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 $L^2$ 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 $h^{r+2}$ and even $h^{2r}$ , comparing with $h^{r+1}$ rate of the counterpart finite element method. Here $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.