On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are \(\sigma\) the viscoelastic part of the extra stress tensor,\(u\) the velocity and\(p\) the pressure. We suppose that the solution\((\sigma,u,p)\) is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp.\(P_1\) discontinuous,\(P_2\) continuous,\(P_1\) continuous FE. Upwinding needed for convection of\(\sigma\) , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of\((\sigma,u,p)\) ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds.