[論文レビュー] Avoiding stabilization terms in virtual elements for eigenvalue problems: The Reduced Basis Virtual Element Method

Silvia Bertoluzza, Fabio Credali+1 more|arXiv (Cornell University)|Feb 12, 2026

Advanced Numerical Methods in Computational Mathematics被引用数 0

ひとこと要約

This paper introduces the Reduced Basis Virtual Element Method (rbVEM) for Laplace eigenvalue problems, achieving stabilization-free, fully conforming discretization by representing the non-polynomial VEM contribution via reduced basis functions and proving optimal spectral convergence.

ABSTRACT

We present the novel Reduced Basis Virtual Element Method (rbVEM) for solving the Laplace eigenvalue problem. This approach is based on the virtual element method and exploits the reduced basis technique to obtain an explicit representation of the virtual (non-polynomial) contribution to the discrete space. rbVEM yields a fully conforming discretization of the considered problem, so that stabilization terms are avoided. We prove that rbVEM provides the correct spectral approximation with optimal error estimates. Theoretical results are supplemented by an exhaustive numerical investigation.

研究の動機と目的

• Motivate and address stabilization issues in virtual element methods for eigenvalue problems.
• Develop a stabilization-free, fully conforming discretization by representing non-polynomial VEM contributions via reduced basis functions.
• Prove optimal convergence rates for both source and eigenvalue problems within the rbVEM framework.
• Provide a robust theoretical spectral analysis guaranteeing correct eigenvalue approximation and absence of spurious modes.
• Validate theoretical results through comprehensive numerical experiments.

提案手法

• Formulate the Laplace eigenvalue problem and its standard lowest order VEM discretization.
• Construct the rbVEM space by approximating the non-polynomial VEM contributions with a reduced basis on a reference element.
• Define a new computable discrete space where bilinear forms are fully computable without stabilization terms.
• Prove optimal a priori error estimates for the source problem and spectral convergence for the eigenproblem via Babuška–Osborn theory.
• Demonstrate that rbVEM yields a fully conforming method that avoids stabilization terms and preserves the eigenstructure.

実験結果

リサーチクエスチョン

• RQ1 Can stabilization terms be completely avoided in VEM for eigenvalue problems without compromising accuracy?
• RQ2 Does the rbVEM framework provide optimal convergence for both eigenvalues and eigenfunctions and prevent spurious eigenvalues?
• RQ3 How does the reduced basis approximation of non-polynomial VEM contributions affect robustness across mesh types and problem parameters?
• RQ4 What are the theoretical and numerical implications of interpreting rbVEM as a conforming discretization equivalent to a stabilized VEM?
• RQ5 How does the rbVEM performance compare to standard VEM and rb-stab–VEM on benchmark domains?

主な発見

• rbVEM yields a fully conforming discretization where stabilization terms are not required.
• The non-polynomial VEM contribution is explicitly represented via a reduced basis, enabling computable bilinear forms.
• The method achieves correct spectral approximation with optimal convergence rates for the source and eigenvalue problems.
• Theoretical results are supported by extensive numerical experiments on polygonal meshes and various domains.
• rbVEM mitigates risk of spurious eigenvalues by avoiding stabilization parameters in the mass and stiffness bilinear forms.
• Convergence results hold under standard regularity assumptions and mesh families.