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

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. The cornerstone of the 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 constructive proofs. We also show how to modify the construction to preserve homogeneous boundary conditions. The material is presented using the language of vector calculus, and links to the formalism of finite element exterior calculus are provided.