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

TL;DR

The paper addresses spurious eigenmodes and parameter-sensitive stabilization in virtual element methods for the Laplace eigenvalue problem. It introduces the Reduced Basis Virtual Element Method (rbVEM), which computes a reduced-basis representation of the non-polynomial VEM component on each element to obtain a fully conforming discretization without stabilization terms. The authors prove optimal a priori error bounds for the source problem, establish correct spectral approximation via the Babuška–Osborn framework, and demonstrate robust, stabilization-free eigenvalue convergence across standard benchmarks (unit square, L-shaped domain, and piecewise-constant coefficients). Numerical experiments confirm that rbVEM yields stable eigenvalues independent of the reduced-basis size and matches the best convergence rates, while avoiding spurious modes characteristic of stabilized VEM. The work thus offers a practical, parameter-free, high-fidelity polygonal discretization for eigenvalue problems with potential impact on simulations using complex geometries and anisotropic materials.

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.

Figures (10)

• Figure 1: Parameters dependence of the eigenvalues.
• Figure 2: An example of affine mapping $\mathscr{L}_E$ between a star shaped pentagon and its reference configuration.
• Figure 3: The robustness of rbVEM with respect to the number of reduced basis functions $M$. In the top line we depict the three different considered meshes, whereas in the bottom line we plot the computed eigenvalues.
• Figure 4: Comparison between VEM (first and second line) rbVEM (third line) and rb--stab--VEM (fourth and fifth line) on a sequence of Voronoi meshes. First column: plots of the first 20 computed and exact eigenvalues. Second column: error plots for the first 220 eigenvalues with respect to their index. Third column: error plots for the first 10 eigenvalues when refining the mesh size.
• Figure 5: Comparison between VEM (first and second line) rbVEM (third line) and rb--stab--VEM (fourth and fifth line) on a sequence of dyadic meshes. First column: plots of the first 20 computed and exact eigenvalues. Second column: error plots for the first 220 eigenvalues with respect to their index. Third column: error plots for the first 10 eigenvalues when refining the mesh size.

Theorems & Definitions (12)

• Remark 1
• Proposition 1
• proof
• Theorem 1
• Theorem 2
• proof
• Remark 2
• Theorem 3
• Lemma 1
• proof