Effect of fluid elasticity and shear-thinning viscosity on the chaotic mixing between two alternately rotating cylinders has been studied. The h-p finite element method is used to obtain high accurate solutions of the steady flow. The unsteady, periodic flow is simulated using the piecewise-steady approximation. Characteristics of the chaotic mixing are analyzed by examining the asymptotic coverage of a passive tracer and the lineal stretching of the fluid elements in the annulus. For the viscoelastic fluids modeled by the upper-convected Maxwell constitutive equation (UCM), our computation predicts little effect of the fluid elasticity on the mixing patterns. On the other hand, the shear-thinning viscosity, modeled by the Carreau equation, has a large impact on the advection of a passive tracer and the distribution of lineal stretching. We find that the zones of the lowest stretching match remarkably well with the regular zones in the tracer-coverage plotting. Our study reveals the vital importance of reducing the discretization errors of the velocity field in the numerical simulation of chaotic flows.