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Natural superconvergence points for splines

Peng Yang, Zhimin Zhang

TL;DR

This work develops a comprehensive theory of natural superconvergence points for polynomial spline approximations to second-order elliptic problems, starting from a one-dimensional Poisson model and extending to higher-dimensional simplicial and tensor-product meshes. Central to the methodology is a local error decomposition $(u-u_h)^{(s)}(x_0)=(owtie_{B_d}u-u_h)^{(s)}(x_0)+(u-owtie_{B_d}u)^{(s)}(x_0)$, paired with negative-norm estimates and symmetry arguments that link parity with superconvergence. A key outcome is that, at a locally symmetric center, the error derivative $(u-u_h)^{(s)}(x_0)$ exhibits superconvergence when $k-s$ is even, with the abundance of high-smoothness B-spline derivatives yielding a constructive asymptotic error expansion and, in the 1D case, a full characterization of all superconvergence points as zeros of $ig(ig( ext{F}ig)^{k-s}ig)(L_1)$. The results extend to high dimensions, with precise rates for both simplicial and tensor-product meshes, and are validated numerically on 1D and 2D problems, confirming robust superconvergence patterns even in highly localized symmetric regions. These findings offer practical guidance for post-processing and isogeometric collocation methods, enabling optimized point selections for gradient recovery and enhanced accuracy without increased computation.

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.

Natural superconvergence points for splines

TL;DR

This work develops a comprehensive theory of natural superconvergence points for polynomial spline approximations to second-order elliptic problems, starting from a one-dimensional Poisson model and extending to higher-dimensional simplicial and tensor-product meshes. Central to the methodology is a local error decomposition , paired with negative-norm estimates and symmetry arguments that link parity with superconvergence. A key outcome is that, at a locally symmetric center, the error derivative exhibits superconvergence when is even, with the abundance of high-smoothness B-spline derivatives yielding a constructive asymptotic error expansion and, in the 1D case, a full characterization of all superconvergence points as zeros of . The results extend to high dimensions, with precise rates for both simplicial and tensor-product meshes, and are validated numerically on 1D and 2D problems, confirming robust superconvergence patterns even in highly localized symmetric regions. These findings offer practical guidance for post-processing and isogeometric collocation methods, enabling optimized point selections for gradient recovery and enhanced accuracy without increased computation.

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 is a local symmetric center of the partition, the numerical error exhibits superconvergence whenever the polynomial degree and the derivative order 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.
Paper Structure (13 sections, 14 theorems, 134 equations, 3 figures, 3 tables)

This paper contains 13 sections, 14 theorems, 134 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Natural superconvergence points in one dimension
  4. Pointwise error estimates between global and local solutions
  5. Symmetry-induced superconvergence points
  6. From symmetry-induced superconvergence points to all superconvergence points when $k\geq 2$ and $\mu=k-1$
  7. Natural superconvergence points in high dimensions
  8. Symmetry-induced superconvergence points on simplicial meshes
  9. Natural superconvergence points on tensor-product meshes
  10. Numerical experiments
  11. Superconvergence of one-dimensional B-spline elements
  12. Superconvergence of triangular quadratic Lagrange and cubic Hermite finite elements at a highly localized symmetric center
  13. Conclusion

Key Result

Lemma 3.1

\newlabellemma, inverse property0 For $w_h\in S_{h}^{k,\mu}(B_d)$ satisfying and any integer $l\geq 0$, there holds the estimate

Figures (3)

  • Figure 1: Numerical results of the pointwise convergence rate $r_{s,m,\Omega_{in}}$ on the reference element [-1,1], where $N=60$ and $k=3$.
  • Figure 2: Numerical results of the pointwise convergence rate $r_{s,m,\Omega_{in}}$ on the reference element [-1,1], where $N=40$ and $k=4$.
  • Figure 3: Illustration of triangular meshes with $N_{tri}$ elements. The red dot marks the mesh vertex $\bm{x}_0=(0.3,0.4)$, enclosed by a red frame indicating the local symmetric region around it; the black dot denotes the mesh vertex $\bm{x}'_0 = (0.7,0.6)$, included for comparison.

Theorems & Definitions (31)

  • Lemma 3.1
  • Lemma 3.2
  • Proof 1
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • Proof 2
  • Lemma 3.7
  • Proof 3
  • ...and 21 more