chapter  2
36 Pages

Exact hp Sequences, Projection-Based Interpolation, De Rham Diagrams

The fundamental property of the discussed finite element methodology is the possibility of a simultaneous discretization of H1-, H(curl)-, and H(div)- conforming fields using the so-called compatible discretizations. The phrase “compatible” refers here to the fact that the corresponding Finite Element spaces, both for a single element, and a global mesh (for simply connected domains), form an exact sequence. This chapter is devoted to the discussion of exact polynomial sequences corresponding to three of four Ne´de´lec’s elements: Ne´de´lec’s tetrahedra of the second and first types, and the Ne´de´lec’s hexahedron. The type number refers here simply to the two fundamental papers of Ne´de´lec. The elements of the first type were described in his 1980 paper [115], whereas the elements of the second type correspond to the contribution [116] from 1986. The exact sequence property is absolutely crucial, the Ne´de´lec’s hexahedron of the second type does not satisfy it, and the corresponding discretization of Maxwell eigenvalues exhibits spurious, nonphysical eigenvalues, see e.g., Reference [34].