Stabilized finite element method for the viscoelastic Oldroyd fluid flows

In this paper, a new stabilized finite element method based on two local Gauss integrations is considered for the two‐dimensional viscoelastic fluid motion equations, arising from the Oldroyd model for the non‐Newtonian fluid flows. This new stabilized method presents attractive features such as being parameter‐free, or being defined for non‐edge‐based data structures. It confirms that the lowest equal‐order P 1 − P 1 triangle element and Q 1 − Q 1 quadrilateral element are compatible. Moreover, the long time stabilities and error estimates for the velocity in H ¹‐norm and for the pressure in L ²‐norm are obtained. Finally, some numerical experiments are performed, which show that the new method is applied to this model successfully and can save lots of computational cost compared with the standard ones.