A pre-conditioner for the GMRES iterative solver is designed specifically tailored to the characteristics of the DEVSS/DG algorithm. The performance is demonstrated for the falling-sphere-in-a-tube benchmark problem. Furthermore, for the four-to-one axi-symmetric contraction flow of an upper convected Maxwell fluid stress fields near the corner singularity are investigated, corroborating the analytical results of Hinch [1] in a core region away from the up-and downstream walls.