Differential complexes and stability of finite element methods. I. The de Rham complex

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Finite element approximation of eigenvalue problems (2010, methodology): “…∈ L2(Ω,Λk), 1 The external derivative we are going to use can be naturally defined as follows: given ω ∈ C∞(Ω, Λk), dkωx(v1, . . . , vk+1) = Pk+1 j=1 (−1)j+1∂vj ωx(v1, . . . , v̂j , . . . , vk+1) (Arnold et al. 2006b, p. 15) where denotes the Hodge star operator mapping k-forms to (n − k)forms.”
Mixed finite element methods for linear elasticity with weakly imposed symmetry (2007, methodology): “We begin by recalling the two principal families of piecewise polynomial spaces of differential forms, following the presentation in [7].”
A discrete de Rham complex with enhanced smoothness (2006, unknown): “In order to show that elements v ∈ V (T ) andw ∈ W (T ) are uniquely determined by the degrees of freedom specified by (12), or by (14) and (15), respectively, we need preliminary results in two dimensions.”
Locking-free Reissner-Mindlin elements without reduced integration (2007, methodology): “This can be checked by using the definition of the projection operators and integration by parts, and is a special case of the commutativity properties of projections presented, e.g., in [ 7 ] and [8].”
Finite element exterior calculus, homological techniques, and applications (2006, methodology): “Some of these have appeared in the literature previously, but the systematic derivation of all of them first appeared in Arnold et al. (2006b).”