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The grad-div conforming virtual element method for the quad-div problem in three dimensions

Xiaojing Dong, Yibing Han, Yunqing Huang

TL;DR

This work develops a stable, high-order grad-div conforming virtual element method for the 3D quad-div operator on general polyhedral meshes. By formulating a weak problem within an enhanced de Rham-based complex and constructing three interacting VEM spaces (H^1, H(curl), and H(grad-div)) that form an exact discrete complex, the authors obtain interpolation estimates, stability, and optimal convergence. They provide a complete discretization framework with computable projections and stabilizations, prove well-posedness of the discrete problem, and validate the approach through numerical experiments on diverse mesh families. The results generalize conforming finite-element ideas to polygonal/polyhedral meshes, enabling robust, accurate fourth-order PDE discretizations in elasticity and related fields.

Abstract

We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results.

The grad-div conforming virtual element method for the quad-div problem in three dimensions

TL;DR

This work develops a stable, high-order grad-div conforming virtual element method for the 3D quad-div operator on general polyhedral meshes. By formulating a weak problem within an enhanced de Rham-based complex and constructing three interacting VEM spaces (H^1, H(curl), and H(grad-div)) that form an exact discrete complex, the authors obtain interpolation estimates, stability, and optimal convergence. They provide a complete discretization framework with computable projections and stabilizations, prove well-posedness of the discrete problem, and validate the approach through numerical experiments on diverse mesh families. The results generalize conforming finite-element ideas to polygonal/polyhedral meshes, enabling robust, accurate fourth-order PDE discretizations in elasticity and related fields.

Abstract

We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the -conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of -conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results.

Paper Structure

This paper contains 15 sections, 20 theorems, 224 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. The quad-div problem
  4. Virtual element spaces
  5. $H^1$-conforming virtual element spaces
  6. $\boldsymbol{H}(\operatorname{curl})$-conforming virtual element space
  7. $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual element space
  8. The discrete complex
  9. Interpolation and stability results
  10. Discretization
  11. The discrete bilinear forms
  12. The discrete problem
  13. Convergence analysis
  14. Numerical experiments
  15. The dual complex

Key Result

Lemma 2.1

If $\Omega$ is a contractible Lipschitz polyhedron, there exists $s>\frac{1}{2}$ such that In particular, for divergence-free or curl-free fields, the Friedrichs inequalities hold: The regularity exponent $s$ typically satisfies $s = \frac{1}{2}$ for general Lipschitz domains, and improves to $s = 1$ when $\Omega$ is convex.

Figures (2)

  • Figure 1: The lowest-order ($r = k = 1$) virtual element complex \ref{['discomplex']} on a polyhedral element.
  • Figure 2: Three representatives of the three families of meshes

Theorems & Definitions (47)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3: The regularity of the solution
  • Remark 3.4
  • Definition 3.5
  • ...and 37 more