Natural superconvergence points for splines

TL;DR

This paper presents a unified theory of superconvergence points in polynomial spline approximations for elliptic problems, revealing systematic distribution patterns and extending to higher dimensions.

Contribution

It introduces a unified framework for natural superconvergence points in spline approximations, including higher-dimensional extensions and error characterization.

Findings

Superconvergence occurs at symmetric partition centers when degree and derivative parity match.

Superconvergence points are abundant for B-splines, enabling detailed error expansion.

Numerical experiments confirm superconvergence persists in localized symmetric regions.

Abstract

This paper develops a unified theory of natural superconvergence points for polynomial spline approximations to second-order elliptic problems. Beginning with the one-dimensional case, we establish that when a point $x_0$ is a local symmetric center of the partition, the numerical error $(u-u_h)^{(s)}(x_0)$ exhibits superconvergence whenever the polynomial degree $k$ and the derivative order $s$ share the same parity. In particular, for the smoothest spline (B-spline) solution, the abundance of superconvergence points allows us to construct asymptotic expansion of the error within the element that fully characterize all superconvergence points, for both function values and derivatives. The theoretical framework is then extended to higher-dimensional settings on simplicial and tensor-product meshes, and the essential conclusions are preserved, with one-dimensional derivatives generalized to mixed derivatives. Numerical experiments demonstrate that superconvergence persists even in extremely localized symmetric regions, revealing that superconvergence points are both readily attainable and follow systematic distribution patterns.