Virtual Element Approximations in PDEs

• Virtual element approximations are a Galerkin-type method using implicit basis functions on arbitrary polygonal and polyhedral meshes.
• They decompose the space into computable polynomial components and non-explicit parts controlled via projection and moment-based degrees of freedom.
• Recent innovations, including lightning and neural approximations, enhance basis evaluation and reduce stabilization requirements for improved convergence.

Virtual element approximations refer to a broad class of Galerkin-type discretization techniques for partial differential equations (PDEs), which generalize the finite element method (FEM) to general polygonal and polyhedral meshes. The defining feature is the use of implicit (“virtual”) local shape functions—enforcing polynomial consistency and stability through projection and moment-based degrees of freedom—without requiring explicit analytic form for all basis functions. This paradigm supports arbitrary mesh geometries, high global regularity, flexible polynomial degree, and facilitates the solution of both conforming and nonconforming, primal and mixed variational problems. Recent innovations include computational approaches to the virtual space using rational (“lightning”) and neural approximations, which further relax analytic constraints on basis construction and stabilization.

1. Fundamental Construction of Virtual Element Spaces

The virtual element approach constructs, for each element $K$ in a shape-regular polygonal (2D) or polyhedral (3D) tessellation, a finite-dimensional local space $V_h^k(K)$ containing $\mathbb P_k(K)$ and suitably defined non-polynomial functions. The general template for second-order PDEs is

$$V_h^k(K) = \{ v_h \in H^1(K) : \Delta v_h \in \mathbb P_{k-2}(K),\ v_h|_e \in \mathbb P_k(e)\ \forall e \subset \partial K\},$$

with degrees of freedom consisting of vertex values, edge moments, and (for $k \geq 2$) internal moments against monomials up to degree $k-2$ (Trezzi et al., 2023). On each global mesh, the local objects are then glued together with appropriate (strong or weak) interelement continuity constraints to obtain the entire virtual element space.

This construction generalizes:

• To higher regularity ($H^m$-conforming, $C^{p}$-globally), by incorporating higher-order normal derivatives and multi-derivative vertex jets into the degrees of freedom (Antonietti et al., 2021, Antonietti et al., 2021).
• To non-conforming settings, by imposing inter-element continuity only in mean moments over faces/edges (Dios et al., 2014, Cangiani et al., 2016).
• To diverse function spaces (e.g., $H(\text{div})$, $H(\text{curl})$), via corresponding elementwise constraints and dof design (Veiga et al., 2016, Dassi et al., 2022).
• To domains with curved boundaries and interfaces, by extending the definition of the trace spaces and associated moments to curved edges or faces (Prada et al., 26 Sep 2025, Brezzi et al., 2023).

2. Polynomial and Virtual Decomposition: Projections and Stabilization

A cornerstone of the VEM is the polynomial/virtual decomposition—the splitting of any $v_h \in V_h^k(K)$ as

$v_h = \Pi_k^\nabla v_h + (I - \Pi_k^\nabla) v_h,$

where $\Pi_k^\nabla$ is the $H^1$-orthogonal projection onto $\mathbb P_k(K)$, determined through element boundary and volume moments (Trezzi et al., 2023, Dios et al., 2014). The polynomial part is computable explicitly; the “virtual part” is not known analytically but is controlled and accessed through the dofs.

The discrete bilinear forms central to VEM are constructed as

$$a_h^K(u_h, v_h) = a^K(\Pi_k^\nabla u_h, \Pi_k^\nabla v_h) + S^K((I-\Pi_k^\nabla) u_h, (I-\Pi_k^\nabla) v_h),$$

where $S^K$ is a computable, symmetric, positive semidefinite stabilization acting on the “virtual” kernel and scales like the energy of the problem. “Dofi–dofi” (sum of squared dofs) and $D^{\perp}$-type stabilizations are standard and yield similar accuracy and optimal convergence; the choice affects only condition numbers and pre-asymptotic robustness (Antonietti et al., 2021, Credali et al., 2023).

Polynomial projection, L$^2$ projections, and, in higher/mixed regularity cases, projections in $H^m$, $H(\text{div})$, or $H(\text{curl})$ play crucial roles in enabling all necessary scalar products using only accessible dofs, circumventing the need for explicit basis function evaluation on the element interior (Dios et al., 2014, Dassi et al., 2018).

3. Recent Advances: Lightning and Neural Virtual Elements

Two modern directions address the core challenge of basis evaluation and stabilization for VEM:

Lightning VEM replaces the unknown, non-polynomial “virtual” component of the basis with a rational function expansion, where poles are exponentially clustered near the element corners. Specifically, the virtual part is approximated as

$$\widehat\phi_i(z) = \Re\left\{\sum_{j=1}^{N_P} \frac{a_j}{z - z_j} + \sum_{m=0}^{N_Z} b_m (z - z_*)^m \right\}$$

with coefficients fit by least squares to enforce boundary dof constraints up to relative tolerance $\epsilon$ in $H^{1/2}(\partial K)$. This approach: (i) allows direct pointwise evaluation of virtual basis functions anywhere in the element; (ii) renders the full energy bilinear form computable exactly, thus eliminating the need for any stabilization term; (iii) enables a Strang-type error analysis in which the error reduces to the sum of best approximation and $O(\epsilon)$ jump terms, and yields optimal convergence rates as in FEM or standard VEM for $\epsilon \ll h^k$ (Trezzi et al., 2023).

Neural Approximated VEM (NAVEM) uses neural networks (MLPs) to parameterize the virtual