Fortin operator and discrete compactness for edge elements

The basic properties of the edge elements are proven in the original papers by Nédélec [22,23]. In the two-dimensional case the edge elements are isomorphic to the face elements (the well-known Raviart-Thomas elements [24]), so that all known results concerning face elements can be easily formulated for edge elements. In three-dimensional domains this is not the case. The aim of the present paper is to show how to construct a Fortin operator which converges uniformly to the identity in the spirit of [5,4]. The construction is given for any order tetrahedral edge elements in general geometries. We relate this result to the well-known commuting diagram property and apply it to improve the error estimate for a mixed problem which involves edge elements. Finally we show that this result can be applied to the analysis of the approximation of the time-harmonic Maxwell's system.