We present the essential instruments to deal with Virtual Element Method (VEM) for the resolution of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. Finally, we exploit such `briks' to construct virtual element approximations of Stokes, Darcy and Navier-Stokes problems and we provide a series of examples to numerically verify the theoretical behavior of high-order VEM.

中文翻译：

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混合高阶虚元方法的积木：投影仪和微分算子

我们介绍了处理虚拟元法 (VEM) 以解决混合形式的偏微分方程的基本工具。描述了函数空间、自由度、投影仪和微分算子，强调了如何在虚拟元素框架中构建它们，并为一般的近似阶。为了实现这个目标，有必要对多项式空间和分解进行深入分析。最后，我们利用这种“briks”来构建斯托克斯、达西和纳维-斯托克斯问题的虚拟元素近似，并提供了一系列示例来数值验证高阶 VEM 的理论行为。