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

TL;DR

This paper introduces locally defined, L2-bounded commuting projections for finite element spaces on shape-regular meshes, using novel local weight functions based on Alfeld splits, with full L2-stability proof and boundary condition preservation.

Contribution

It provides the first fully L2-stable local projections for finite element spaces on Alfeld splits, extending to boundary conditions and linking to finite element exterior calculus.

Findings

Projections are locally defined, L2-bounded, and commute with differential operators.

Constructive proof of discrete Poincaré inequalities on local stars.

Extension of projections to preserve boundary conditions.

Abstract

We construct projections onto the classical finite element spaces based on Lagrange, N\'ed\'elec, 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 Poincar\'e 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.