Error growth and a posteriori error estimates for conservative Galerkin approximations of periodic orbits in Hamiltonian systems

We prove an a posteriori error estimate for approximations of periodic orbi ts in Hamiltonian systems using Galerkin methods which conserve the Hamilto nian. The error is estimated in terms of the local time step, the residual obtained by inserting the approximate solution into the differential equati on, and a stability factor describing relevant stability properties of the adjoint linearized problem. The quantitative growth of the stability factor as a function of time is of particular interest. We show that the stabilit y factor grows linearly with time for a certain class of problems when the conservative scheme is used, in contrast to the quadratic growth of the sta bility factor, expected for non-conservative schemes in general.