In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the $\mathit{H}$(div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

中文翻译：

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对流扩散反应问题的最小二乘虚拟单元法

在本文中，我们针对混合形式的对流扩散反应问题引入了最小二乘虚拟单元法。我们使用$\mathit{H}$（div）虚拟元素和连续虚拟元素分别逼近通量和原始变量。该方法允许使用非常普通的多边形网格。建立通量和原始变量在适当范数中的最优先验误差估计。最小二乘法可提供高效的后验误差估计器，而无需付出额外的努力。而且，悬挂的节点自然用虚拟元素方法进行处理，因为永远不需要局部网格后处理，因此在网格细化方面提供了很高的灵活性。这两个吸引人的特征促使我们开发该方法的后验误差估计。