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A grad-curl conforming virtual element method for a grad-curl problem linking the 3D quad-curl problem and Stokes system

Xiaojing Dong, Yibing Han, Yunqing Huang

TL;DR

The paper develops a grad-curl conforming virtual element method on polyhedral meshes to discretize a grad-curl problem that links the 3D quad-curl problem with the Stokes system through vector potential formulations. It constructs three families of $oldsymbol{H}( ext{grad-curl})$-conforming VEM spaces with arbitrary order, establishing an exact discrete Stokes complex and commutative interpolation diagrams. The discretization yields a stable, convergence-proven scheme and, notably, a pressure-decoupled symmetric positive definite reduced system that can offer computational advantages over traditional velocity-pressure formulations. Numerical experiments on cube and Voronoi meshes validate the theoretical convergence rates and demonstrate the practicality of the vector-potential approach for Stokes-related quad-curl problems on general polyhedral meshes.

Abstract

Based on the Stokes complex with vanishing boundary conditions and its dual complex, we reinterpret a grad-curl problem arising from the quad-curl problem as a new vector potential formulation of the three-dimensional Stokes system. By extending the analysis to the corresponding non-homogeneous problems and the accompanying trace complex, we construct a novel $\boldsymbol{H}(\operatorname{grad-curl})$-conforming virtual element space with arbitrary approximation order that satisfies the exactness of the associated discrete Stokes complex. In the lowest-order case, three degrees of freedom are assigned to each vertex and one to each edge. For the grad-curl problem, we rigorously establish the interpolation error estimates, the stability of discrete bilinear forms, and the convergence of the proposed element on polyhedral meshes. As a discrete vector potential formulation of the Stokes problem, the resulting system is pressure-decoupled and symmetric positive definite. Some numerical examples are presented to verify the theoretical results.

A grad-curl conforming virtual element method for a grad-curl problem linking the 3D quad-curl problem and Stokes system

TL;DR

The paper develops a grad-curl conforming virtual element method on polyhedral meshes to discretize a grad-curl problem that links the 3D quad-curl problem with the Stokes system through vector potential formulations. It constructs three families of -conforming VEM spaces with arbitrary order, establishing an exact discrete Stokes complex and commutative interpolation diagrams. The discretization yields a stable, convergence-proven scheme and, notably, a pressure-decoupled symmetric positive definite reduced system that can offer computational advantages over traditional velocity-pressure formulations. Numerical experiments on cube and Voronoi meshes validate the theoretical convergence rates and demonstrate the practicality of the vector-potential approach for Stokes-related quad-curl problems on general polyhedral meshes.

Abstract

Based on the Stokes complex with vanishing boundary conditions and its dual complex, we reinterpret a grad-curl problem arising from the quad-curl problem as a new vector potential formulation of the three-dimensional Stokes system. By extending the analysis to the corresponding non-homogeneous problems and the accompanying trace complex, we construct a novel -conforming virtual element space with arbitrary approximation order that satisfies the exactness of the associated discrete Stokes complex. In the lowest-order case, three degrees of freedom are assigned to each vertex and one to each edge. For the grad-curl problem, we rigorously establish the interpolation error estimates, the stability of discrete bilinear forms, and the convergence of the proposed element on polyhedral meshes. As a discrete vector potential formulation of the Stokes problem, the resulting system is pressure-decoupled and symmetric positive definite. Some numerical examples are presented to verify the theoretical results.

Paper Structure

This paper contains 22 sections, 23 theorems, 260 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The dual complex
  5. The trace complex
  6. the grad-curl problem
  7. Homogeneous boundary conditions
  8. Non-homogeneous boundary conditions
  9. Virtual element spaces
  10. Scalar $H^1$-conforming virtual element space
  11. Vector $\boldsymbol{H}^1$-conforming virtual element space
  12. $\boldsymbol{H}(\operatorname{grad-curl})$-conforming virtual element space
  13. Scalar $L^2$-conforming space
  14. The discrete complex
  15. Interpolation error estimates
  16. ...and 7 more sections

Key Result

Lemma 2.1

If $\Omega$ is a Lipschitz polyhedron with a connected boundary, there exists $s>\frac{1}{2}$ and positive constant $C_F$ such that The regularity exponent $s$ is typically $\frac{1}{2}$ for the general Lipschitz domains and increases 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: Two representatives of the two families of meshes

Theorems & Definitions (54)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 44 more