Error analysis for a fully-discrete finite element approximation of the unsteady p(.,.)-Stokes equations

n this paper, we examine a fully-discrete finite element approximation of the unsteady p(.,.)-Stokes equations ({i.e.}, p(.,.) is time- and space-dependent), employing a backward Euler step in time and conforming discretely inf-sup stable finite elements in space, for error decay rates for the velocity vector field. More precisely, we derive error decay rates that are optimal with respect to natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. The optimality of the derived error decay rates is confirmed via numerical experiments.