We discuss the viscoelastic flow of the matrix fluid in an Extensional Flow Mixer (EFM) obtained by a numerical simulation. The effect of the fluid viscoelasticity on the flow state in an EFM was considered by the comparison of the flow states of the purely viscous fluid and the viscoelastic fluid. The mixing performance of the EFM is also considered. The Phan-Thien Tanner model was used as the constitutive equation of the viscoelastic fluid and the Carreau model was used as that of the purely viscous fluid. Results obtained are as follows: 1) Stagnation flows occur in the converging-diverging plates (c-d plates) in the case of viscoelastic fluid. The size of stagnation flow region becomes large when the fluid has a strain-thickening elongational viscosity. 2) The absolute value of the extensional strain and the pressure drop in the c-d region increase with fluid elasticity. Meanwhile, the residence time in the c-d region decreases with that. 3) There is no stagnation region in the c-d plates for a purely viscous fluid. The stagnation flow has an effect on the estimation value of EFM's performance. The prediction by the purely viscous fluid is not useful for the flow in an EFM because of extremely low accuracy.