Local bounded cochain projections

We construct projections from HΛk(Ω), the space of differential k forms on Ω which belong to L2(Ω) and whose exterior derivative also belongs to L2(Ω), to finite dimensional subspaces of HΛk(Ω) consisting of piecewise polynomial differential forms defined on a simplicial mesh of Ω. Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom. Moreover, these projections have the properties that they commute with the exterior derivative and are bounded in the HΛk(Ω) norm independent of the mesh size h. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clément interpolant. A double complex structure is introduced as a key tool to carry out the construction.