Exact de Rham Sequences of Spaces Defined on Macro-Elements in Two and Three Spatial Dimensions

This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H-conforming, H(curl) conforming, H(div) conforming and piecewise constant spaces associated with an unstructured “fine” mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest order Nédélec spaces, lowest order Raviart–Thomas spaces, and for piecewise linear H-conforming spaces, all in three-dimensions. The resulting V -cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case.