Conservation And Symplectic Properties Of Continuous Finite Element Methods For Hamiltonian Systems

Hamiltonian system is the most important dynamic system. Feng Kang has pointed out, all the real physical processes with negligible dissipation can be expressed, in some way or orther, in suitable Hamiltonian forms. They are ordinary differential or partial differential equations. Hamiltonian systems have the two most important characteristics: conserved properties and preserve symplectic structures of flow. These characteristics can be maintained in the numerical calculation is of great significance. Feng Kang researched on this question in 1983 stated that he was surprised to discover that previously was non-existent here, and many classic algorithmdo not meet these characteristics. It has been already changed unrecognizable after tens of thousands of steps calculated. He first put forward symplectic difference algorithm research in 1984, and the depth study opened up a vast new areas of research. Many domestic and international scholars did the work in many aspects and extensive application afterwards. The first of Feng kang create work has been recognized by the international acknowledge unanimously.However, most numerical methods can not maintain the two properties simultaneously: symplectic and energy conservative in general(Ge-Masden theorem). Symplectic difference algorithm can very good conserve symptectic properties. But only in the sense of accuracy of energy conservation format. Many scholars believe that energy conservation is more important sometimes. So we turn to the finite element methods. The study has so far found little. Our research shows that the finite element always conserve energy, as well as for linear Hamiltonian systems are proved to be symplectic and for nonlinear Hamiltonian systems are proved high accuracy conservate their symplectic structure. So finite element methods from the other to make up the symplectic difference algorithm, can preserve long-term stability properties and precision of the calculated phase space track, the effect is very good. There are important advance to symplectic algorithm.In this paper, the main innovations are as follows:(1). The first system depth study of arbitrary m-th order finite element to solve nonlinear Hamiltonian system and prove that the energy is always conserved in any nodes, as well as utilizing superconvergence analysis methods study symplectic properties of the finite element methods.(2). A profound new high order superconvergence estimate O(h^2m+1)are proved to arbitrary m-th order finite element methods for linear Hamiltonian systems, The first prove that m-th order finite element at nodes is 2m-th order diagonal Páde approximants, thus is symplectic scheme. The conclusions of this study are in agreement with Feng Kang's symplectic difference scheme conclusions.(3). For every step of arbitrary m-th order finite element methods of nonlinear Hamiltonian systems are first proved high accuracy conservate their symplectic structure, through construct a new auxiliary equation and negative norm estimated of error.(4). We study track properties in a long time computate on physical plane, but found little change in shape of the track while in track translation. We first prove that translation of track from the cycle and directly proportional to calculated steps. Such an undesirable nature of the past did not draw attention to and study.(5). Base on Feng Kang put forword some important numerical test problem, such as harmonic oscillator, high-low frequency hybrid linear system, nonlinear Huygens oscillator, the Henon-Heiles system with chaos phenomenon and the classical trajectory calculation of A₂B molecule reflect system, we respectively utilizing the quadratic element and 4-th order symplectic difference scheme to computate and compare, the results coincide with the theoretical. The numerical results confirmed that utilizing the finite element methods to solve Hamiltonian systems have an excellent prospect.