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Superconvergence of differential structure for finite element methods on perturbed surface meshes

Guozhi Dong, Hailong Guo, Ting Guo

TL;DR

This work addresses gradient recovery and superconvergence for finite element methods on discretized surfaces when exact geometry is unavailable. By introducing geometric supercloseness, it bridges the gap between deviated mesh geometry and differential-structure accuracy, enabling rigorous superconvergence results for recovered gradients via a two-level isoparametric framework (recovering the Jacobian and then the gradient). Theoretical results are supported by numerical experiments on spheres and general surfaces, showing that gradient recovery can achieve $O(h^2)$ convergence under mild mesh conditions, even when vertices are not on the exact surface. The framework answers open questions about recovering differential structures without exact geometry and offers a practical approach for gradient recovery on perturbed surface meshes with potential extensions to higher-order discretizations and vector fields.

Abstract

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical results.

Superconvergence of differential structure for finite element methods on perturbed surface meshes

TL;DR

This work addresses gradient recovery and superconvergence for finite element methods on discretized surfaces when exact geometry is unavailable. By introducing geometric supercloseness, it bridges the gap between deviated mesh geometry and differential-structure accuracy, enabling rigorous superconvergence results for recovered gradients via a two-level isoparametric framework (recovering the Jacobian and then the gradient). Theoretical results are supported by numerical experiments on spheres and general surfaces, showing that gradient recovery can achieve convergence under mild mesh conditions, even when vertices are not on the exact surface. The framework answers open questions about recovering differential structures without exact geometry and offers a practical approach for gradient recovery on perturbed surface meshes with potential extensions to higher-order discretizations and vector fields.

Abstract

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical results.

Paper Structure

This paper contains 16 sections, 8 theorems, 44 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Structure
  4. Geometric setting and counter examples
  5. Geometric setting
  6. Examples of $\mathcal{O}(h^2)$ deviated surfaces where superconvergence fails
  7. Supercloseness of geometric approximation
  8. Superconvergent gradient recovery on deviated surfaces
  9. Parametric gradient recovery schemes on surfaces
  10. Superconvergence analysis on deviated geometry
  11. Numerical results
  12. Examples on a sphere with deviations
  13. Verification of geometric supercloseness
  14. Superconvergence of gradient recovery on deviated sphere
  15. The Laplace-Beltrami equation on general surface
  16. ...and 1 more sections

Key Result

Lemma 3.4

Let $\tau_h$ and $\tau^*_h$ be shape regular triangle pair of diameter $h$ from $\mathcal{M}^*_{h}$ and $\mathcal{M}_{h}$ respectively, and the distance of each of their vertex pair satisfies condition eq:close_condition1, for $h \leq 1$ sufficiently small. In addition, if the bound in eq:close_con where $\left\{ l_{k} | {k=1,2,3}\right\}$ (or $\left\{ l_{k}^* | {k=1,2,3}\right\}$) denotes the le

Figures (3)

  • Figure 1: Here $\tau_{h,j}$ and $\tau^*_{h,j}$ are planar triangles, while $\tau_j$ and $\tilde{\tau}_{h,j}$ are curved triangles, $\Omega_i$ is the parameter domain for the surrounding patches at $x^*_{n,i}$, which we show only one triangle part of it. This parameter domain is used to define the local geometric maps for triangles surrounding the vertex $x^*_{n,i}$ , i.e., $\mathbf{r}_i:\Omega_i \to \tau_j$, $\mathbf{r}_{h,i}:\Omega_i \to \tau_{h,j}$, $\mathbf{r}_{h,i}^* :\Omega_i \to \tau^*_{h,j}$. On the planar triangles we have the local maps $\mathbf{r}_{\tau_{h,j}}:\tau_{h,j}\to \tau_j$, $\mathbf{r}^*_{\tau_{h,j}}: \tau^*_{h,j}\to \tau_j$ and $\tilde{\mathbf{r}}_{\tau_{h,j}}:\tau_{h,j}\to \tilde{\tau}_{h,j}$.
  • Figure 2: Counter example of superconvergence of the recovered gradient : (a) random $\mathcal{O}(h^2)$ in the tangential direction; (b) random $\mathcal{O}(h^2)$ in the normal direction.
  • Figure 3: Initial mesh on a general surface

Theorems & Definitions (17)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • proof
  • ...and 7 more