Nonconforming vector valued finite elements

In the realm of finite element discretizations of elliptic problems a basic distinction can be made between conforming and nonconforming methods. The latter rely on approximations not contained in the spaces in which the continuous variational problem is posed. This paper investigates nonconforming finite element approximations of the spaces H(div;Ω) and H(curl;Ω) of vector valued functions. First, we extend the `generalized patch test' which has been developed to provide necessary and sufficient conditions for the viability of nonconforming schemes for standard Sobolev spaces. Then we use the calculus of differential forms to derive coupling conditions sufficient for success in the patch test. Based on this result we design convergent nonconforming methods for a number of variational problems involving the above-mentioned function spaces. Finally, we point out that a naive approach leads to an inconsistent scheme.