In this paper, a least-squares virtual element method is presented for approximating the vector and scalar variables of second-order elliptic problems. The $\mathit{H}\left(\mathrm{div}\right)$-conforming and scalar-conforming virtual elements are used to approximate the vector and scalar variables, respectively. The method allows the use of very general polygonal meshes and leads to a symmetric positive definite system. The optimal a priori error estimates are established for the vector variable in $\mathit{H}\left(\mathrm{div}\right)$ norm and the scalar variable in $H^1$ norm. A simple a posteriori error estimator is also presented and proved to be reliable and efficient. The virtual element method handles the hanging nodes naturally, thus the local mesh post-processing to remove hanging nodes is not required. Numerical experiments are conducted to verify the accuracy of the method, and show the effectiveness and flexibility of the adaptive strategy driven by the proposed estimator and suitable mesh refinement strategy.

本文提出了一种最小二乘虚拟单元法来逼近二阶椭圆问题的向量和标量变量。的$\mathit{H}（\mathrm{div}）$符合标准和符合标量的虚拟元素分别用于近似向量和标量变量。该方法允许使用非常通用的多边形网格，并导致一个对称的正定系统。为向量变量建立最佳先验误差估计$\mathit{H}（\mathrm{div}）$ 规范和中的标量变量 $H^{\mathrm{1个}}$规范。还提出了一种简单的后验误差估计器，并证明了该方法可靠且有效。虚拟元素方法自然地处理了悬挂节点，因此不需要局部网格后处理来去除悬挂节点。进行了数值实验，验证了该方法的准确性，并表明了所提出的估计器和合适的网格细化策略驱动的自适应策略的有效性和灵活性。