Stable Discrete Projection Operators

• Stable discrete projection operators are well-posed, norm-bounded projectors on finite element spaces that preserve key properties such as commutation with differential operators and boundary conditions.
• They are constructed using local weight functions on refined Alfeld splits, ensuring uniform L2-stability and reliable approximation in numerical PDE discretizations.
• Their design supports practical applications including multigrid preconditioning, parallel implementations, and robust finite element exterior calculus-based analyses.

A stable discrete projection operator is a mathematically well-posed, norm-bounded projector—typically local—acting on discrete (often finite element) function spaces, designed to preserve key structural properties such as commutation with differential operators, locality, and prescribed boundary conditions. These operators play an essential role in numerical analysis, finite element exterior calculus (FEEC), multigrid preconditioning, and the construction of discretizations that mirror topological and analytic features of continuous partial differential equation (PDE) complexes.

1. Mesh Preliminaries, Alfeld Splits, and Discrete Complexes

Discrete projection operators are defined over structured meshes. Let $\Omega\subset\mathbb{R}^3$ be a polyhedral Lipschitz domain, subdivided into a shape-regular simplicial mesh $\mathcal{T}_h$. For each simplex $\sigma\in\Delta_h^l$ ($l=0,1,2,3$), the extended star $\es(\sigma)$ comprises the union of all tetrahedra sharing at least one vertex with $\sigma$. The Alfeld split of each tetrahedron introduces a barycenter connected to each vertex, yielding a refined local mesh $\mathcal{T}_h^a$.

The family of standard finite element subspaces includes:

• Lagrange $V_p^0(\mathcal{T}_h)$, Nédélec $\bV_p^1(\mathcal{T}_h)$, Raviart–Thomas $\bV_p^2(\mathcal{T}_h)$, and discontinuous $V_p^3(\mathcal{T}_h)$ spaces.
• These form a de Rham subcomplex:

$\mathbb{R}\subset V_p^0(\mathcal{T}_h)\xrightarrow{\nabla} \bV_p^1(\mathcal{T}_h)\xrightarrow{\mathrm{curl}} \bV_p^2(\mathcal{T}_h)\xrightarrow{\mathrm{div}} V_p^3(\mathcal{T}_h)\to 0$

as required for robust discretizations in computational PDEs (Ern et al., 5 Feb 2025).

2. Local Weight Functions and Discrete Construction

The stable discrete projection is enabled by the construction of local weight functions $\ZZ_p^l(\sigma)$, built on the Alfeld split $\mathcal{T}_h^a$ and supported within $\es(\sigma)$. Each $\ZZ_p^l(\sigma)\in\mVp{3-l}(\mathcal{T}^a_{\es(\sigma)})$ satisfies, for all test functions $v \in V_p^l(\mathcal{T}_h)$, the moment reproduction property:

$\int_{\es(\sigma)} \ZZ_p^l(\sigma)\,\cdot\,v = \phi_{\sigma}(v)$

where $\phi_{\sigma}$ denotes canonical degrees of freedom (e.g., vertex evaluation, edge integrals).

The key technical properties are:

• Support: $\supp\ZZ_p^l(\sigma)\subset\overline{\es(\sigma)}$
• Norm control: $\|\ZZ_p^l(\sigma)\|_{L^2(\es(\sigma))}\leq C_Zh_\sigma^{-3/2+l}$
• Cochain commutation relations: e.g., for edges, $-\mathrm{div}\,\ZZ_p^1(e) = \sum_{v\in e}\iota_{ev}\,\ZZ_p^0(v)$, enforcing structural compatibility.

Explicit constructions are given for each simplex dimension, involving local Poisson, curl, and divergence solutions with polynomial weights (Ern et al., 5 Feb 2025).

3. Discrete Poincaré Inequalities and Stability Analysis

To guarantee uniform $L^2$-stability, a set of discrete Poincaré inequalities on extended stars is established. For every $\sigma\in\Delta_h^l$ and $u\in\Zz^{\perp}V_p^l(\mathcal{T}_\omega)$ ($\omega = \es(\sigma)$), the following holds:

$$\|u\|_{L^2(\omega)} \leq C_P\,h_\sigma\,\|d^l u\|_{L^2(\omega)}$$

where $d^l$ is the relevant differential operator, $C_P$ depends only on shape-regularity and $p$, and this result is robust under affine mappings and standard finite element reference-geometry arguments. The $L^2$-boundedness of the overall projection follows directly (Ern et al., 5 Feb 2025).

4. Global Projections: Locality, Commutativity, and Boundary Prescription

Let $\{W_{\sigma}\}$ denote basis elements dual to canonical DOFs. The global stable projection in the lowest-order case ($p=0$) takes the form:

$\Pi_0^l(u) = \sum_{\sigma\in\Delta_h^l} \left( \int_{\es(\sigma)} \ZZ_0^l(\sigma) u \right) W_{\sigma}$

and is extended to higher order via local corrections leveraging higher-order bubble moments.

Fundamental properties include:

• Locality: The projection on an element $\tau$ depends only on data from a fixed neighborhood $\es^2(\tau)$.
• Commutation: $\nabla\circ\Pi_0^0 = \bPi_0^1\circ\nabla$, $\mathrm{curl}\circ\bPi_0^1 = \bPi_0^2\circ\mathrm{curl}$, $\mathrm{div}\circ\bPi_0^2 = \Pi_0^3\circ\mathrm{div}$.
• $L^2$-boundedness: $\|\Pi_0^l(u)\|_{L^2(\tau)}\leq C_\Pi\|u\|_{L^2(\es(\tau))}$, uniform in $h$.
• Boundary Adaptation: For entities on or adjacent to $\partial\Omega$, the construction of $\ZZ_p^l(\sigma)$ is modified to ensure vanishing trace or flux, yielding projections that enforce homogeneous boundary data without loss of stability (Ern et al., 5 Feb 2025).

5. Connection to Finite Element Exterior Calculus and Applications

The assembled operators $\{\Pi_p^l\}_{l=0}^3$ form a bounded cochain projection of the continuous de Rham complex onto its discrete analog:

$0\to H^1(\Omega)\xrightarrow{d^0} \bH(\mathrm{curl},\Omega)\xrightarrow{d^1} \bH(\mathrm{div},\Omega)\xrightarrow{d^2} L^2(\Omega)\to 0$

onto

$0\to V_p^0(\mathcal{T}_h)\xrightarrow{d^0} \bV_p^1(\mathcal{T}_h)\xrightarrow{d^1} \bV_p^2(\mathcal{T}_h)\xrightarrow{d^2} V_p^3(\mathcal{T}_h)\to 0$

with $d^l\circ\Pi_p^l = \Pi_p^{l+1}\circ d^l$ for $l=0,1,2$, and uniformly bounded projection norms. The locality and commutativity directly support best-approximation error estimates, facilitate the construction of stable multigrid preconditioners, and support parallel implementation.

Further, these projections can be adapted for specific applications requiring boundary conditions, such as enforcing essential boundary constraints in electromagnetic or fluid problems, and their structure-preserving nature is essential for a posteriori analysis and FEEC-based discretizations (Ern et al., 5 Feb 2025).

6. Comparison with Related Stable Projection Frameworks

The Alfeld-split-based stable discrete projection operators complement approaches developed for:

• general finite element de Rham complexes, where local bounded projections were constructed via patch-based harmonic projection and DOF re-corrections (Hu et al., 2023),
• high-regularity complexes (e.g., Argyris and Hermite) and discrete gradgrad complexes (elasticity, general relativity) with explicit local bounded commuting projectors built from bubble-complex inverses and patchwise Galerkin lifts (Hu et al., 2023),
• multipatch spline spaces with non-matching interfaces, using $L^p$-stable tensor-product projections enhanced with interface corrections to enforce conformity and commutation globally (Pinto et al., 2023).

The Alfeld split construction is particularly notable for providing constructive, purely local ingredients (weight problem solves on small patches) and uniform $L^2$-norm control independent of the mesh size, while retaining all critical FEEC commutation properties (Ern et al., 5 Feb 2025).