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Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

This paper demonstrates that for elliptic self-adjoint eigenvalue problems, stabilization of the mass matrix is unnecessary when using lower-order standard Virtual Element Method spaces, with numerical evidence suggesting this finding extends to higher-order schemes as well.

Original authors: Daniele Boffi, Francesca Gardini, Lucia Gastaldi

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniele Boffi, Francesca Gardini, Lucia Gastaldi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to tune a giant, complex musical instrument made of thousands of tiny, irregularly shaped pieces (like a puzzle with triangles, squares, and weird blobs). Your goal is to find the specific notes (eigenvalues) this instrument can play. In the world of mathematics and engineering, this is called an eigenvalue problem.

The paper you asked about is written by three mathematicians (Daniele Boffi, Francesca Gardini, and Lucia Gastaldi) who are experts in a technique called the Virtual Element Method (VEM). This method is a super-flexible way to solve these problems on computer models, even when the shapes are messy and irregular.

Here is the core story of their paper, explained simply:

The Problem: The "Ghost Notes"

When you try to calculate these musical notes on a computer, you have to build a mathematical model. Usually, this model involves two main parts:

The Stiffness (The String): How hard it is to bend the material.
The Mass (The Weight): How heavy the material is.

In the VEM method, calculating the "Mass" part is tricky. To make the computer work, mathematicians usually add a "safety net" called stabilization. Think of this like adding a little bit of glue or a shock absorber to keep the system from wobbling apart.

However, there's a catch. If you tune this "glue" (the stabilization parameter) wrong, you don't just get the right notes; you start hearing ghost notes (spurious eigenvalues). These are fake sounds that don't exist in reality.

If you tune the glue for the "Stiffness" wrong, the ghost notes hide at the very top of the scale (high frequencies), which is annoying but usually harmless.
If you tune the glue for the "Mass" (the right-hand side) wrong, the ghost notes sneak down to the bottom of the scale. This is a disaster because the "bottom notes" are usually the most important ones (like the fundamental tone of a guitar string).

The Big Question

For years, everyone assumed you had to use this glue (stabilization) for the Mass part to prevent the system from falling apart. The standard rule was: "Add glue to both sides, or the math breaks."

The authors of this paper asked a bold question: "Is the glue actually necessary for the Mass part, or are we just being overly cautious?"

The Discovery: "The Glue Wasn't Needed!"

They proved mathematically that for low-order calculations (simple, basic approximations), you do not need to add that extra glue to the Mass part.

The Analogy:
Imagine you are building a house of cards.

The Old Way: Everyone thought you needed to tape the cards together (stabilization) to keep them from falling. If you didn't tape them, the house would collapse, or you'd get weird, unstable structures.
The New Discovery: The authors found that if you build the house using a specific, sturdy design (standard VEM spaces with low complexity), the cards naturally hold themselves up! You don't need the tape. Even better, if you don't use the tape, you avoid the risk of accidentally taping a "ghost card" into the structure that messes up your final design.

What They Found

For Simple Shapes (Low Order): They proved that if you use simple polynomial degrees (k=1 or k=2), the math works perfectly without the "Mass glue." The computer finds the right notes, and the "ghost notes" don't appear, even though the mathematical matrix (the list of numbers) technically has some "holes" (a non-trivial kernel) that used to scare people.
For Complex Shapes (High Order): They ran computer simulations with more complex shapes (k=3, k=4) and on all sorts of weird meshes (triangles, squares, honeycombs, Voronoi patterns). Even though they couldn't prove it with a simple formula for these complex cases, the computer results showed the same thing: It still works without the glue! The method converges to the right answer, and no ghost notes appear.

Why This Matters

This is a big deal for engineers and scientists because:

Simplicity: You don't have to spend hours "tuning" a mysterious parameter to find the perfect amount of glue.
Safety: You avoid the risk of accidentally creating "ghost notes" that ruin your simulation of a bridge, a plane wing, or a medical scan.
Efficiency: It makes the code cleaner and potentially faster.

The Bottom Line

The paper is like a mechanic telling you: "You've been carrying a heavy toolkit of extra stabilizers in your trunk for years, thinking your car needs them to run. But actually, if you drive the car normally (using standard settings), it runs perfectly fine without them, and you'll get better mileage (better accuracy) because you aren't dragging that extra weight around."

They showed that for many common problems, the "stabilization" of the right-hand side is unnecessary, and removing it leads to cleaner, more reliable results.

1. Problem Statement

The paper addresses the Virtual Element Method (VEM) approximation of elliptic self-adjoint eigenvalue problems. In standard VEM formulations, the bilinear forms associated with the stiffness matrix ( $A$ , derived from $a(\cdot, \cdot)$ ) and the mass matrix ( $B$ , derived from $b(\cdot, \cdot)$ ) are often stabilized to ensure positive definiteness and avoid singularities.

The Core Issue: While stabilization of the stiffness matrix ( $A$ ) is widely accepted as necessary, the stabilization of the mass matrix ( $B$ ) is more controversial. The authors note that increasing the stabilization parameter for $B$ can shift spurious eigenvalues toward the lower part of the spectrum, potentially polluting the fundamental modes of interest.
The Question: Is it strictly necessary to stabilize the right-hand side matrix $B$ (the mass matrix) when using standard VEM spaces to approximate eigenvalues?
Context: Previous works suggested that without stabilization, $B$ might have a non-trivial kernel, leading to infinite eigenvalues or spurious modes. This paper investigates whether convergence can be achieved without stabilizing $B$ , particularly for lower-order schemes.

2. Methodology

2.1 Abstract Framework

The authors utilize the Descloux–Nassif–Rappaz (DNR) theory for the convergence analysis of non-compact operators. This framework is chosen because standard Galerkin analysis is difficult to apply directly to VEM due to the difficulty of defining a discrete solution operator for all functions in $L^2(\Omega)$ .

The analysis relies on two key properties for the discrete solution operator $T_h$ approximating the continuous operator $T$ :

Property P1 (Operator Consistency): $\|(T - T_h) f_h\|_V \to 0$ as $h \to 0$ for $f_h$ in a specific subspace.
Property P2 (Approximability): The ability to approximate the continuous solution space within the discrete space $K_b^\perp$ (the orthogonal complement of the kernel of $B$ ).

2.2 VEM Formulation

Problem: The Laplace eigenvalue problem $-\Delta u = \lambda u$ with Dirichlet boundary conditions.
Discretization:
- Stiffness Matrix ( $A$ ): Stabilized using a standard term $S_E$ (e.g., "doF-doF" stabilization) to ensure coercivity.
- Mass Matrix ( $B$ ): Not stabilized. It is constructed purely using the $L^2$ -projection $\Pi^0_k$ onto polynomial spaces: $b_h(u_h, v_h) = \sum_E (\Pi^0_k u_h, \Pi^0_k v_h)_E$ .
Kernel Handling: Since $B$ is not stabilized, it may possess a non-trivial kernel $K_{b,h} = \{v_h \in V_h : \Pi^0_k v_h = 0\}$ . The authors restrict the analysis to the subspace $K_{b,h}^\perp$ (orthogonal with respect to the stiffness form $a_h$ ) to avoid infinite eigenvalues.

2.3 Theoretical Proofs

The authors prove that Properties P1 and P2 hold even without stabilizing $B$ , provided the polynomial degree $k$ is low ( $k=1, 2$ ).

Key Insight for $k=1, 2$ : For these degrees, the gradient projection $\Pi^\nabla_k$ and the $L^2$ projection $\Pi^0_k$ coincide on the virtual element space. This property allows the authors to bound the error terms involving the unstabilized mass matrix, proving that the kernel elements do not interfere with the convergence of the physical eigenmodes.
Assumptions: The proof assumes standard mesh regularity (star-shaped elements, bounded aspect ratios) and relies on standard approximation properties of the virtual element space.

3. Key Contributions

Theoretical Proof for Low Orders: The paper provides a rigorous proof that for standard VEM spaces of degree $k=1$ and $k=2$ , the stabilization of the mass matrix $B$ is not necessary. The discrete eigenvalues converge to the exact eigenvalues even if $B$ has a non-trivial kernel.
Resolution of Kernel Issues: The authors demonstrate that the presence of a kernel in $B$ does not introduce spurious eigenvalues in the window of interest, provided the analysis is restricted to the appropriate subspace $K_{b,h}^\perp$ .
Extension to Higher Orders (Numerical): While the theory is proven for $k=1, 2$ , the numerical experiments suggest that the result holds for higher orders ( $k=3, 4$ ) as well, despite the lack of a theoretical proof for these cases.

4. Numerical Results

The authors conducted extensive numerical tests on the unit square using four different mesh types:

T: Triangular meshes.
S: Square meshes.
V: Voronoi meshes (irregular polygons).
H: Hexagonal meshes.
D: Dyadic meshes (a specific non-standard mesh).

Findings:

Kernel Dimensions: Tables 1–21 show that for $k=1$ and $k=2$ , the dimension of the kernel of $B$ is often zero (especially for T, S, V, and H meshes). For $k \ge 3$ , non-trivial kernels appear for many mesh types (e.g., Voronoi and Hexagonal).
Convergence Rates:
- For $k=1$ and $k=2$ , the computed eigenvalues converge with optimal rates ( $O(h^2)$ for $k=1$ , $O(h^4)$ for $k=2$ ) across all mesh types, even when the kernel is non-trivial.
- For $k=3$ and $k=4$ , despite the presence of non-trivial kernels and the lack of theoretical proof, the numerical results show optimal convergence rates and no spurious eigenvalues in the spectrum.
Robustness: The method remains stable and accurate across various polygonal decompositions, confirming that the "tuning" of stabilization parameters for the mass matrix is unnecessary for standard VEM spaces.

5. Significance

Simplification of Implementation: This work removes the need for users to tune stabilization parameters for the mass matrix $B$ . Since improper tuning of $B$ 's stabilization can degrade accuracy (by polluting low-frequency modes), eliminating this requirement simplifies the VEM implementation for eigenvalue problems.
Theoretical Justification: It resolves a long-standing ambiguity in the VEM community regarding the necessity of mass matrix stabilization, proving that for the most common low-order cases, the standard projection-based formulation is sufficient.
Robustness on Polygonal Meshes: The results confirm that VEM retains its robustness on complex polygonal meshes (Voronoi, Hexagonal) without the need for ad-hoc stabilization of the $L^2$ inner product.

In conclusion, the paper establishes that stabilization of the right-hand side matrix is not necessary for the VEM approximation of elliptic eigenvalue problems when using standard spaces of degree $k=1$ and $k=2$ , and numerical evidence strongly suggests this holds for higher orders as well.