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Local L2-bounded commuting projections using discrete local problems on Alfeld splits

Alexandre Ern, Johnny Guzman, Pratyush Potu, Martin Vohralik

TL;DR

This work resolves the absence of fully established local $L^2$-stable commuting projections in FEEC by introducing fully discrete local weight functions based on the Alfeld split and proving constructive discrete Poincaré inequalities on extended stars. The authors construct a hierarchy of local weights $\mathsf{Z}_p^l(\sigma)$ that link to canonical degrees of freedom and to the differential operators via incidence numbers, enabling local $L^2$-bounded commuting projections for all polynomial degrees. A key novelty is extending the construction to preserve homogeneous boundary conditions on a subset $\Gamma$ of $\partial\Omega$, with corresponding boundary-prescribing weight functions and projections. The results provide a robust theoretical foundation for stability and error analysis in FEEC-based primal and mixed formulations, with potential impact on eigenvalue problems and PDE discretizations requiring local, boundary-aware projections.

Abstract

We construct projections onto the classical finite element spaces based on Lagrange, Nédélec, Raviart-Thomas, and discontinuous elements on shape-regular simplicial meshes. Our projections are defined locally, are bounded in the L2-norm, and commute with the corresponding differential operators. Such projections are a fundamental tool in finite element stability and error analysis. Moreover, to the best of our knowledge, the present construction is the first in the literature where local $L^2$-stability is fully established. The cornerstone of our construction are local weight functions which are piecewise polynomials built using the Alfeld split of local patches from the original simplicial mesh. This way, the L2-stability of the projections is established by invoking discrete Poincaré inequalities on these local stars, for which we provide here an original, constructive proof. Another important novelty is that we extend the construction of the projections so as to preserve homogeneous boundary conditions on a subset of the domain boundary. Altogether, the material is presented using the language of vector calculus, and links to the formalism of finite element exterior calculus are provided.

Local L2-bounded commuting projections using discrete local problems on Alfeld splits

TL;DR

This work resolves the absence of fully established local -stable commuting projections in FEEC by introducing fully discrete local weight functions based on the Alfeld split and proving constructive discrete Poincaré inequalities on extended stars. The authors construct a hierarchy of local weights that link to canonical degrees of freedom and to the differential operators via incidence numbers, enabling local -bounded commuting projections for all polynomial degrees. A key novelty is extending the construction to preserve homogeneous boundary conditions on a subset of , with corresponding boundary-prescribing weight functions and projections. The results provide a robust theoretical foundation for stability and error analysis in FEEC-based primal and mixed formulations, with potential impact on eigenvalue problems and PDE discretizations requiring local, boundary-aware projections.

Abstract

We construct projections onto the classical finite element spaces based on Lagrange, Nédélec, Raviart-Thomas, and discontinuous elements on shape-regular simplicial meshes. Our projections are defined locally, are bounded in the L2-norm, and commute with the corresponding differential operators. Such projections are a fundamental tool in finite element stability and error analysis. Moreover, to the best of our knowledge, the present construction is the first in the literature where local -stability is fully established. The cornerstone of our construction are local weight functions which are piecewise polynomials built using the Alfeld split of local patches from the original simplicial mesh. This way, the L2-stability of the projections is established by invoking discrete Poincaré inequalities on these local stars, for which we provide here an original, constructive proof. Another important novelty is that we extend the construction of the projections so as to preserve homogeneous boundary conditions on a subset of the domain boundary. Altogether, the material is presented using the language of vector calculus, and links to the formalism of finite element exterior calculus are provided.

Paper Structure

This paper contains 31 sections, 11 theorems, 147 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discrete setting
  3. Simplicial mesh
  4. Piecewise polynomial spaces, canonical degrees of freedom, and Whitney forms
  5. Incidence matrices
  6. Local weight functions
  7. Alfeld split
  8. Local polynomial spaces and discrete Poincaré inequalities
  9. Main result
  10. Proof of Theorem \ref{['th:weights']} on the local weight functions
  11. Construction and properties of $\mathsf{Z}_p^0(v)$
  12. Construction and properties of $\bm{\mathsf{Z}}_p^1(e)$
  13. Construction and properties of $\bm{\mathsf{Z}}_p^2(f)$
  14. Construction and properties of $\mathsf{Z}_p^3(\tau)$
  15. Local $L^2$-bounded commuting projections
  16. ...and 16 more sections

Key Result

Proposition 3.1

For all $\sigma \in \Delta_h$, under the assumption that $\operatorname{cl}(\operatorname{es}(\sigma))$ is contractible, the following discrete sequences are exact:

Figures (4)

  • Figure 1: Example of $\operatorname{st}(\sigma)$ and $\operatorname{es}(\sigma)$ for $\sigma\in\mathcal{F}_h$. Here, $\sigma$ is highlighted in red, and the star (resp. extended star) of $\sigma$ is shaded in blue.
  • Figure 2: Example showing an extended star $\operatorname{es}(\sigma)$ (shaded in blue) of a face $\sigma\in\mathcal{F}_h$ (hatched in red) where $\operatorname{cl}(\operatorname{es}(\sigma))$ is non-contractible. Here, the gray region represents a hole in the domain $\Omega$. A similar 2D example (where $\sigma$ highlighted in red is now an edge) is also included for clarity.
  • Figure 3: Two-dimensional illustration of a non-contractible closure of an extended star without a hole in the domain $\Omega$ (here, the white region is part of the domain $\Omega$). Subfigure (a) additionally illustrates the difference between considering the closure or not of the extended star. Extended star $\operatorname{es}(\sigma)$ (shaded in blue) of an edge $\sigma$ (highlighted in red).
  • Figure 4: Example of an Alfeld split

Theorems & Definitions (15)

  • Remark 2.1: Non-contractible extended stars
  • Remark 2.2: Link to FEEC formalism
  • Proposition 3.1: Exact sequences
  • Proposition 3.2: Discrete Poincaré inequalities on extended stars
  • Proposition 3.3: Discrete Poincaré inequalities on Alfeld splits of extended stars
  • Theorem 3.4: Local weight functions
  • Theorem 5.1: Lowest-order projections
  • proof
  • Theorem 5.2: Higher-order projections
  • Theorem 6.1: Local weight functions with boundary prescription
  • ...and 5 more