H 1 -conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes.

A finite element cochain complex on Cartesian meshes of any dimension based on the H 1 -inner product is introduced. It yields H 1 -conforming finite element spaces with exterior derivatives in H 1 . We use a tensor product construction to obtain L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.