Extended Virtual Element Method for Elliptic Problems with Weakly Singular Solutions

핵심 개념

The extended virtual element method (X-VEM) is designed to handle highly generic enrichment functions, including singularities arising from fractured domains, and achieves arbitrary approximation orders even in the presence of singular solutions.

초록

The paper introduces a novel extended virtual element method (X-VEM), which is an extension of the conforming virtual element method (VEM). 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.

통계

The paper does not contain any explicit numerical data or statistics.

인용구

"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."

핵심 통찰 요약

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

by Jerome Droni... 게시일 arxiv.org 04-05-2024

https://arxiv.org/pdf/2402.02902.pdf

더 깊은 질문

How can the X-VEM method be extended to three-dimensional problems

To extend the X-VEM method to three-dimensional problems, several key considerations need to be taken into account. Firstly, the construction of the local X-VEM spaces on polyhedral elements would need to be adapted to three-dimensional polyhedral elements. This involves defining appropriate enrichment spaces on each element and edge, similar to the two-dimensional case. The formulation of the stabilisation term and the elliptic projector would also need to be adjusted to account for the additional dimension. Additionally, the consistency and stability properties of the method would need to be re-evaluated in the three-dimensional setting to ensure optimal convergence rates. Overall, the extension to three dimensions would involve generalising the concepts and techniques used in the two-dimensional X-VEM method to the three-dimensional case.

What are the limitations of the current analysis and how can they be addressed, such as the requirement for a globally conforming enrichment space

The current analysis of the X-VEM method has certain limitations that could be addressed in future research. One limitation is the requirement for a globally conforming enrichment space, which restricts the method from incorporating local enrichment functions. To address this limitation, future work could focus on developing a framework for local enrichment in the X-VEM method, allowing for more flexibility in handling singularities in complex geometries. Additionally, the current analysis could be extended to include a more detailed investigation of the stability and consistency properties of the method, especially in the presence of highly singular solutions. By refining the analysis and addressing the limitations related to the enrichment space, the X-VEM method could be further enhanced in terms of its applicability and efficiency.

Can the X-VEM method be combined with local mesh refinement techniques to efficiently handle singularities in complex geometries

The X-VEM method can be combined with local mesh refinement techniques to efficiently handle singularities in complex geometries. By incorporating local mesh refinement, the method can adapt the mesh resolution to capture the singular behavior of the solution in specific regions, leading to more accurate results. Local mesh refinement allows for the concentration of computational resources in areas of interest, such as near fractures or re-entrant corners, while maintaining a coarser mesh in less critical regions. This adaptive approach can significantly improve the accuracy and efficiency of the X-VEM method when dealing with problems involving singularities in complex geometries. Additionally, the combination of X-VEM with local mesh refinement can help mitigate issues related to ill-conditioning that may arise from enriching elements with singular functions.