  In this paper, Particle-in-Cell algorithms for the Vlasov-Poisson system are
presented based on its Poisson bracket structure. The Poisson equation is
solved by finite element methods, in which the appropriate finite element
spaces are taken to guarantee that the semi-discretized system possesses a well
defined discrete Poisson bracket structure. Then, splitting methods are applied
to the semi-discretized system by decomposing the Hamiltonian function. The
resulting discretizations are proved to be Poisson bracket preserving.
Moreover, the conservative quantities of the system are also well preserved. In
numerical experiments, we use the presented numerical methods to simulate
various physical phenomena. Due to the huge computational effort of the
practical computations, we employ the strategy of parallel computing. The
numerical results verify the efficiency of the new derived numerical
discretizations.
