The eXtended Virtual Element Method for elliptic problems with weakly singular solutions

Abstract: 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.