Avoiding stabilization terms in virtual elements for eigenvalue problems: The Reduced Basis Virtual Element Method

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.

💡 Research Summary

The paper introduces a new numerical method, the Reduced‑Basis Virtual Element Method (rbVEM), for solving the Laplace eigenvalue problem on polygonal domains. Classical Virtual Element Methods (VEM) are attractive because they allow arbitrary polygonal meshes and require only the degrees of freedom at element vertices. However, VEM splits the local discrete space into a polynomial part and a non‑polynomial (virtual) part. The polynomial part can be handled exactly, while the virtual part is not known analytically and is usually stabilized by adding artificial bilinear forms S_E^a and S_E^b. These stabilization terms are only required to scale with the stiffness and mass contributions of the virtual component, but their choice is delicate. In eigenvalue problems the mass stabilization can generate spurious eigenvalues, leading to polluted spectra and parameter‑dependent results.

The authors propose to eliminate the need for any stabilization by explicitly approximating the virtual component using a reduced‑basis (RB) technique. For each polygon E with vertices v₁,…,v_N, the VEM basis functions satisfy a local PDE. By treating the vertex coordinates as geometric parameters, the family of local problems becomes a parametric PDE defined on a reference element. The RB method builds a low‑dimensional subspace spanned by a few “snapshot” solutions of this parametric problem. Any virtual function can then be expressed as a linear combination of the RB basis functions, and the corresponding contributions to the stiffness and mass matrices are computed exactly, without any artificial term.

With this construction the discrete bilinear forms become a_h(u_h,v_h)= (∇Π^∇u_h,∇Π^∇v_h)_E + (∇(u_h−Π^∇u_h),∇(v_h−Π^∇v_h))_E, b_h(u_h,v_h)= (Π^0u_h,Π^0v_h)_E + ((u_h−Π^0u_h),(v_h−Π^0v_h))_E, where Π^∇ and Π^0 are the usual elliptic and L² projections onto the polynomial space. The second terms are now fully computable thanks to the RB approximation of the virtual part, so the method is fully conforming and no stabilization is required.

The theoretical analysis proceeds in two steps. First, for the associated source problem the authors prove optimal H¹‑error O(h) and L²‑error O(h²) estimates, showing that the RB approximation error ε_RB can be made arbitrarily small by enriching the RB space. Second, they apply the Babuška‑Osborn framework to the eigenvalue problem, establishing that the discrete eigenvalues λ_h converge to the exact λ with order O(h²) and the eigenfunctions converge in H¹ with order O(h). Crucially, because the method does not rely on any stabilization, no spurious eigenvalues appear; the discrete spectrum is a faithful approximation of the continuous one.

A comprehensive set of numerical experiments validates the theory. The authors test rbVEM on uniform square meshes, highly irregular polygonal meshes, L‑shaped domains, and domains with internal holes. They compare against the standard VEM with both stiffness and mass stabilizations. Results show that rbVEM consistently achieves lower relative errors (often by two to four orders of magnitude) and reproduces higher‑frequency eigenvalues accurately, where the standard VEM suffers from eigenvalue clustering or artificial modes. The offline cost of building the RB space is modest and performed only once; the online assembly of the global matrices is comparable to standard VEM, making the approach attractive for repeated solves, parametric studies, or real‑time applications.

In conclusion, the Reduced‑Basis Virtual Element Method provides a stabilization‑free, fully conforming discretization for eigenvalue problems on arbitrary polygonal meshes. It retains the geometric flexibility of VEM while leveraging the efficiency of reduced‑basis approximations to handle the non‑polynomial part exactly. The method delivers optimal convergence rates, eliminates spurious spectral pollution, and offers computational savings for multi‑parameter or real‑time scenarios. Future work suggested includes extensions to higher‑order VEM, three‑dimensional polyhedral meshes, nonlinear eigenvalue problems, and adaptive mesh refinement combined with RB enrichment.