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

TL;DR

This work addresses elliptic problems with weak singularities by introducing the eXtended virtual element method (X-VEM), which enriches the VEM space with a generic enrichment space $Ψ(Ω)$ to capture singular components while preserving high-order accuracy. The discrete scheme uses a_h defined via extended elliptic projections $Π^{∇,Ψ}_{k,E}$ and stabilization $S^E$, with the right-hand side projected as $f_h=Π^{Δ}_{l,E}f$, and proves an energy-error bound $\|u_h-\mathcal{I}_{k,h}u\|_{1,h} \lesssim h^k |u^r|_{H^{k+1}(Ω_h)}$. A complete convergence analysis under mild mesh-regularity assumptions is provided, establishing stability, consistency, and optimal rates in $H^1$-type norms, with continuous-energy error bounds. Numerical experiments on fractured and L-shaped domains demonstrate optimal convergence in $L^2$ and $H^1$ norms for the enriched method; however, ill-conditioning is observed for globally enriched schemes, while local enrichment offers robust performance. The results suggest that X-VEM is a flexible and effective framework for high-order discretization on polytopal meshes in the presence of singularities, with potential extensions to three dimensions and broader elliptic problems.

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.

Figures (5)

• Figure 1: Plots of meshes 2 and 3 used in the fractured domain test. The fracture is coloured blue and enriched elements are coloured red.
• Figure 2: Tests on fractured domain
• Figure 3: Plots of meshes 2 and 4 used in the L-shaped domain test of Section \ref{['sec:Lshape.1']}. The enriched elements are coloured red.
• Figure 4: Tests on L-shaped domain, hexagonal meshes
• Figure 5: Tests on L-shaped domain, Cartesian meshes

Theorems & Definitions (17)

• Remark 2.2: Conforming enrichment space and local enrichment
• Lemma 2.3: Unisolvence of boundary values
• Remark 2.4: Choice of boundary degrees of freedom
• Remark 2.5: Computation of the boundary degrees of freedom
• Lemma 2.6: Unisolvence
• Lemma 2.7: Computability of the elliptic projector
• Remark 2.8: Example of stabilisation and choice of local semi-norm
• Theorem 2.9: Discrete Energy Error
• Remark 2.10: Alternate interpolator and local enrichment
• Theorem 2.11: $H^1$ Error