Stability and Numerical Analysis of Micropolar Viscoelastic Systems - Bazarra, Noelia; Fernández, José R.; Fernández Sare, Hugo D.; Quintanilla, Ramón (Journal of elasticity, 2025)
In this work, we consider two dynamic systems arising in micropolar viscoelasticity. In this sense, the material structure is assumed to have macroscopic and microscopic levels. First, an existence and uniqueness result is proved by using the theory of linear semigroups and, secondly, the decay of the solutions to the equilibrium state is shown. Then, the polynomial energy decay is obtained applying a characterization of the system operator. In a second part, we consider the numerical approximation of a variational version of the above problem. This is done by using the 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 provided. The linear convergence of the approximations is deduced under some additional regularity conditions on the continuous solution. Finally, some numerical simulations are shown to demonstrate numerical convergence and the behavior of the discrete energy.