New superconvergent structures developed from the finite volume element method in 1D

Abstract

New superconvergence structures are introduced by the finite volume element method (FVEM), which gives us the freedom to choose the superconvergent points of the derivative (for odd order schemes) and the superconvergent points of the function value (for even order schemes) for $k\geq 3$. The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the superconvergence properties of the new structures. In addition, the relationships between the orthogonal condition and the convergence properties for the FVE schemes are carried out in Table 1. The above results are also valid to $k=1,2$. For these cases, the new superconvergence structures are the same with the Gauss-Lobatto structure, which yields to the FVE schemes with Gauss-points-dependent dual meshes. Numerical results are given to illustrate the theoretical results.