Cohomology of Finite Element Stokes Complexes on Alfeld Splits
This work provides theoretical guarantees for the accuracy of finite element methods for fluid dynamics problems, particularly for researchers and practitioners using Alfeld split meshes and minimal conforming elements.
This paper demonstrates that the cohomology of finite element Stokes complexes on Alfeld split meshes is isomorphic to the cohomologies of continuous Stokes and de Rham complexes. Additionally, it constructs novel minimal conforming finite element complexes with the same isomorphic cohomology properties.
We show that the cohomology of the finite element Stokes complex consisting of piecewise polynomials spaces on an Alfeld split mesh from Fu, Guzmán, & Neilan (2020, Math. Comp., 89, 1059--1091) is isomorphic to the cohomologies of the continuous Stokes and de Rham complexes. We also construct novel "minimal" conforming finite element complexes where the $H^1$-conforming space is the lowest-order space from Guzmán & Neilan (2018, SIAM J. Numer. Anal., 56, 2826--2844) and the $L^2$-conforming space is piecewise constants. These minimal complexes also have cohomologies isomorphic to the continuous Stokes and de Rham complexes. We further construct local, bounded, cochain projections for the minimal complexes. All the results hold for strongly Lipschitz domains with nontrivial topologies and in the presence of mixed boundary conditions.