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：本文提出了一种新型扩展虚拟元方法，该方法是对协调虚拟元方法的扩展。XVEM通过在局部空间中引入适当的增强函数来构建。该方法旨在处理高度通用的增强函数，包括由断裂区域引起的奇异性。通过在增强空间上实现一致性，该方法被证明即使在奇异解存在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，展示了在断裂或L形区域上$L^2$范数和$H^1$范数中的最优收敛率。