An a posteriori error analysis for dynamic viscoelastic problems

¹ Departamento de Matemática Aplicada I, ETSE de Telecomunicación, Universidade de Vigo, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain. jose.fernandez@uvigo.es
² Departamento de Matemática Aplicada, Escola Politécnica Superior, Campus Univ. s/n, Universidade de Santiago de Compostela, 27002 Lugo, Spain. duarte.santamarina@usc.es

Received: 8 September 2010
Revised: 25 December 2010

Abstract


In this paper, a dynamic viscoelastic problem is numerically studied. The variational problem is written in terms of the velocity field and it leads to a parabolic linear variational equation. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. An a priori error estimates result is recalled, from which the linear convergence is derived under suitable regularity conditions. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and quasistatic viscoelastic problems. Upper and lower error bounds are obtained. Finally, some two-dimensional numerical simulations are presented to show the behavior of the error estimators.