On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization

In this paper, we investigate the two-dimensional viscoelastic fluid motion problem arising in the Oldroyd model. By applying the dual method, the Helmholtz decomposition and some other techniques, we deduce the long-time optimal error estimates for its penalty method. Furthermore, the global optimal error bounds in the L2 and H1-norms for the time semi-discrete of the penalty system are derived under the backward Euler implicit scheme, which improves the best estimate available.