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.

翻译：我们针对三维四散度问题提出了一种新的稳定变分形式，并证明了其适定性。基于此弱形式，我们在多面体网格上发展并分析了任意逼近阶的$\boldsymbol{H}(\operatorname{grad-div})$协调虚拟元方法。通过利用具有增强光滑性的de Rham子复形结构，我们构造了三族$\boldsymbol{H}(\operatorname{grad-div})$协调虚拟元，从而得到一个精确的离散虚拟元复形。在最低阶情形中，最简单的单元在每个顶点和每个面上分别仅有一个自由度。我们严格证明了插值误差估计、离散双线性形式的稳定性、离散格式的适定性以及最优误差估计。数值算例验证了理论结果。