Local interpolation in ${{{\varvec{H}}(\text {div})}}$ and ${{{\varvec{H}}(\text {curl})}}$ (II)

Abstract

In this chapter, we continue our investigation of the interpolation operators associated with the finite elements introduced in Chapters 14 and 15. We consider a shape-regular sequence of affine simplicial meshes with a generation-compatible orientation. The key idea here is to extend the degrees of freedom on the faces and the edges by requiring some integrability of the divergence or the curl of the function to be interpolated. This approach is useful when such integrability properties can be extracted from a PDE solved by the function in question, as it is often the case in applications (e.g., for Darcy’s equations and for Maxwell’s equations). The crucial advantage of the present approach over that from the previous chapter based on the scale of Sobolev spaces is that interpolation error estimates with lower smoothness requirements can be obtained. On the way, we also devise a face-to-cell lifting operator that is useful in the analysis of nonconforming approximations of elliptic problems.