Strong Discontinuous Finite Element Methods For Ordinary Differential Equations

Ordinary differential equations (systems) with Dirichlet boundary arise widely in the fields of science and technology and economic. There are a lot of effective algorithms, such as difference methods and the finite element methods. Nowadays special attention has been put on the discontinuous finite element methods because of its high accuracy and its loose demand of the smoothness of the solution to the equations (systems).In this paper,a kind of discontinuous finite element method has been discussed. In 1981, M.Delfour etc found that for even degree discontinuous finite finite element method, numerical flux at nodes U_j^* = (U_j^- + U_j⁺)/2 have the highest order superconvergence O(h^2k+2), but the proof is unavailable.We study this problem intensively.First,through constructing a new local discrete Green function,we prove the highest order maximum norm estimation of the solution of discontinuous finite element methods;Second, using Legendre expansion in an element to construct a comparative function and duality arguments and other new skills ,we get the optimal order superconvergence at nodes for the first time .H(x_j) -U_j^*|< Ch^2k+2||u||_k+2, k = 0, 2, 4,....This is highest order superconvergence results we know for all discontinuous finite element methods nowadays. The numerical results confirm this result. In addition,we get the momentum conservation for nonlinear Hamilton systems by using this type of discontinuous finite element method.This is the first momentum conservation in numerical experiments.