On a posteriori estimates for a linearized Oldroydʹs problem

We study and give some a posteriori estimates for finite element approximation of a linearized Oldroydʹs problem for the approximation of viscoelastic fluid flows. The major difficulty lies in the coupling between two different kinds of equations: a hyperbolic equation associated to the constitutive equation and Stokes problem. These estimates are then obtained combining two different techniques used for each kind of equation: the first technique is given in (First ICIAM, Paris, 1987) for Stokesʹs equations and the second in (C.R. Acad. Sci. Paris. Ser. I. 322 (1996) 493) for Friedrichsʹs system.