Cubic superconvergent finite volume element method for one-dimensional elliptic and parabolic equations

In this paper, a cubic superconvergent finite volume element method based on optimal stress points is presented for one-dimensional elliptic and parabolic equations. For elliptic problem, it is proved that the method has optimal third order accuracy with respect to H1H1 norm and fourth order accuracy with respect to L2L2 norm. We also obtain that the scheme has fourth order superconvergence for derivatives at optimal stress points. For parabolic problem, the scheme is given and error estimate is obtained with respect to L2L2 norm. Finally, numerical examples are provided to show the effectiveness of the method.