  Periodic solutions of delay equations are usually approximated as continuous
piecewise polynomials on meshes adapted to the solutions' profile. In practical
computations this affects the regularity of the (coefficients of the)
linearized system and, in turn, the effectiveness of assessing local stability
by approximating the Floquet multipliers. To overcome this problem when
computing multipliers by collocation, the discretization grid should include
the piecewise adapted mesh of the computed periodic solution. By introducing a
piecewise version of existing pseudospectral techniques, we explain why and
show experimentally that this choice is essential in presence of either strong
mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is
unrelated to that of the periodic solution.
