Stabilization-Free H(curl) and H(div)-Conforming Virtual Element Method

Technical Summary: Stabilization-Free $H(\text{curl})$ and $H(\text{div})$-Conforming Virtual Element Method

Problem Statement
The Virtual Element Method (VEM) offers significant advantages over the Finite Element Method (FEM) by handling general star-shaped polytopes, making it ideal for complex geometries and adaptive mesh refinement. However, standard VEM implementations for $H(\text{curl})$ and $H(\text{div})$ spaces require stabilization terms to ensure coercivity. This necessity arises because the virtual element space $V(K)$ is larger than the polynomial space $P_k(K)$, causing the standard $L^2$-projection to be unstable (i.e., $\|\Pi_k v\|_K \not\gtrsim \|v\|_K$). These stabilization terms introduce several drawbacks: they negatively impact numerical stability and condition numbers, require problem-specific design (particularly for nonlinear problems), and exhibit isotropic behavior that struggles with anisotropic coefficients or meshes. Existing stabilization-free approaches either require excessive internal degrees of freedom (DoFs) that increase computational costs or modify the projection space in ways that dilute VEM's geometric flexibility or fail to preserve essential properties like commutativity with differential operators.

Methodology
The authors propose a novel, stabilization-free VEM for general order $H(\text{curl})$ and $H(\text{div})$-conforming spaces in $\mathbb{R}^3$. The core of the methodology involves constructing novel serendipity projectors and corresponding serendipity spaces with a minimized number of DoFs.

1. Serendipity Projectors: Unlike previous works that rely on trivial projectors or internal DoFs, the authors define specific projectors ($\Pi^S_k$) for face, edge, and node spaces. These projectors map virtual functions to polynomial spaces using only "preserved" boundary DoFs and a specific set of internal moments. The projectors are defined via orthogonality conditions involving the normal/tangential traces and internal moments (e.g., $(v - \Pi^S_k v, x_K \times p_{kr})_K = 0$).
2. Serendipity Spaces: The authors define serendipity spaces ($V^S$) by constraining the standard virtual element spaces such that the "reduced" DoFs are determined by the preserved DoFs via the serendipity projector. This ensures that the $L^2$-projection onto a sufficiently high-order polynomial space $P_l$ is stable and computable using only the boundary DoFs.
3. De Rham Complex: The method is designed to handle the full De Rham complex chain in $\mathbb{R}^3$ ($H^1 \to H(\text{curl}) \to H(\text{div}) \to L^2$). The construction ensures exact commutativity with differential operators ($\nabla, \nabla\times, \nabla\cdot$), preserving the topological structure of the problem.
4. Stability Analysis: The authors prove that for a sufficiently large projection order $l$, the norm equivalence $\|\Pi_l v\|_K \approx \|v\|_K$ holds for functions in the serendipity space. This eliminates the need for stabilization terms while maintaining optimal approximation properties.

Key Contributions

• Complete De Rham Treatment: The work provides a unified treatment for the entire De Rham complex in 3D, ensuring seamless integration across different differential operators, a feature not present in previous stabilization-free works limited to $H^1$.
• Preservation of Mathematical Properties: The proposed serendipity spaces rigorously maintain boundary continuity, operator conformity, exact commutativity with differential operators, and B-compatibility.
• Computational Efficiency: By minimizing the number of DoFs and utilizing serendipity projections computed solely from boundary data, the method significantly reduces computational overhead. This is particularly beneficial for eigenvalue problems where internal DoFs cannot be eliminated via static condensation.
• Rigorous Error Analysis: The paper provides a theoretical proof of optimal approximation properties and stability for the high-order polynomial projections.

Results
The theoretical claims are validated through numerical experiments on Maxwell eigenvalue problems using Voronoi polyhedral meshes.

• Convergence Rates: The numerical results confirm the theoretical convergence rates. For infinitely regular solutions, the method achieves $O(h^{m+1})$ convergence in the $L^2$-norm and $O(h^{2m+2})$ for eigenvalues. For singular solutions (where regularity is limited by $s$), the rates are $O(h^s)$ and $O(h^{2s})$, respectively.
• Computational Cost: The assembly time for the proposed Stabilization-Free VEM (SFVEM) is shown to be only moderately larger than conventional FEM, verifying the method's efficiency. The required projection order $l$ is found to be close to the polynomial order $k$ in practice.
• Comparison: The method successfully eliminates stabilization-related complications while matching the accuracy of conventional FEM and stabilized VEM.

Significance and Claims
The authors claim that this work offers a fundamentally different approach to stabilization-free VEM that overcomes the limitations of existing strategies (excessive DoFs or loss of geometric flexibility). By eliminating stabilization terms, the method improves numerical stability and condition numbers, making it particularly valuable for:

• Mixed formulations.
• Nonlinear problems.
• Applications requiring robust numerical behavior across diverse mesh configurations.

The paper concludes that while the technique can theoretically be generalized to higher dimensions, practical implementation complexities and the rapid growth of boundary DoFs currently limit the discussion to the 3D setting. The proposed method represents a theoretically rigorous and computationally efficient alternative for solving partial differential equations, advancing the applicability of VEM in computational mathematics and engineering.