Title: Local bounded cochain projections
Authors: Richard S. Falk and Ragnar Winther
Source: Math. Comp. 83 (2014), no. 290, pp. 2631-2656.
Status: Published
Abstract: We construct projections from H \Lambda^k(\Omega), the space of
differential k-forms on \Omega which belong to L^2(\Omega) and
whose exterior derivative also belongs to L^2(\Omega), to finite
dimensional subspaces of H \Lambda^k(\Omega) consisting of
piecewise polynomial differential forms defined on a simplicial
mesh of \Omega. 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 H \Lambda^k(\Omega) 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 Clement interpolant. A double complex structure is
introduced as a key tool to carry out the construction.