Local and Global Constraints in Finite Element Modeling and the Benefits of Nodal and Edge Elements Coupling

Various constraints can be encountered in partial differential problems. On the one hand, there are local constraints, which locally act on fields. These are usually boundary conditions, fixing components of fields, as well as interface conditions, connecting such components. On the other hand, global behaviors of fields can be constrained, leading to define global constraints. It is the case when vector field fluxes and circulations have to be defined, again in order to be either fixed or connected.
A finite element model of a partial differential problem, through weak formulations, then leads to split up the considered constraints in two families, known as essential and natural constraints. This means that some constraints are strongly satisfied while others are only weakly satisfied.
It is the aim of this paper to make a survey of local and global constraints encountered in finite element models of electromagnetic systems. It particularly points out the benefits of using both nodal and edge finite elements to achieve their consistent discrete definitions. There are indeed properties that are worth to be kept from the continuous to the discrete level. The constraints are defined in the frame of dual formulations in order to point out their dual, or complementary, natures. Systematic explicit characterizations of constrained function spaces are shown to be quite convenient.