The eXtended Virtual Element Method for elliptic problems with weakly singular solutions
CoRR(2024)
摘要
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.
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