Any order superconvergence finite volume schemes for 1D general elliptic equations

TL;DR

This paper introduces a high-order finite volume scheme for 1D elliptic equations, demonstrating optimal convergence and superconvergence properties at Gauss points, supported by theoretical analysis and numerical validation.

Contribution

The paper develops a unified high-order finite volume scheme for 1D elliptic equations with superconvergence at Gauss points, extending existing methods with rigorous proofs.

Findings

Optimal convergence rates under energy and L2 norms

Superconvergence of derivatives at Gauss points

Convergence rate can reach h^{2r} in special cases

Abstract

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 case, the convergence rate can reach $h^{2r}$, where $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.