Stabilization-Free Virtual Element Method (SFVEM)

• SFVEM is a numerical method that discretizes PDEs on arbitrary polygonal and polyhedral meshes by systematically eliminating traditional stabilization terms.
• It leverages intrinsic polynomial approximations and enhanced projection operators, including serendipity constructions, to guarantee coercivity, rank-sufficiency, and optimal convergence.
• SFVEM demonstrates robust performance in applications such as diffusion, elasticity, and multiphysics simulations while enabling precise error estimation and adaptive refinement.

A Stabilization-Free Virtual Element Method (SFVEM) is a class of numerical discretization techniques for partial differential equations (PDEs) on general polygonal and polyhedral meshes, characterized by the elimination—or systematic reduction—of classical stabilization terms from the formulation. These methods rely on the intrinsic polynomial approximation and enhanced projection operators of the virtual element space to attain coercivity, rank-sufficiency, and convergence without relying on ad hoc remedy terms. Modern developments encompass elliptic, mixed, elasticity, H(curl), H(div) and multiphysics formulations, with broad applications from diffusion to thermomechanical analysis on arbitrarily complex, possibly nonconforming, multiscale meshes.

1. Foundations and Principle of Stabilization-Free VEM

SFVEM designs depart from the standard VEM paradigm where the local bilinear form a_E^h(·,·) is split into a “polynomial-consistent” part and a stabilization term:

$$a_E^h(v, w) = a_E(\Pi_E v, \Pi_E w) + s_E\bigl((I-\Pi_E)v, (I-\Pi_E)w\bigr),$$

with $\Pi_E$ a computable projection onto the polynomial subspace. Classical $s_E(\cdot, \cdot)$ contributions are required because the full set of virtual basis functions is not polynomial and only their boundary values are available. However, several theoretical studies (Veiga et al., 2016, Berrone et al., 2021, Berrone et al., 2023, Xu et al., 2023, Lin et al., 2023) establish that, by carefully choosing the enhanced polynomial projection (for instance, increasing the projection degree or constructing appropriate serendipity subspaces), one can guarantee rank-sufficiency and coercivity directly via the projection; $s_E$ may be restricted to only boundary DoFs or dropped entirely.

SFVEM thus exploits the following principles:

• The discrete bilinear form is defined entirely through polynomial projections, e.g., for scalar diffusion or elasticity,

$$a_E^h(u_h, v_h) = \bigl(\Pi_\ell^E \nabla u_h, \Pi_\ell^E \nabla v_h\bigr)_E,$$

with the order $\ell$ linked to the number of vertices (or mesh topology), ensuring injectivity and stability.

• The virtual space is enlarged or "trimmed" (e.g., by a serendipity construction, (Liao et al., 25 Jan 2025)), so that the projection is computable and the degrees of freedom suffice for unique polynomial recovery.
• The stabilization term is either omitted completely, or reduced to utilize only boundary (as opposed to internal) DoFs; see (Veiga et al., 2016) for rigorous stability proofs under this scenario.

2. Mathematical Structure and Error Analysis

The stability of SFVEM is underpinned by coercivity and consistency of the projected forms. For any element $E$, one establishes the existence of constants $C_* > 0, C^* < \infty$ such that (Veiga et al., 2016, Berrone et al., 2021, Berrone et al., 2023):

$$C_* |v_h|_E^2 \le a_E^h(v_h, v_h) \le C^* |v_h|_E^2, \quad \forall v_h \in V_h|_E,$$

where the norm $$|v|_E^2 := s_E((I-\Pi_E)v, (I-\Pi_E)v) + a_E(\Pi_E v, \Pi_E v)$$ (with $s_E$ possibly set to zero).

To ensure injectivity of the projection, SFVEM requires a sufficiently rich polynomial subspace. For E$^2$VEM and related approaches, the minimal polynomial degree $\ell$ for the L²-type projection $\Pi_\ell^E$ must satisfy tailored element-dependent rank conditions, e.g.,

$$(\ell + 1)(\ell + 2) - \mathrm{dim}(\mathrm{BadPoly}_\ell^E) \geq N_E - 1,$$

where $N_E$ is the number of vertices and "BadPoly" measures bad polynomial directions lost in the projection (Berrone et al., 2021).

Optimal a priori error estimates are obtained:

• For scalar problems, if $u \in H^{k+1}(\Omega)$, then

$\|u - u_h\|_1 \le C h^{k} |u|_{k+1},$

and

$\|u - u_h\|_0 \le C h^{k+1} |u|_{k+1},$

where $k$ is the approximation order (Lin et al., 2023, Xu et al., 2023).

For adaptive SFVEM, a posteriori error estimators are available and exhibit reliability and efficiency constants robust across mesh types and strong coefficient jumps (Canuto et al., 2023, Berrone et al., 22 Jun 2025). The absence of stabilization facilitates an exact equivalence between the error norm and residual estimators:

$$C_1 \sum_E [\eta_E^2 + \mathcal{F}_E^2] \le \lVert u - u_h \rVert^2_{\mathcal{E}} \le C_2 \sum_E [\eta_E^2 + \mathcal{F}_E^2],$$

with $\eta_E$ based on volume residuals and flux jumps, and $\mathcal{F}_E$ quantifying data oscillation (Berrone et al., 22 Jun 2025).

3. Algorithmic Realizations and Projection Operators

SFVEM schemes use advanced projection operators:

• Polynomial projections: $\Pi_\ell^E$ maps the gradient or strain to $[\mathbb{P}_\ell(E)]^d$, computed via local inner products and the element's degrees of freedom.
• Harmonic projections: e.g., (Berrone et al., 2023, Borio et al., 2023), which project onto the gradient of harmonic polynomials, leveraging Green's identities to reduce volume integrals to boundary terms.

On general polygons and polyhedra, the projection may be designed using macro-triangle or macro-tetrahedron subdivisions (e.g., Hsieh–Clough–Tocher splits (Xu et al., 2023, Lin et al., 2023)), ensuring that the virtual space is a continuous $P_k$ on each sub-simplex and that all required projection moments are available.

For vector-valued problems (e.g., plane elasticity (Chen et al., 2022), H(curl), H(div) (Liao et al., 25 Jan 2025)), the enhanced space is constructed so that the energy projection (with respect to the elasticity or Maxwell bilinear form) is exactly computable. The resulting spaces fit into a discrete de Rham sequence for full conformity and commutativity (e.g., $\nabla \circ V^n \subset V^e$, $\nabla \times \circ V^e \subset V^f$).

A representative construction table is:

Element Type	Space Construction	Projection Degree
Scalar, polygonal	Macro-triangle $P_k$-interpolation	$\ell$ from vertex-count
H(div), quadrilateral/polyhedral	Continuous $P_k$ on sub-triangle/tetrahedra	$\ell$ from face/edge-count
H(curl), general	Serendipity with L$^2$-projection	Satisfy exact sequence

The careful selection of the projection order and enriched basis is crucial for stability and convergence.

4. Applications and Numerical Evidence

SFVEM principally serves:

• Second order elliptic problems (Poisson, diffusion, anisotropic diffusion) (Berrone et al., 2021, Berrone et al., 2022, Berrone et al., 2023)
• Plane elasticity (linear and higher-order) (Chen et al., 2022, Chen et al., 2022)
• Acoustic vibration eigenproblems (Alzaben et al., 9 Jan 2024)
• Mixed and H(div)/H(curl) conforming problems (Borio et al., 2023, Liao et al., 25 Jan 2025)
• Thermomechanical multi-field simulations with complex, multiscale, non-matching meshes (Gong et al., 15 Aug 2025)

Empirical validation on convex, non-convex, Voronoi, star-concave, highly distorted, and nonconforming meshes demonstrates:

• Robust stability for arbitrary mesh geometries, even with arbitrarily small edges (Veiga et al., 2016).
• Elimination of spurious oscillations close to geometric singularities when moving from classical to derivative-based (or purely projection-based) stabilization (Veiga et al., 2016).
• Quasi-optimal or superconvergent rates for higher order approximations, and improved conditioning (smaller system condition numbers) due to the full use of all degrees of freedom (Xu et al., 2023, Lin et al., 2023).

In electronic packaging thermomechanics (Gong et al., 15 Aug 2025), SFVEM, together with non-matching polygonal mesh generation, achieves accurate stress and temperature predictions at material interfaces for structures with Through-Silicon Vias and Ball Grid Arrays, validated against analytical and standard FEM solutions.

5. Advantages and Implementation Considerations

The principal advantages of SFVEM, systematically demonstrated across the literature, are:

• Absence of arbitrary stabilization parameters, streamlining implementation and removing parameter-tuning from the solution process (Berrone et al., 2021, Gong et al., 15 Aug 2025).
• Improved error and convergence rates in anisotropic (and multi-field) problems, as stabilization terms in standard VEM can introduce isotropic error components or cause error propagation between fields (Berrone et al., 2022, Gong et al., 15 Aug 2025).
• Enhanced conditioning of the linear system, especially for higher degree spaces and extreme mesh distortion (Xu et al., 2023, Lin et al., 2023).
• Clean a posteriori error analysis and adaptive refinement: residual-type indicators are robust, as no additional stabilization penalty obscures the error-residual relationship (Canuto et al., 2023, Berrone et al., 22 Jun 2025).
• Flexibility for arbitrary polytopal, nonconforming, and multi-scale mesh generation, enabling non-matching meshes and efficient local refinement (Gong et al., 15 Aug 2025).

Potential challenges include:

• For some mesh types and higher-order elements, care must be taken to choose the projection degree high enough to preserve coercivity and avoid kernel degeneracy (Berrone et al., 2021, Berrone et al., 2023).
• Computation of higher-order or harmonic polynomial projections can introduce additional local algebraic solves per element, but this cost is typically offset by improved accuracy and stability.

6. Extensions, Variants, and Future Directions

Advanced SFVEM variants include:

• Serendipity SFVEM for H(curl) and H(div) problems using L²-serendipity projectors and reduced DoF spaces (Liao et al., 25 Jan 2025), ensuring the exactness of discrete de Rham complexes and strong approximation properties.
• The “lightning” VEM, which directly computes the virtual component using rational function approximations, fully eliminating the stabilization and permitting pointwise evaluation (Trezzi et al., 2023).
• Reduced-basis approaches for post-processing and stabilization design, approximating the virtual part of the basis and providing H$^1$-conforming reconstructions (Credali et al., 2023).
• Stabilization-free adaptive methods with provable contraction, facilitating complex, high-accuracy, and computationally efficient adaptive loops (Canuto et al., 2023).

Future research directions include extending these methods to:

• Nonlinear and time-dependent PDEs
• Coupled multiphysics and evolution equations
• High-order, three-dimensional applications, and mesh adaptivity in difficult geometric settings.

7. Comparative Impact and Significance

The stabilization-free paradigm represents a substantial advancement in the theory and practice of virtual element discretizations. By leveraging enhanced projections, serendipity reductions, and explicit basis reconstructions, SFVEMs overcome fundamental limitations of classic VEM—arbitrariness in artificial stabilization, parameter dependence, and error contamination. They open the way to robust, highly flexible, parameter-free discretization on general polytopal meshes, with demonstrated benefits across model problems and in complex, real-world engineering and scientific applications.