The rheological behavior of viscoelastic fluids in the vicinity of singular points, where stress and pressure tend to infinity due to the abrupt change in boundary conditions, such as die lips or sharp edge corners of re-entrant flow, is not yet fully understood. We have developed sector singular elements, that are placed in a small core around the singularity, and are suitable for finite element analysis of viscoelastic fluids; such finite elements have been previously reported for Newtonian flow problems. Each element has special interpolation functions along its radial direction which take into account the nature of the singularity, and conventional elements are used in the rest of the domain. The problem was solved by a 'decoupled method', which alternatively solves the velocity-pressure fields and the stress field, in combination with several numerical techniques to enhance the upper limits of the Weissenberg number. Compared to the conventional FEM, the sector singular finite element method is better at yielding smoother results even near the singularity, but poorer at convergency, and requires a larger amount of system core memory due to its more complicated topology. These disadvantages represent serious problems for multi-mode and/or 3D flow simulation of viscoelastic fluids. Therefore, we developed a modified version of the sector singular finite element method for the purpose of improving the simulation performance.