We introduce and analyse the first order Enlarged Enhancement Virtual Element Method (E$^2$VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement (from which comes the prefix of the name E$^2$) property of local virtual spaces. The polynomial degree of local projections is chosen based on the number of vertices of each polygon. We provide a proof of well-posedness and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the criterium for well-posedness and the theoretical convergence rates.

翻译：本文介绍并分析了一阶扩展增强虚拟元方法（E$^2$VEM）用于求解泊松问题。该方法通过适当扩展局部虚拟空间的增强性质（由此得名前缀E$^2$），使得高阶多项式投影可计算，从而定义了无需稳定项的双线性形式。局部投影的多项式次数根据每个多边形的顶点数确定。我们给出了问题的适定性证明以及最优阶先验误差估计。在凸与非凸多边形网格上的数值测试验证了适定性判据及理论收敛速率。