In this paper, the upper convected Maxwell (UCM) model, which is used to describe viscoelastic fluid behavior, is converted into a velocity-pressure-total-stress formulation, instead of the usual velocity-pressure-extra-stress formulation. The VPTS formulation couples pressure and stress directly and convective terms of the pressure appear in this new formulation. The equations in the new formulation are approximated using a spectral element method, with both Jacobi and Legendre polynomials to create the spectral element approximation. The use of this method will be demonstrated by solving the viscoelastic flow through an axisymmetric four-to-one contraction. The difficulties arising from the discontinuity in the solution, which appears at the reentrant corner of such a geometry, are overcome by introducing a variable order spectral element grid. This grid contains spectral elements, using polynomials of a different order in different directions.