The performance of two low-order discretization schemes in combination with the Discontinuous Galerkin method for the analysis of viscoelastic flows is investigated. An (extended) linear interpolation of the velocity-pressure variables is used in combination with a piecewise discontinuous constant and linear approximation of the extra stresses. Galerkin-leastsquares methodology is applied to stabilize the velocity-pressure discretization. As test problems, the falling sphere in a tube and the stick-slip configuration are studied. The constant stress triangular element converges to high Deborah numbers for a wide variety of material parameters of the Phan-Thien-Tanner model. In particular, for the upper convected Maxwell model, the falling sphere problem converges at least up to Deborah number of 4, while the stick-slip problem converges up to a Deborah number of 25.5.