The spectral element method is used on periodic flows containing change of type of the vorticity equation. On slightly perturbed viscometric flows in a channel, the method is compared to the so-called EVSS finite element method and to analytical results when they are available. It is found that the method performs very well on all flows examined and that the computational cost is reduced compared to EVSS. The analysis of the uniform perturbed flow of a Maxwell fluid has revealed an unusual condition on the polynomial space containing the extra-stress representations. The technique is also applied to study the increase of the flow resistance in a periodically constricted tube. Results are compared with existing ones produced by means of the mixed pseudospectral-finite difference method (PCFD). Excellent agreement is found when comparing flow resistance. Explanation of the observed numerical difficulties is also investigated by inspection of the vorticity patterns corresponding to several pairs of Weissenberg and Reynolds numbers.