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Periodic solutions of delay equations are usually approximated as continuouspiecewise polynomials on meshes adapted to the solutions' profile. In practicalcomputations this affects the regularity of the (coefficients of the)linearized system and, in turn, the effectiveness of assessing local stabilityby approximating the Floquet multipliers. To overcome this problem whencomputing multipliers by collocation, the discretization grid should includethe piecewise adapted mesh of the computed periodic solution. By introducing apiecewise version of existing pseudospectral techniques, we explain why andshow experimentally that this choice is essential in presence of either strongmesh adaptation or nontrivial multipliers whose eigenfunctions' profile isunrelated to that of the periodic solution.

Piecewise discretization of monodromy operators of delay equations on adapted meshes