An a Posteriori Error Analysis for Dynamic Viscoelastic Problems

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. Mathematics Subject Classification. 74H15, 65M15, 74D05, 74S05, 65M60. Received September 8, 2010. Revised December 25, 2010. Published online 26 April, 2011. . Introduction In this paper, a dynamic problem involving a viscoelastic body is considered from the numerical point of view. Viscoelastic materials have been studied in the past thirty years and they are interesting because many metals or crystals can be modeled by using viscoelasticity theory. We recall, for instance, the well-known Kelvin-Voigt viscoelastic constitutive law. Since the first results provided by [13], many works dealing with mathematical problems including viscoelastic materials have been published (see, for instance [6,11,12,14–16,23,24]) or with their numerical analysis (see, e.g., [1,3,20,22,26,29]). Recently, a large number of quasistatic contact problems including a more general constitutive law have been analyzed from both points of view (see the monograph [18] and the numerous references cited therein). In this paper, we revisite a well-known dynamic problem involving a linear viscoelastic body. An a priori analysis is recalled (to our knowledge, it was not published yet), by using some ideas employed in [7] for the