The Long-time Properties Of Finite Element Method For Hamiltonian System

Hamiltonian system is one of the most important dynamic systems, which has two important properties:energy conservation and symplectic structure. Algorithms for Hamiltonian system should be designed to keep these original properties as well as possible. This paper works on the long time properties of continuous finite element method (CFE) for Hamiltonian system, which preserves energy exactly. The main results and innovations of this paper are as follows.Firstly, the long time properties of symplectic method and CFE are stud-ied in this paper. Based on a large number of numerical results, three conjec-tures about orbit, energy and symplectic are proposed, respectively. We firstly prove that the error of the CFE solution for Hamiltonian system grows linearly with time over a long time, which is called Feng Kang conjecture. The proof includes several parts.(a). We prove the convergence and super convergence results of the solution of CFE over a short time.(b). Based on two basic assumptions, we derive three important uniform estimates.(c). Based on (a),(b), and the energy conservation property of CFE, this paper has proved that Feng Kang conjecture for CFE is still valid in the long time.The equivalence of a kind of implicit RK method and CFE is found out and firstly proved rigorously. Research shows that some m-stage RK is equivalent to m-degree CFE with the corresponding m-point quadrature rule. For exam-ple, m-stage Gauss RK is equivalent to m-degree CFE with m-point Gaussian quadrature rule, and they are both symplectic.We focus on the growth way of the orbit error, and define a class of reg-ular method for Hamiltonian system, whose orbit error has a linear growth with time. The numerical experiments show that besides symplectic methods and energy conservation methods, there are many algorithms belong to regu-lar methods, such as, trapezoid rule, simpson method, LobattoIIIA, average discontinuous finite element method, and so on. Feng Kang conjecture also holds for these regular methods, which extend the scope of the algorithms for Hamiltonian system.In view of the error asymptotic expansion and the extrapolation technique, this paper proposes the pushing-on algorithm for the time-space Hamiltonian system. The main idea is that the solution U(tj＋1) could be obtained by the approximate value based on the solutions U(tk)(k<j＋1) and the extrap-olation technique, and a few iterations to achieve the demanding accuracy. Therefore, the computation cost in each layer could be reduced by the better initial value. Numerical results demonstrate that as far as now, the pushing-on algorithm could at least shorten the computing time by one half if the proper extrapolation formula is used.Finally, we consider the two-dimensional Poisson equation on a rectan-gular domain. Based on Element Orthogonality Analysis(EOA), correction techniques and tensor product, this paper has proved that the bi-k degree rectangular CFE solution has the highest super convergence rate2k at nodes in each element for arbitrary positive integer k. This conclusion is also the foundation of the research of convergence property of CFE for the time-space Hamiltonian system.