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This paper is devoted to studying the well-posedness and optimal solution techniques for a full discretization scheme, proposed by Lee and Xu (2006), for a large class of viscoelastic flow models. By using some special properties of the scheme such that it is stable in the energy norm and it preserves the positivity of the conformation tensor, the global existence and uniqueness of the solution of the discrete scheme is established. Furthermore, it is shown that the solution of the discrete scheme at each time step can be obtained by an iterative procedure that only requires operations (with N being the number of nodes of the underlying finite element grid).