Natural superconvergence points for splines

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.