Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a c onvex polygon, existence of a solution is proved for small relaxation times . Continuous piecewise linear finite elements together with a Galerkin Leas t Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point th eorem, a posteriori error estimates are also derived. An Elastic Viscous Sp lit Stress (EVSS) scheme related to the GLS method is introduced. Numerical results confirm the theoretical predictions.