In this paper, we present the H(curl2)-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of H(curl2)-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the H1-conforming and H(curl2)-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.

中文翻译：

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四卷曲问题的卷曲-卷曲符合虚元法

在本文中，我们介绍了H(卷曲2)-二维四卷曲问题的符合虚拟元素（VE）方法。基于de Rham复合体的思想，我们首先构建了三个家族H(卷曲2)-符合 VEs，其中最简单的 VEs 只有一个自由度，分别与最低阶情况下的每个顶点和每条边相关联。之间建立了一个精确的离散复合体H1- 符合和H(卷曲2)- 符合标准的 VE。我们严格证明了插值误差估计、离散双线性形式的稳定性、相应离散公式的矫顽力和 inf-sup 条件。我们表明，符合条件的 VE 具有最佳收敛性。给出了一些数值例子来证实理论结果。