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：本文提出了一种闪电虚元法，通过利用闪电逼近技术实际计算局部VEM基函数中的虚部分量，从而消除了稳定化项。具体而言，闪电VEM使用极点指数级密集分布在多边形剖分每个单元角点附近的有理函数来逼近基函数的虚部，这带来了两大优势。首先，先验误差估计的数学分析变得极为简便，本质上与任何其他非协调伽辽金离散方法一致。其次，由于闪电VEM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。局部构建VEM基函数的代价是为获得闪电VEM方法优势所必须付出的实现成本，但该操作的高度并行化特性最终将产生一种成本效益接近标准VEM和FEM的高效方案。