Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reach and even , comparing with rate of the counterpart finite element method. Here is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.