Ultraconvergence For Averaging Discontinuous Finite Elements And Its Applications

In the fields of science technology and economic etc., ordinary differential equations (systems) with initial value problem are widely applied. There are lots of effective numerical algorithms, such as difference method and finite element method. In recent years, the discontinuous finite element method becomes more and more popular for scholars, not only because its accuracy is much higher but also the demand on the solution's smoothness is lower.This paper studies a averaging discontinuous finite element method with ultraconvergence, the main work and innovation are as follows:(1) For the linear and nonlinear ordinary differential equations with ini-tial value problem, the k-degree averaging discontinuous finite element method is discussed. When k is even, we prove that the averaging numerical flux Ujs={Uj-+Uj+)/2 (the average of left and right limits for discontinuous fi-nite element at nodes) has the highest order ultraconvergence O(h2k+2) for the first time. This is the highest order superconvergence result for all the finite element methods nowadays. In 1981, M. Delfour etc. found this result by nu-merical computation, but the proof is unavailable and since then no proof. The numerical experiments confirm this highest order superconvergence result and show the superconvergence points in elements.(2) Hamiltonian system is one of the most important dynamic system. Hamiltonian system has three important properties:energy conservation, sym-plectic structure and periodic solution. For the long-time calculation problems, how to construct the numerical scheme to keep these properties is of great impor-tance. For Hamiltonian system, this paper discusses the long-time properties of averaging discontinuous finite element for the first time. Numerical experiments show that the deviation of orbit grows linearly with time (Feng's conjecture is valid) and energy preserves approximately conservation (the deviation of energy does not grow with time).(3) For nonlinear Hamiltonian systems with momentum conservation (such as Kepler system, ordinary differential and partial differential Schrodinger equa-tions systems), we first time find that the averaging discontinuous finite element method preserves momentum conversation at nodes. Previous continuous finite element method preserves energy conversation, other kinds of discontinuous fi-nite element method neither preserve energy conversation, nor preserve momen-tum conversation. These properties are confirmed by numerical experiments.