中文摘要：

构造了求解两点边值问题的一类修改的Lagrange型三次有限体积元法．试探函数空间取以四次Lobatto多项式的零点作为插值节点的Lagrange型三次有限元空间．将插值多项式的导数超收敛点（应力佳点）作为对偶单元的节点，检验函数空间取相应于对偶剖分的分片常数函数空间．证明了新方法具有最优的H^1模和L^2。模收敛阶，讨论了在应力佳点导数的超收敛性，并通过数值实验验证了理论分析结果．

英文摘要：

In this paper, a class of modified Lagrangian cubic finite volume element method is presented for solving two-point boundary value problems. The trial function space is taken as the Lagrangian cubic finite element space which uses the zero points of quartic Lobatto polynomial as the interpolation nodes. We use the superconvergence points （optimal stress points） of the derivative by the interpolation polynomial as the nodes of the dual unit, the test function space is defined as the piecewise constant function space. It is proved that the method has optimal convergence orders of H^1 and L^2 norms. The superconvergence of numerical derivatives at optimal stress points is discussed. Finally, the numerical experiments show the validity of theoretical analysis