This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The XVEM is formulated by incorporating appropriate enrichment functions in the local spaces. The method is designed to handle highly generic enrichment functions, including singularities arising from fractured domains. By achieving consistency on the enrichment space, the method is proven to achieve arbitrary approximation orders even in the presence of singular solutions. The paper includes a complete convergence analysis under general assumptions on mesh regularity, and numerical experiments validating the method's accuracy on various mesh families, demonstrating optimal convergence rates in the $L^2$- and $H^1$-norms on fractured or L-shaped domains.

翻译：本文提出了一种新颖的扩展虚拟单元方法，这是对协调虚拟单元方法的推广。该方法通过在当前局部空间中引入适当的富集函数进行构造，旨在处理高度通用的富集函数（包括由断裂区域引起的奇异性）。通过在富集空间上实现一致性，该方法被证明即使在存在奇异解的情况下也能达到任意逼近阶。本文在网格正则性的一般假设下给出了完整的收敛性分析，并通过数值实验验证了该方法在不同网格族上的精度，在断裂或L形域上展示了$L^2$和$H^1$范数下的最优收敛速率。