A finite element elasticity complex in three dimensions

A finite element elasticity complex on tetrahedral meshes and the corresponding commutative diagram are devised. The H 1 H^1 conforming finite element is the finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an H ( inc ) H(inc) -conforming finite element of minimum polynomial degree 6 6 for symmetric tensors is the focus of this paper. Our construction appears to be the first H ( inc ) H(inc) -conforming finite elements on tetrahedral meshes without further splitting. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the inc inc operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition. The trace of the inc inc operator is induced from a Green’s identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Two-dimensional smooth finite element Hessian complex and div ⁡ div divdiv complex are constructed.