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

TL;DR

This paper introduces a novel virtual element method for a grad-curl problem linked to the 3D quad-curl and Stokes systems, providing a stable, convergent, and pressure-decoupled formulation with numerical validation.

Contribution

It develops a new $oldsymbol{H}( ext{grad-curl})$-conforming virtual element space that ensures exactness of the discrete Stokes complex and applies to non-homogeneous problems on polyhedral meshes.

Findings

Established error estimates and stability for the new element.

Proved convergence and pressure decoupling in the discretized system.

Numerical examples confirm theoretical accuracy and stability.

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.