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.

翻译：基于具有零边界条件的Stokes复形及其对偶复形，我们将四旋度问题中产生的grad-curl问题重新解释为三维Stokes系统的一种新向量势形式。通过将分析推广至相应的非齐次问题及伴随的迹复形，我们构造了一种具有任意逼近阶数的新型$\boldsymbol{H}(\operatorname{grad-curl})$-协调虚拟元空间，该空间满足关联离散Stokes复形的精确性。在最低阶情形中，每个顶点分配三个自由度，每条边分配一个自由度。对于grad-curl问题，我们严格建立了插值误差估计、离散双线性形式的稳定性以及所提单元在多面体网格上的收敛性。作为Stokes问题的离散向量势形式，所得系统具有压力解耦特性且为对称正定。数值算例验证了理论结果。