  We prove the existence of weak solutions to a viscoelastic phase separation
problem in two space dimensions. The mathematical model consists of a
Cahn-Hilliard-type equation for two-phase flows and the Peterlin-Navier-Stokes
equations for viscoelastic fluids. We focus on the case of a polynomial-like
potential and suitably bounded coefficient functions. Using the
Lagrange-Galerkin finite element method complex behavior of solution for
spinodal decomposition including transient polymeric network structures is
demonstrated.
