A continuous interior penalty method for viscoelastic flows.

In this paper we consider a finite element discretization of the Oldroyd-B model of viscoelastic flows. The method uses standard continuous polynomial finite element spaces for velocities, pressures, and stresses. Inf-sup stability and stability for convection-dominated flows are obtained by adding a term penalizing the jump of the solution gradient over element faces. To increase robustness when the Deborah number is high, we add a nonlinear artificial viscosity of shock-capturing type. The method is analyzed on a linear model problem, and optimal a priori error estimates are proven that are independent of the solvent viscosity $\\eta_s$. Finally we demonstrate the performance of the method on some known benchmark cases.