Superconvergence Of Time Continuous Fully Discrete Finite Element For Parabolic Problem

Based on the finite element ideal for ordinary differential equation that proposed by Delfour.we take advantage of duality argument and orthogonal expand in the element.simplify prove superconvergence of the m-degree continuous finite element and discontinuous finite element at the nodes and some characteristic points for 1-degree initial value problem of ordinary differential equation.as well as use tensor product decompose generalized to parabolic equation indude superconvergence of fully discrete schemes for parabolic at nodes and some characteristic points.when continuous finite element method is applied to nonlinear Schrodinger equation, energy conservation law is obtained.Main results follows:(1)We take advantage of two type of orthogonal expand in the element.simplify prove superconvergence of the m-degree continuous and discontinuous finite element at the nodes and some characteristic points for 1-degree ordinary differential equation.(2) Base on the finite element ideal for parabolic problem that proposed bv Thomee.and use the tensor product ideal proposed by Douglas in solve elliptie to parabolic equation.we proved linear parabolic problem continuous finite element solution in time and space have superconvergence at m+1-degree the Lobatto characteristic point in the element Where Sk is m-degree finite element space in time.Sh0 is n-degree finite element space in space.and Zh is the n+1-degree the Lobatto characteristic point of the space.(3)We use continuous finite element in time fully discrete scheme to solve nonlinear Schrodinger equation,vertify finiteelement solution have energe inte-geration conservation and the numerical results show the theory.