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A finite element cochain complex on Cartesian meshes of any dimension basedon the H1-inner product is introduced. It yields H1-conforming finite elementspaces with exterior derivatives in H1. We use a tensor product construction toobtain L2-stable projectors into these spaces which commute with the exteriorderivative. The finite element complex is generalized to a family of arbitraryorder.

H1-conforming finite element cochain complexes and commuting quasi-interpolation operators on cartesian meshes