Stabilized finite element method for the approximation of linearized viscoelastic fluid flow models

In this article, we present a stabilized finite element (FE) method for the linearized viscoelastic fluid flow. The FE spaces for the unknown variables are chosen as P 1-P 0-P 1, where the fluid velocity and the pressure are discretized by the lowest-order Lagrange elements and the stress tensor is discretized by piecewise P 1 polynomial. In order to get a stable scheme, we added a stabilization term. This method has some prominent features: parameter-free, avoiding calculation of higherorder derivatives and its behaviour towards pressure is totally local. We obtained optimal error estimates and presented several numerical experiments to verify the proposed scheme.