We study conforming and nonconforming methods that preserve the Helmholtz structure of mixed problems at the discrete level. On the conforming side we essentially gather classical tools to design numerical approximations that are compatible with an underlying Helmholtz decomposition. We then show that a recent approach developped for the time-dependent Maxwell equations allows to design new nonconforming methods based on fully discontinuous finite element spaces, that share the same stability and compatibility properties, with no need of penalty terms.