analysis of a Type II Thermal Problem Involving a Viscoelastic Beam

Abstract

In this work, we will study, from both analytical and numerical points of view, a nonlocal problem involving a thermoviscoelastic beam which has been modeled by using the type II thermal law. In the first part, we will show that this problem has a unique solution and that the solutions decay exponentially by using the theory of linear semigroups. We will also prove that the semigroup of contractions is not differentiable and the impossibility of localization, that is, we will obtain that the unique solution which can vanish in an open nonempty set is the null solution. In the second part, we will focus on the numerical approximation of a variational formulation of the thermomechanical problem. By using the finite element method and the implicit Euler scheme to approximate the spatial variable and to discretrize the time derivatives, respectively, a fully discrete scheme will be introduced. Then, we will prove a discrete stability property and we will provide an a priori error analysis. The linear convergence of the approximations will be deduced whenever the continuous solution is regular enough. Finally, some numerical results will be presented to demonstrate the numerical convergence and the exponential decay of the discrete energy.