Mixed stress‐velocity‐pressure finite element techniques are frequently used to solve differential viscoelastic flow problems of the Maxwell (Oldroyd‐B) type. In the present paper, simplified models are introduced to analyze the stability and the accuracy of a class of methods used to solve differential viscoelastic problems. In particular, a one‐dimensional equation is presented which exhibits most of the difficulties of the coupled Maxwell flow problem but requires only a fraction of the CPU time. The performance of some formulations (EEME, EVSS, 4×4, 4×4 SU, 4×4 SUPG, MIX1, ... ) are analyzed on the basis of this equation. New numerical techniques which enhance both the stability and the accuracy of those mixed formulations are also presented. Those techniques are based on a combination of two streamline integration techniques, in such a way as to maintain the elliptic character to the coupled system.