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Continuous Linear Finite Element Method for Biharmonic Problems on Surfaces

Ying Cai, Hailong Guo, Zhimin Zhang

TL;DR

This work develops a $C^0$ continuous, linear finite element method for biharmonic equations on smooth surfaces by leveraging a surface gradient recovery operator to approximate second-order surface derivatives on an approximate surface. A residual stabilization enriches the numerical gradient and a weak edge-consistency term ensures stability without requiring a discrete Poincaré inequality, enabling optimal a priori error estimates in both the energy and $L^2$ norms despite geometric approximation errors. The authors establish stability, weak and strong gradient recovery properties, and precise error bounds, supported by numerical experiments on a sphere, a torus, and a heart-shaped surface that corroborate the theory and reveal gradient superconvergence. The approach offers a computationally efficient pathway for solving fourth-order PDEs on curved surfaces using standard $C^0$ elements while maintaining rigorous convergence guarantees and practical robustness to mesh quality.

Abstract

This paper presents an innovative continuous linear finite element approach to effectively solve biharmonic problems on surfaces. The key idea behind this method lies in the strategic utilization of a surface gradient recovery operator to compute the second-order surface derivative of a piecewise continuous linear function defined on the approximate surface, as conventional notions of second-order derivatives are not directly applicable in this context. By incorporating appropriate stabilizations, we rigorously establish the stability of the proposed formulation. Despite the presence of geometric error, we provide optimal error estimates in both the energy norm and $L^2$ norm. Theoretical results are supported by numerical experiments.

Continuous Linear Finite Element Method for Biharmonic Problems on Surfaces

TL;DR

This work develops a continuous, linear finite element method for biharmonic equations on smooth surfaces by leveraging a surface gradient recovery operator to approximate second-order surface derivatives on an approximate surface. A residual stabilization enriches the numerical gradient and a weak edge-consistency term ensures stability without requiring a discrete Poincaré inequality, enabling optimal a priori error estimates in both the energy and norms despite geometric approximation errors. The authors establish stability, weak and strong gradient recovery properties, and precise error bounds, supported by numerical experiments on a sphere, a torus, and a heart-shaped surface that corroborate the theory and reveal gradient superconvergence. The approach offers a computationally efficient pathway for solving fourth-order PDEs on curved surfaces using standard elements while maintaining rigorous convergence guarantees and practical robustness to mesh quality.

Abstract

This paper presents an innovative continuous linear finite element approach to effectively solve biharmonic problems on surfaces. The key idea behind this method lies in the strategic utilization of a surface gradient recovery operator to compute the second-order surface derivative of a piecewise continuous linear function defined on the approximate surface, as conventional notions of second-order derivatives are not directly applicable in this context. By incorporating appropriate stabilizations, we rigorously establish the stability of the proposed formulation. Despite the presence of geometric error, we provide optimal error estimates in both the energy norm and norm. Theoretical results are supported by numerical experiments.
Paper Structure (19 sections, 22 theorems, 160 equations, 4 figures, 6 tables)

This paper contains 19 sections, 22 theorems, 160 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notation and setting
  3. Surfaces and differential operators on surfaces
  4. Model equations
  5. Discrete surface
  6. A recovery based linear surface finite element method
  7. Stability analysis
  8. Error estimate
  9. Interpolation
  10. Weak approximation of gradient recovery operator
  11. Approximation property of gradient recovery operator
  12. Energy error estimate
  13. $L^2$ error estimate
  14. Error estimate on discrete surfaces
  15. Numerical Experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

The following estimates hold provided that the mesh size $h$ is sufficiently small:

Figures (4)

  • Figure 1: Two elements and their conormal vectors on $E$ in a different view.
  • Figure 1: The approximate surface and the mesh of Example 1.
  • Figure 2: The approximate surface and the mesh of Example 2.
  • Figure 3: The approximate surface and the mesh of Example 3.

Theorems & Definitions (45)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Proof 2
  • Lemma 4.2
  • ...and 35 more