We propose a lightning Virtual Element Method that eliminates the stabilisation term by actually computing the virtual component of the local VEM basis functions using a lightning approximation. In particular, the lightning VEM approximates the virtual part of the basis functions using rational functions with poles clustered exponentially close to the corners of each element of the polygonal tessellation. This results in two great advantages. First, the mathematical analysis of a priori error estimates is much easier and essentially identical to the one for any other non-conforming Galerkin discretisation. Second, the fact that the lightning VEM truly computes the basis functions allows the user to access the point-wise value of the numerical solution without needing any reconstruction techniques. The cost of the local construction of the VEM basis is the implementation price that one has to pay for the advantages of the lightning VEM method, but the embarrassingly parallelizable nature of this operation will ultimately result in a cost-efficient scheme almost comparable to standard VEM and FEM.

翻译：我们提出了一种闪电虚拟元方法，该方法通过利用闪电近似实际计算局部虚拟元基函数中的虚拟分量，从而消除了稳定项。具体而言，闪电虚拟元使用有理函数近似基函数的虚拟部分，这些有理函数的极点指数级聚集在多边形剖分每个单元的角点附近。这带来两大优势。首先，先验误差估计的数学分析变得简单得多，且本质上与非协调伽辽金离散化一致。其次，由于闪电虚拟元真正计算了基函数，用户无需任何重构技术即可直接获取数值解的逐点值。局部构造虚拟元基函数的代价是为了获得闪电虚拟元方法的优势所必须付出的实现成本，但这一操作的高度并行化特性最终将使该方案的成本效率几乎与标准虚拟元方法和有限元方法相当。