A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

In this paper, a fully discrete finite element penalty method is considered for the two-dimensional linearized viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newton fluid flows. With the finite element method for the spatial discretization and the backward Euler scheme for the temporal discretization, the velocity and pressure are decoupled in this method, which leads to a large reduction of the computational scale. Under some realistic assumptions, the unconditional stability of the fully discrete scheme is proved. Moreover, the optimal error estimates are obtained, which are better than the existing results. Finally, some numerical results are given to verify the theoretical analysis. The difference between the motion of the Newton and non-Newton fluid is also observed.