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A stabilized nonconforming finite element method for the surface biharmonic problem

Shuonan Wu, Hao Zhou

Abstract

This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its $H^1$-relative conformity and weak $H({\rm div})$ conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only $H^3$ regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken $H^1$ norm. Numerical experiments are provided to support the theoretical results.

A stabilized nonconforming finite element method for the surface biharmonic problem

Abstract

This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its -relative conformity and weak conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken norm. Numerical experiments are provided to support the theoretical results.

Paper Structure

This paper contains 22 sections, 17 theorems, 109 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Differential operators and function spaces
  4. Discretization
  5. Operators and function spaces on $\Gamma_h$
  6. Extensions and lifts
  7. Surface Piola transforms
  8. New Zienkiewiz-type element
  9. Shape function space
  10. Degrees of freedom
  11. Finite element method on surface
  12. Surface NZT space
  13. Approximation properties
  14. Stabilized nonconforming FEM
  15. Error estimates
  16. ...and 7 more sections

Key Result

Lemma 2.2

\newlabellm:Piola-equivalence0 For any $K \in \mathcal{T}_h$, let $\bm{g}$ be a vector field on the $C^4$ surface $K^\gamma$, and denote its corresponding Piola transform according to def:Piola_invp by $\breve{\bm g} = \mathscr{P}_{\bm{p}^{-1}} \bm{g}: K \to \mathbb{R}^3$. If $\bm{g} \in \bm{H}^m(

Figures (4)

  • Figure 1: Composition of surface NZT space $V_h$ (right) from the NZT element (left), where the DoF of tangential derivative at vertex $a$ in $K_a$ is mapped to $K$ via the Piola transform with respect to the closest point projection on $K_a$.
  • Figure 1: The numerical approximation of $u = r^{-3}(3x^2y - y^3)$ with 7686 DoFs on the unit sphere.
  • Figure 2: The numerical approximation of $u = \sin(3\phi) \cos(3\theta+\phi)$ with 6144 DoFs on the torus.
  • Figure 3: The numerical approximation of $u = y$ with 3462 DoFs on an implicitly defined surface.

Theorems & Definitions (35)

  • Remark 2.1: inapplicable with $\nabla_\gamma^2$
  • Lemma 2.2: norm equivalence of the Piola transform
  • Proof 1
  • Lemma 2.3: $\mathcal{O}(h^2)$ approximation of Piola transform
  • Proof 2
  • Remark 2.4
  • Lemma 2.5: Lemma 2.5 of demlow2024tangential
  • Lemma 3.1: weak $\bm{H}({\rm div}_{\Gamma_h}; \Gamma_h)$ conformity of $\nabla_{\Gamma_h}v$
  • Proof 3
  • Lemma 3.2: local jump estimates
  • ...and 25 more