The eXtended virtual element method for elliptic problems with weakly singular solutions

Technical Report · Tue Jun 04 04:00:00 EDT 2024

DOI:https://doi.org/10.2172/2377294· OSTI ID:2377294

Droniou, Jérôme ^[1]; Gianmarco, Manzini ^[2]; Yemm, Liam ^[3]

---

1. Univ. of Montpellier (France); Monash Univ., Melbourne, VIC (Australia)
2. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
3. Monash Univ., Melbourne, VIC (Australia)

This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The X-VEM 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 L²- and H¹- norms on fractured or L-shaped domains.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• DOE Contract Number: 89233218CNA000001
• OSTI ID: 2377294
• Report Number(s): LA-UR--24-21141
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
L-shaped domain
X-VEM
corner singularities
enriched method
error analysis
polytopal method